Displaying 1-10 of 16 results found.
Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.
+20
7
1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 6, 4, 6, 4, 2, 4, 1, 2, 6, 4, 10, 6, 6, 4, 8, 12, 10, 8, 22, 4, 8, 2, 1, 2, 6, 4, 6, 2, 6, 2, 18, 10, 8, 6, 18, 12, 16, 4, 26, 16, 24, 8, 20, 14, 4, 6, 26, 16, 14, 8, 30, 6, 8, 4, 1, 2, 6, 4, 14, 12, 12, 8, 18, 12, 24, 4, 8, 12, 14, 4, 24, 20, 28, 20, 26, 16, 16, 12, 32, 26, 24, 14, 28, 16
COMMENTS
From David A. Corneth's Feb 27 2019 comment in A276150 follows that the only odd terms in this sequence are 1's occurring at 0 and at two's powers. Subsequences starting at each n = 2^k are slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, etc.. Compare also to the behaviors of A324342 and A342463.
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)};
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
a(1) = 0, and for n > 1, a(n) = 1 if G( A329886(n)) >= G( A329886(floor(n/2))), otherwise 0, where G(n) = sigma(n) / (n*log(log(n))), where sigma is the sum of the divisors.
+20
6
0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
COMMENTS
Equivalently, for n > 3, a(n) = 1 if the value of H( A329886(n)) <= H( A329886(n\2)), where H(n) = log(n)^(n/sigma(n)) [= exp(1/G(n))], otherwise 0. The ratio G(n) = sigma(n) / (n*log(log(n))) comes from Grönwall's theorem (listed, for example, as Theorem 1 in the Caveney, Nicolas and Sondow paper; see also the other papers linked at A073751). This sequence gives the positions of those points in the A329886-tree (Primorial inflation of Doudna-tree) where this ratio doesn't decrease when going downwards (which it mostly does, but see also A342455 for a counterexample). There are 11355 such increasing cases among the first 65536 terms, and 113134 among the first 2^20, although overall the ratio of such cases on each row k (that has 2^k terms) seems to start decreasing after the seventh row. See A342020. It seems that the tree has infinitely long leftward branches that contain either only zeros or only ones after a while: The leftmost edge (that are primorials, A002110, in A329886) appears to consist of zeros only after its single 1 at the second term. This depends on the (so far conjectural) observation that (log( A002110(n)) + log(prime(1+n)))^(prime(1+n)) > log( A002110(n))^(1+prime(1+n)) for all n >= 1. On the other hand, the leftward branch starting from the left child of the 496th term of the tree appears to contain only ones (corresponding to A342455 from its fifth term onward, see comments there).
EXAMPLE
The binary tree begins as:
0
..................../ \.................
1 1
0......./ \........ 0 0 ......./ \........0
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
0 1 0 0 0 0 0 0
/\ /\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \ / \
0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
with 1 marked at each node of A329886 (Primorial inflation of Doudna-tree) if its G(n) = sigma(n) / (n*log(log(n))) ratio is greater than or equal to the corresponding ratio of its parent, and 0 otherwise. For example, A329886(1) = 2, and A329886(3) = 4. The latter has ratio G(4) = 5.357674..., while for the former (the parent of 4), the ratio is G(2) = -4.0926..., which is less than 5.357674..., therefore a(3) = 1. For 130 = 2*65, we have A329886(65) = 60060, A329886(130) = 3063060 = A283980(60060). Here G(3063060) = 1.594960... > 1.56762... = G(60060), in other words, here again the child has a larger G-ratio than its parent, and this is the first case where it is a child obtained with A283980 instead of doubling, thus n=130 is also the first such even number after 2 for which a(n) = 1.
