Displaying 1-10 of 151 results found.
page
1
2
3
4
5
6
7
8
9
10
... 16
Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization ( A005361).
(Formerly M3333)
+20
29
1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
FORMULA
For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.
MATHEMATICA
a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *) With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *) DeleteDuplicates[Table[{n, Times@@FactorInteger[n][[All, 2]]}, {n, 26*10^5}], GreaterEqual[#1[[2]], #2[[2]]]&][[All, 1]] (* Harvey P. Dale, May 13 2022 *)
PROG
(PARI) {prdex(n)=local(s, fac); s=1; fac=factor(n); for(k=1, matsize(fac)[1], s=s*fac[k, 2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1, m, if(prdex(n)>rec, rec=prdex(n); print1(n", ")))}
EXTENSIONS
Hardy and Subbarao give an extensive table.
1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 10, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 12, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 20, 2, 8, 8, 16
COMMENTS
Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^ A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328. Dirichlet inverse b(n) [= A355837(n)] is multiplicative with b(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e > 0.
FORMULA
Multiplicative with a(p^e) = 2*e for prime p and e > 0.
Dirichlet g. f.: (zeta(s))^2 * zeta(2*s) / zeta(4*s).
Sum_{k=1..n} a(k) ~ 15*(log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 - 360*zeta'(4)/Pi^4) * n / Pi^2 + 6*zeta(1/2)^2 * sqrt(n) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 20 2020
MATHEMATICA
a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 2^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)
PROG
(PARI) a(n) = my(f=factor(n)); vecprod(f[, 2])*2^omega(n); \\ Michel Marcus, Dec 04 2018 (Python)
from math import prod
from sympy import factorint
def A322327(n): return prod(e<<1 for e in factorint(n).values()) # Chai Wah Wu, Dec 26 2022
CROSSREFS
Cf. A000005, A000007, A001221, A001620, A005361, A034444, A226602, A227291, A286324, A322328, A355837 (Dirichlet inverse).
1, 1, 1, 2, 1, 2, 4, 6, 1, 2, 4, 6, 8, 12, 18, 24, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480, 600, 720, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480
COMMENTS
a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - Gus Wiseman, Apr 26 2020
MATHEMATICA
Table[Times @@ FactorInteger[#][[All, -1]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 93}] (* Michael De Vlieger, Mar 18 2017 *)
PROG
(PARI)
A034386(n) = prod(i=1, primepi(n), prime(i));
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
CROSSREFS
All of the following pertain to compositions in standard order ( A066099): - Constant compositions are A272919. - Distinct parts are counted by A334028.
1, 4, 4, 8, 4, 16, 4, 12, 8, 16, 4, 32, 4, 16, 16, 16, 4, 32, 4, 32, 16, 16, 4, 48, 8, 16, 12, 32, 4, 64, 4, 20, 16, 16, 16, 64, 4, 16, 16, 48, 4, 64, 4, 32, 32, 16, 4, 64, 8, 32, 16, 32, 4, 48, 16, 48, 16, 16, 4, 128, 4, 16, 32, 24, 16, 64, 4, 32, 16, 64, 4
COMMENTS
Let k be some fixed integer and a_k(n) = A005361(n) * k^ A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see A322327, for k = 3 see A226602 (offset 1), and for k = 4 see this sequence.
FORMULA
Multiplicative with a(p^e) = 4*e for prime p and e > 0.
Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.
Dirichlet inverse is b(n) = a(n) * A008836(n) for n > 0, and b(n) is multiplicative with b(p^e) = 4*e*(-1)^e for prime p and e > 0. Equals Dirichlet convolution of A034444 with itself.
MAPLE
f:= n -> mul(4*t[2], t=ifactors(n)[2]):
MATHEMATICA
a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)
PROG
(PARI) a(n) = my(f=factor(n)); vecprod(f[, 2])*4^omega(n); \\ Michel Marcus, Dec 04 2018 (Python)
from math import prod
from sympy import factorint
def A322328(n): return prod(e<<2 for e in factorint(n).values()) # Chai Wah Wu, Dec 24 2022
Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.
