Displaying 1-10 of 25 results found.
a(n) = A329900( A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.
+20
9
1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 3, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 7, 5, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 1, 2, 3, 8, 5, 3, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 7, 3, 1, 4, 9, 2, 1, 6, 5, 2, 3, 4, 1, 6, 7, 2, 3, 2, 5, 8, 1, 7, 3, 10, 1, 3, 1, 4, 5
COMMENTS
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor ( A006530) and its exponent ( A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example. (End)
FORMULA
a(n^2) = n for all n >= 1.
EXAMPLE
The list of all numbers with image 12 and their corresponding prime factors begins:
144: (3,3,2,2,2,2)
216: (3,3,3,2,2,2)
240: (5,3,2,2,2,2)
288: (3,3,2,2,2,2,2)
336: (7,3,2,2,2,2)
360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - Antti Karttunen, Oct 13 2021 Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - Antti Karttunen, Oct 15 2021
MATHEMATICA
Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]]], 2], {n, 100}] (* Gus Wiseman, Oct 13 2021 *)
PROG
(PARI) A329888(n) = if(1==n, n, my(f=factor(n), m=1, p=0); forstep(k=#f~, 1, -1, while(f[k, 2], m *= f[k, 1]^(p%2); f[k, 2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021
CROSSREFS
The sum of prime indices of a(n) is A346700(n). The odd non-reverse version is A346703. The non-reverse version is A346704. A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.
Least integer of each prime signature A124832; also products of primorial numbers A002110.
+10
601
1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310
COMMENTS
All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b( A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019 The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019
FORMULA
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012 A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
(End)
EXAMPLE
The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...
MAPLE
isA025487 := proc(n)
local pset, omega ;
pset := sort(convert(numtheory[factorset](n), list)) ;
omega := nops(pset) ;
if op(-1, pset) <> ithprime(omega) then
return false;
end if;
for i from 1 to omega-1 do
if padic[ordp](n, ithprime(i)) < padic[ordp](n, ithprime(i+1)) then
return false;
end if;
end do:
true ;
end proc:
option remember ;
local a;
if n = 1 then
1 ;
else
for a from procname(n-1)+1 do
if isA025487(a) then
return a;
end if;
end do:
end if;
end proc:
MATHEMATICA
PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *) (* Second program: generate all terms m <= A002110(n): *) f[n_] := {{1}}~Join~
Block[{lim = Product[Prime@ i, {i, n}],
ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
Map[Block[{w = #, k = 1},
Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
Do[
If[# < lim,
Sow[#]; k = 1,
If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
If[k == 1,
MapAt[# + 1 &, w, k],
PadLeft[#, Length@ w, First@ #] &@
Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
{i, Infinity}] ][[-1]]
PROG
(PARI) isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011 (PARI) factfollow(n)={local(fm, np, n2);
fm=factor(n); np=matsize(fm)[1];
if(np==0, return([2]));
n2=n*nextprime(fm[np, 1]+1);
if(np==1||fm[np, 2]<fm[np-1, 2], [n*fm[np, 1], n2], [n2])}
al(n) = {local(r, ms); r=vector(n);
ms=[1];
for(k=1, n,
r[k]=ms[1];
ms=vecsort(concat(vector(#ms-1, j, ms[j+1]), factfollow(ms[1]))));
(PARI) is(n) = {if(n==1, return(1)); my(f = factor(n)); f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]} \\ David A. Corneth, Feb 14 2019 (PARI) upto(Nmax)=vecsort(concat(vector(logint(Nmax, 2), n, select(t->t<=Nmax, if(n>1, [factorback(primes(#p), Vecrev(p)) || p<-partitions(n)], [1, 2]))))) \\ M. F. Hasler, Jul 17 2019 (PARI)
\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista, 2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista, t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e. v025487 = A025487list(101);
(Haskell)
import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
(_ : b : bs) = a002110_list
h cs s xs'@(x:xs)
| m <= x = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
| otherwise = x : h (x:cs) (s `union` fromList (map (* x) (x:cs))) xs
where (m, s') = deleteFindMin s
(Sage)
def sharp_primorial(n): return sloane. A002110(prime_pi(n)) N = 2310
nmax = 2^floor(log(N, 2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
CROSSREFS
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683. Equals range of values taken by A046523. Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132. Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).
+10
388
1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8
COMMENTS
a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n). Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a( A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022
FORMULA
Other identities. For all n >= 1:
(End)
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a( A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
(End)
EXAMPLE
20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)* A000040(1) = 3*2 = 6.
MAPLE
local a;
a := 1;
for pf in ifactors(n)[2] do
a := a*ithprime(pf[2]) ;
end do:
a ;
end proc:
MATHEMATICA
{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)
PROG
(Haskell)
a181819 = product . map a000040 . a124010_row
(PARI) a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); } \\ Michel Marcus, Nov 16 2015 (Scheme, with memoization-macro definec, two variants)
CROSSREFS
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 ( A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).