MATHEMATICA
Block[{a, b, c, f, nn = 105}, b[0] = c[1] = 1; f[n_] := DivisorSigma[1, n]/(n Log[Log[n]]); Do[b[i] = Prime[1 + BitLength[i] - DigitCount[i, 2, 1]]*b[i - 2^Floor@ Log2@ i]; c[i + 1] = Apply[Times, Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger[b[i]]], Greater]]]; a[i - 1] = Boole[f[c[i]] >= f[c[Floor[(i + 1)/2]]]], {i, nn}]; Array[a, nn - 1]] (* Michael De Vlieger, Mar 07 2021 *)
PROG
(PARI)
default(realprecision, 10001);
G(n) = (sigma(n) / (n*log(log(n))));
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
(PARI)
\\ Alternative program, probably less vulnerable to the loss of precision:
Gie(n) = (log(n)^(n/sigma(n))); \\ = exp(1/G(n)), function H in comments
CROSSREFS
Cf. A000203, A002110, A003961, A004394, A004490, A005940, A025487, A057641, A058209, A073751, A108951, A329886, A342012, A342013, A342455. Cf. also A336834 for a similarly constructed sequence.
Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.
+20
5
1, 2, 5, 9, 19, 21, 37, 75, 139, 267, 535, 539, 555, 1067, 2091, 4139, 8279, 16471, 32855, 32919, 32923, 65691, 131227, 262299, 524599, 1048887, 2097463, 4194615, 4194647, 8388951, 16777559, 33554775, 67109207, 67109463, 134218327, 268436655, 536872111, 536872119, 1073743031, 2147484855, 2147485879, 4294969527, 8589936823
COMMENTS
Like A342012, also this sequence is monotonic. Proof: the doubling step corresponds here to step *2 + 1, and "bumping up" some of the prime factors likewise results a larger A156552-code, thus both steps keep the result growing. The binary length of these numbers ( A070939, = 1+ A000523) grows by 0 or 1 at each step, thus the next colossally abundant number is always found on either on the same row (right of the current CA-number), or the next row of A329886, the row immediately below. The next CA-number will be on the same row only when its factorization contains neither a new prime nor yet another instance of prime 2.
PROG
(PARI)
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
CROSSREFS
Cf. A004490, A005940, A073751, A108951, A156552, A319626, A329886, A329900, A342000, A342010 (the binary weight), A342011, A342012.
Number of nonzero digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.
+20
5
1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 2, 5, 4, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 3, 3, 2, 4, 3, 4, 3, 4, 3, 4, 2, 3, 4, 4, 3, 4, 3, 4, 3, 3, 4
Largest digit value when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.
+20
4
1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 6, 4, 4, 2, 1, 2, 1, 2, 6, 4, 8, 5, 5, 3, 6, 8, 5, 4, 11, 2, 6, 1, 1, 2, 6, 4, 4, 1, 3, 1, 16, 5, 4, 3, 9, 10, 8, 3, 10, 12, 10, 6, 10, 7, 2, 3, 18, 10, 5, 4, 12, 2, 4, 2, 1, 2, 6, 4, 13, 12, 10, 8, 12, 8, 13, 2, 4, 6, 7, 2, 15, 15, 12, 10, 9, 8, 7, 6, 10, 12, 10, 9, 11, 6, 9, 6, 18, 15
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
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
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.
Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001( A276086(n)).
+10
48
0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 2, 9, 11, 37, 16, 47, 3, 11, 14, 43, 19, 53, 4, 13, 17, 49, 22, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 33, 101, 134, 373, 169, 443, 2, 11, 13, 47, 20, 61, 17, 69, 86, 277, 121, 347, 24, 83, 107, 319, 142, 389, 31
COMMENTS
The scatter plot shows an interesting structure.