+20
10
1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000
COMMENTS
All terms are highly powerful ( A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.
EXAMPLE
The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.
MATHEMATICA
nn=1000;
d=Table[Times@@Last/@FactorInteger[n], {n, nn}];
Select[Range[nn], !MemberQ[Take[d, #-1], d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/ A025487/b025487.txt", "Table"], {_, _}][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)
CROSSREFS
These are the positions of first appearances in A005361, counted by A266477. This is the sorted version of A085629. The version for shadows instead of exponents is A353397, firsts in A353394. A130091 lists numbers with distinct prime exponents, counted by A098859.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Index of the first of two successive 2's in A005361.
+20
4
44, 49, 75, 98, 116, 147, 171, 244, 260, 275, 315, 332, 363, 387, 475, 476, 507, 524, 531, 548, 549, 603, 604, 636, 692, 724, 725, 747, 764, 774, 819, 844, 845, 846, 867, 908, 924, 931, 963, 1035, 1075, 1083, 1179, 1196, 1251, 1274, 1275, 1324, 1340, 1395
COMMENTS
Numbers such that bigomega(n) - omega(n) = 1 and bigomega(n+1) - omega(n+1) = 1, where bigomega(n) is the number of primes dividing n (counted with repetition) and omega(n) is the number of distinct primes dividing n. - Michel Lagneau, Dec 17 2011 This sequence has 3548 terms up to 10^5, 35340 up to 10^6, 353147 up to 10^7, and 3531738 up to 10^8, suggesting a natural density around 0.0353.... - Charles R Greathouse IV, Mar 06 2017
MATHEMATICA
Select[Range[1400], PrimeOmega[#]-PrimeNu[#] == 1 && PrimeOmega[#+1] - PrimeNu[#+1] == 1 &] (* Indranil Ghosh, Mar 05 2017 *) SequencePosition[Table[PrimeOmega[n]-PrimeNu[n], {n, 1500}], {1, 1}][[;; , 1]] (* Harvey P. Dale, Jul 17 2024 *)
PROG
(PARI) isok(n) = (bigomega(n)-omega(n) == 1) && (bigomega(n+1)-omega(n+1) == 1); \\ Michel Marcus, Mar 05 2017 (PARI) is(n)=factorback(factor(n)[, 2])==2 && factorback(factor(n+1)[, 2])==2 \\ Charles R Greathouse IV, Mar 06 2017
Product of exponents of prime factorization of n by prime signature: A005361( A025487).
+20
3
1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 4, 4, 2, 6, 6, 5, 3, 7, 8, 4, 6, 1, 9, 4, 8, 10, 6, 7, 2, 12, 5, 9, 12, 8, 8, 3, 15, 8, 6, 10, 9, 14, 4, 16, 10, 9, 4, 18, 12, 7, 11, 12, 16, 1, 6, 20, 12, 10, 5, 21, 16, 8, 12, 15, 18, 2, 8, 24, 18, 14, 11, 8, 16, 6, 24, 20, 9, 9, 25, 13, 18, 20, 3
PROG
(PARI) lista() = {v = readvec("b025487.txt"); for (i=1, #v, f = factor(v[i]); print1(prod(k=1, #f~, f[k, 2]), ", "); ); } \\ Michel Marcus, Nov 02 2014
a(n) = one more than the number of iterations of A005361 needed to reach 1 from the starting value n.
+20
2
1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 2, 3, 2, 2
FORMULA
a(1) = 1, and for n > 1, a(n) = 1 + a( A005361(n)).
EXAMPLE
For n = 2 = 2^1, A005361(2) = 1, so we reach 1 in one step, and thus a(2) = 1+1 = 2. For n = 4 = 2^2, A005361(4) = 2; A005361(2) = 1, so we reach 1 in two steps, and thus a(4) = 2+1 = 3. For n = 6 = 2^1 * 3^1, A005361(6) = 1*1 = 1, so we reach 1 in one step, and thus a(6) = 1+1 = 2. For n = 64 = 2^6, A005361(64) = 6, thus a(64) = 1 + a(6) = 3. For n = 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7*1, A005361(3628800) = 64, thus a(3628800) = 1 + a(64) = 4.