+10
159
1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520
COMMENTS
This sequence is a permutation of A025487. A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015
FORMULA
Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
a(1) = 1, and for n > 1, a(n) = n * a( A064989(n)). Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
Further identities:
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
(End)
(End)
EXAMPLE
a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).
MATHEMATICA
a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *) Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)
PROG
(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)
(Sage)
def sharp_primorial(n): return sloane. A002110(prime_pi(n)) def p(f):
return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1, 51)]
(PARI) primorial(n)=prod(i=1, primepi(n), prime(i))
(Python)
from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
f = factorint(n)
return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
CROSSREFS
Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.
EXTENSIONS
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Primorial deflation of n (numerator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the numerator of g(n).
+10
36
1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 69
COMMENTS
See A319627 for the corresponding denominators. The restriction of f to the natural numbers corresponds to A108951. The function g is completely multiplicative over the positive rational numbers with g(2) = 2 and g(q) = q/p for any pair (p, q) of consecutive prime numbers.
FORMULA
a(n) <= n with equality iff n belongs to A319630. Many of the formulas given in A329900 apply here as well: (End)
EXAMPLE
f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 21.
MATHEMATICA
Array[#1/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI) a(n) = my (f=factor(n)); numerator(prod(i=1, #f~, my (p=f[i, 1]); (p/if (p>2, precprime(p-1), 1))^f[i, 2]))
CROSSREFS
Cf. A006530, A053585, A064989, A181815, A307035, A319627, A319630, A329902, A330749, A330750 (rgs-transform), A330751 (ordinal transform).
a(n) = largest integer such that, when any of its divisors divides A025487(n), the quotient is a member of A025487.
+10
27
1, 2, 4, 3, 8, 6, 16, 12, 5, 32, 9, 24, 10, 64, 18, 48, 20, 128, 36, 15, 96, 7, 27, 40, 256, 72, 30, 192, 14, 54, 80, 512, 144, 60, 384, 28, 108, 25, 160, 1024, 45, 288, 21, 81, 120, 768, 56, 216, 50, 320, 2048, 90, 576, 11, 42, 162, 240, 1536, 112, 432, 100, 640, 4096, 180, 1152
COMMENTS
A permutation of the natural numbers.
The scatter plot looks like a curtain of fractal spray, which is typical for many of the so-called "entanglement permutations". Indeed, according to the terminology I use in my 2016-2017 paper, this sequence is obtained by entangling the complementary pair ( A329898, A330683) with the complementary pair ( A005843, A003961), which gives the following implicit recurrence: a( A329898(n)) = 2*a(n) and a( A330683(n)) = A003961(a(n)). An explicit form is given in the formula section. (End)
FORMULA
If A025487(n) is considered in its form as Product A002110(i)^e(i), then a(n) = Product p(i)^e(i). If A025487(n) is instead considered as Product p(i)^e(i), then a(n) = Product (p(i)/ A008578(i))^e(i). (End)
EXAMPLE
For any divisor d of 9 (d = 1, 3, 9), 36/d (36, 12, 4) is a member of A025487. 9 is the largest number with this relationship to 36; therefore, since 36 = A025487(11), a(11) = 9.
MATHEMATICA
(* First, load the program at A025487, then: *) Map[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[g, ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, Union@ Flatten@ f@ 6] (* Michael De Vlieger, Dec 28 2019 *)
CROSSREFS
If this sequence is considered the "primorial deflation" of A025487(n) (see first formula), the primorial inflation of n is A108951(n), and the primorial inflation of A025487(n) is A181817(n). A181820(n) is another mapping from the members of A025487 to the positive integers.
Cf. A003961, A051282, A108951, A163511, A304886, A319626, A329897, A329898, A329900, A329901 (inverse), A329904, A329905, A329907, A330682 (reduced modulo 2), A330683.
Primorial deflation of n (denominator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the denominator of g(n).
+10
20
1, 1, 2, 1, 3, 1, 5, 1, 4, 3, 7, 1, 11, 5, 2, 1, 13, 2, 17, 3, 10, 7, 19, 1, 9, 11, 8, 5, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 3, 37, 5, 41, 7, 4, 19, 43, 1, 25, 9, 26, 11, 47, 4, 21, 5, 34, 23, 53, 1, 59, 29, 20, 1, 33, 7, 61, 13, 38, 3, 67, 1, 71, 31, 6
COMMENTS
See A319626 for the corresponding numerators and additional comments.
EXAMPLE
f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 5.
MATHEMATICA
Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI) a(n) = my (f=factor(n)); denominator(prod(i=1, #f~, my (p=f[i, 1]); (p/if (p>2, precprime(p-1), 1))^f[i, 2]))
CROSSREFS
Cf. A319626 (numerators, see comments there).