The terms are essentially the "wild" or "unherited" part of the arithmetic derivative ( A003415) of those natural numbers ( A048103) that are not immediately beyond all hope of reaching zero by iteration (as the terms of A100716 are), ordered by the primorial base expansion of n as in A276086. Sequence A342018 shows the positions of the terms here that have just moved to the "no hope" region, while A342019 shows how many prime powers in any term have breached the p^p limit. Note that the results are same as for A327860(n), as the division by "regular part", A328572(n) does not affect the "wild part" of the arithmetic derivative of A276086(n). - Antti Karttunen, Mar 12 2021 I decided to name this sequence in honor of Lithuanian artist Mikalojus Čiurlionis, 1875 - 1911, as the scatter plot of this sequence reminds me thematically of his work "Pyramid sonata", with similar elements: fractal repetition in different scales and high tension present, discharging as lightning. Like Čiurlionis's paintings, this sequence has many variations, see the Formula and Crossrefs sections. - Antti Karttunen, Apr 30 2022
FORMULA
There are several permutations of this sequence. The following formulas show the relations:
(End)
PROG
(PARI)
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
(PARI) A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ Antti Karttunen, Mar 12 2021 (PARI) A342002(n) = { my(s=0, p=2, e); while(n, e = (n%p); s += (e/p); n = n\p; p = nextprime(1+p)); numerator(s); }; \\ (Taking denominator instead would give A328571) - Antti Karttunen, Mar 12 2021
CROSSREFS
Cf. A002110 (positions of 1's), A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001, A342005, A342006, A342016, A342022 (rgs-transform), A342417, A342419. Cf. A351952 (similar definition, but using factorial base, with quite a different look).
EXTENSIONS
Sequence renamed as "Čiurlionis sequence" to honor Lithuanian artist Mikalojus Čiurlionis - Antti Karttunen, Apr 30 2022
Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.
+10
26
1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1
COMMENTS
When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)* A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.
MATHEMATICA
Array[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[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *) Array[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 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)
PROG
(PARI) A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, , if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); }; (PARI)
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1, oo, if(n%prime(i), return(i-1))); }
CROSSREFS
Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.
Primorial deflation (denominator) of Doudna-tree.
+10
9
1, 1, 2, 1, 3, 1, 4, 1, 5, 3, 2, 1, 9, 2, 8, 1, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 11, 7, 14, 5, 21, 5, 20, 3, 5, 3, 2, 1, 9, 2, 8, 1, 49, 25, 50, 9, 15, 3, 4, 1, 125, 27, 18, 2, 81, 8, 32, 1, 13, 11, 22, 7, 33, 7, 28, 5, 55, 21, 14, 5, 63, 10, 40, 3, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 121
COMMENTS
Like A005940, also this irregular table can be represented as a binary tree: 1
|
...................1...................
2 1
3......../ \........1 4......../ \........1
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
5 3 2 1 9 2 8 1
7 5 10 3 3 1 4 1 25 9 6 1 27 4 16 1
etc.
A194602 gives the positions of nodes that have value 1. They correspond to terms of A005940 that are products of primorials ( A025487). The first 2^k nodes contain A000041(k+1) 1's.
MATHEMATICA
Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &[# + 1] &, 96] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
Primorial deflation (numerator) of Doudna-tree.
+10
7
1, 2, 3, 4, 5, 3, 9, 8, 7, 10, 5, 6, 25, 9, 27, 16, 11, 14, 21, 20, 7, 5, 15, 12, 49, 50, 25, 9, 125, 27, 81, 32, 13, 22, 33, 28, 55, 21, 63, 40, 11, 14, 7, 10, 35, 15, 45, 24, 121, 98, 147, 100, 49, 25, 25, 18, 343, 250, 125, 27, 625, 81, 243, 64, 17, 26, 39, 44, 65, 33, 99, 56, 91, 110, 55, 42, 275, 63, 189, 80, 13, 22
COMMENTS
Tree with both numerators (this sequence) and denominators ( A337377) shown starts as: 1/1
|
2
-
1
3 / \ 4
- ................. ................. -
2 1
5 / \ 3 9 / \ 8
- ....... ....... - - ....... ....... -
3 1 4 1
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7 10 5 6 25 9 27 16
- -- - - -- - -- --
5 3 2 1 9 2 8 1
/ \ / \ / \ / \ / \ / \ / \ / \
11 14 21 20 7 5 15 12 49 50 25 9 125 27 81 32
-- -- -- -- - - -- -- -- -- -- - --- -- -- --
7 5 10 3 3 1 4 1 25 9 6 1 27 4 16 1
etc.
FORMULA
If A007814(n+1) < A337821(n+1) then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n). If A337377(n) mod 2 = 0 then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n).
MATHEMATICA
Array[#1/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &, 82] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
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