MATHEMATICA
ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] expr[x_] := Apply[Times, ep[x]] Table[Length[FixedPointList[expr, w]]-1, {w, 2, 128}]
(* Second program: *)
Table[Length@ NestWhileList[Apply[Times, FactorInteger[#][[All, -1]]] &, n, # != 1 &], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)
PROG
(PARI)
(PARI) first(n) = my(v = vector(n)); v[1] = 1; for(i=2, n, v[i] = v[factorback(factor(i)[, 2])] + 1); v \\ David A. Corneth, Jul 28 2017
EXTENSIONS
Term a(1)=1 prepended, Name and Example sections edited by Antti Karttunen, Jul 28 2017
First position of a plateau of length n in the product-of-exponents function A005361.
+20
2
2, 4, 843, 74848, 671345, 8870024
COMMENTS
a(8) is conjectured to be 1770019255373287038727484868192109228823.
Each term a(n) satisfies four properties: 1, divisible by all prime factors of n; 2, divisible by only the prime factors of n; 3, not equal to any of the terms a(1), a(2), ... a(n-1); 4, smallest number satisfying 1-3 if A005361(n) is even, or second smallest number satisfying 1-3 if A005361(n) is odd.
+20
2
1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 18, 361, 10, 63, 44, 529, 36, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 24, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 48, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721, 124, 21, 32
COMMENTS
This sequence is permutation of the positive integers.
The prime p occurs at n = p^2.
Multiples of a number x have density 1/x.
Conjecture: this permutation of positive integers is self-inverse. Compare with A358971. The principal distinction between this sequence and A358971 is that fixed points aside from A358971(1) = 1 are explicitly ruled out in the latter. - Michael De Vlieger, Dec 10 2022
REFERENCES
Brad Klee, Posting to Sequence Fans Mailing List, Dec 21, 2014.
LINKS
Michael De Vlieger, Log log scatterplot of a(n) <= 12000, n = 1..2^10 showing primes in red, other prime powers (in A246547) in gold, squarefree composites (in A120944) in green, numbers neither squarefree nor prime power (in A120706) in blue and magenta. The terms in magenta are products of composite prime powers (in A286708). Michael De Vlieger, Log log scatterplot of a(n) <= 2^14, n = 1..2^14, showing a(n) such that rad(n) = 6 in red, and A358971(n) such that rad(n) = 6 in blue for comparison. This is an example of a self-inverse relation among terms a(n) in A003586.
MAPLE
A253288div := proc(a, n)
local npr, d, apr ;
npr := numtheory[factorset](n) ;
for d in npr do
if modp(a, d) <> 0 then
return false;
end if;
end do:
apr := numtheory[factorset](a) ;
if apr minus npr = {} then
true;
else
false;
end if;
end proc:
option remember;
local a, i, prev, act, ev ;
if n =1 then
1;
else
act := 1 ;
ev := true;
else
ev := false;
end if;
for a from 1 do
prev := false;
for i from 1 to n-1 do
if procname(i) = a then
prev := true;
break;
end if;
end do:
if not prev then
if A253288div(a, n) then
if ev or act > 1 then
return a;
else
act := act+1 ;
end if;
end if;
end if;
end do:
end if;
end proc:
MATHEMATICA
nn = 1000; c[_] = False; q[_] = 1; f[n_] := f[n] = Map[Times @@ # &, Transpose@ FactorInteger[n]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], SquareFreeQ[n], k = First@ f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, First@ f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]; k *= m, True, t = 0; Set[{k, s}, {First[#], 1 + Boole@ OddQ@ Last[#]} &[f[n]]]; m = q[k]; Until[t == s, If[m > q[k], m++]; While[Nand[! c[k m], Divisible[k, First@f[m]]], m++]; t++]; If[s == 1, While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 10 2022 *)
CROSSREFS
Cf. A005361 (Product of exponents of prime factorization of n), A358971.
Search completed in 0.126 seconds
|