Primorial inflation of Doudna-tree: a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).
+10
17
1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, 900, 72, 216, 16, 2310, 420, 1260, 120, 6300, 360, 1080, 48, 44100, 1800, 5400, 144, 27000, 432, 1296, 32, 30030, 4620, 13860, 840, 69300, 2520, 7560, 240, 485100, 12600, 37800, 720, 189000, 2160, 6480, 96, 5336100, 88200, 264600, 3600, 1323000, 10800, 32400, 288, 9261000
FORMULA
a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).
EXAMPLE
This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A283980 to the parent, and each child to the right is obtained by doubling the parent: 1
|
...................2...................
6 4
30......../ \........12 36......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
210 60 180 24 900 72 216 16
etc.
MATHEMATICA
Block[{a}, a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@#1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[a, 57, 0]]
(* or, via Doudna *)
Map[Times @@ Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@#1, #2] & @@@ FactorInteger[#]], Greater]] &, Nest[Append[#1, Prime[1 + BitLength[#2] - DigitCount[#2, 2, 1]]*#1[[#2 - 2^Floor@ Log2@ #2 + 1]]] & @@ {#, Length@ #} &, {1}, 57] ] (* Michael De Vlieger, Mar 05 2021 *)
PROG
(PARI)
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).
+10
16
1, 2, 4, 3, 6, 12, 9, 24, 10, 20, 15, 40, 30, 60, 28, 21, 56, 42, 84, 63, 168, 126, 336, 140, 66, 189, 280, 132, 99, 264, 198, 528, 220, 396, 297, 440, 792, 156, 117, 312, 234, 624, 260, 468, 351, 520, 936, 390, 1040, 1872, 780, 585, 306, 1560, 340, 612, 459, 680, 1224, 510, 1360, 2448, 1020, 765, 342, 2040, 1530, 684, 513
FORMULA
A056239(a(n)) = A112778(n). [Number of prime factors, counted with multiplicity]
A001222(a(n)) = A112779(n). [Largest exponent in the prime factorization]
MATHEMATICA
Map[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 69] ] (* Michael De Vlieger, Jan 13 2020, imports b-file at A002182 *)
CROSSREFS
Cf. A002182, A002183, A056239, A108951, A112778, A112779, A306802, A319626, A324381, A324382, A324581, A324582, A324888, A329040, A329605, A329900, A330743, A330748.
The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.
+10
11
1, 6, 120, 19399380, 195534950863140268380, 538938984694949877040715541221415046162838700, 216487559804430601784907786655491617909711008142914104790481010259258659171900
COMMENTS
For n > 0, the least k such that for at least n-1 iterations of map x -> A003961(x), starting from x=k, x stays nondeficient. In other words, from each a(n) starts a chain of at least n nondeficient numbers ( A023196) obtained by successive prime shifts, e.g, for a(3) we have: 19399380 -> 334639305 -> 5391411025, where -> stands for applying A003961, the prime shift towards larger primes. After 1 all other terms here are even, because if an odd number k is nondeficient, then A064989(k) is nondeficient also, where A064989 is the prime shift towards smaller primes. Moreover, because A047802 is defined for every n >= 0, also this sequence is. Upper bounds for a(4) and a(5) are:
a(5) <= 538938984694949877040715541221415046162838700 = A064989^4(( A047802(4)*17*19)/137). (End)
Let prime(n)# be the n-th primorial number, A002110(n) = A034386(prime(n)). Then: a(6) <= 191# * 7#;
a(7) <= 311# * 5#;
a(8) <= 457# * 5#.
(End)
That each term occurs in A025487 follows because (1), the abundancy index of prime(i)^e is larger than that of prime(i+1)^e, that is, sigma(prime(i)^e)/prime(i)^e > sigma(prime(i+1)^e)/prime(i+1)^e, and (2) because the abundancy index of p^(e+d) * q^e is larger than that of p^e * q^(e+d), where p and q are distinct primes, p < q, and e, d > 0. Thus, for any n, we can first find a "prime-factorization compressed version" of it, A071364(n), and then sort the exponents to the non-ascending order with A046523 (and actually, A046523( A071364(n)) = A046523(n), so we need to apply just A046523), to get a term x of A025487, that certainly have the abundancy index >= n [and this inequivalence stays same for their successive prime shifts as well, the abundancy index of A003961(x) being at least that of A003961(n), etc.], and as A046523(n) <= n for all n, it is guaranteed that the least k for which A336835(k) >= n are found from A025487, which is the range of A046523.
PROG
(PARI)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
CROSSREFS
Cf. A003961, A005100, A023196, A025487, A047802, A108951, A246277, A329900, A336834, A336835, A337202, A337205, A337477, A337478. From term a(2) = 120 onward a subsequence of A337386.
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