A013664 -id:A013664 - cp's OEIS frontend

A085966 Decimal expansion of the prime zeta function at 6.

0, 1, 7, 0, 7, 0, 0, 8, 6, 8, 5, 0, 6, 3, 6, 5, 1, 2, 9, 5, 4, 1, 3, 3, 6, 7, 3, 2, 6, 6, 0, 5, 9, 3, 9, 9, 2, 0, 9, 5, 8, 5, 9, 4, 1, 8, 7, 4, 5, 4, 4, 2, 4, 4, 7, 3, 3, 1, 6, 3, 3, 6, 8, 8, 3, 6, 9, 6, 9, 7, 3, 6, 7, 4, 7, 1, 7, 2, 4, 3, 6, 6, 7, 1, 8, 6, 0, 3, 5, 0, 0, 7, 8, 1, 8, 0, 6, 2, 3, 0, 2, 8, 8, 2, 3

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

			0.0170700868506365129541...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1802
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), A085965 (at 5), this sequence (at 6), A085967 (at 7) to A085969 (at 9).

Cf. A013664, A030516.

R := RealField(106);
PrimeZeta := func;
[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(6,57)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[6*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n = 200]; While[s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 6], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,6) \\ Hugo Pfoertner, Feb 03 2020

P(6) = Sum_{p prime} 1/p^6 = Sum_{n>=1} mobius(n)*log(zeta(6*n))/n
Equals 1/2^6 + A085995 + A086036. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A030516(k). - Amiram Eldar, Jul 27 2020

A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

Original entry on oeis.org

1, 65, 47449, 3037465, 47463376609, 47464376609, 5584183099672241, 357389058474664049, 260537105518334091721, 52107472322919827957, 92311616995117182948130877, 92311647383100199924330877, 445570781131605573859221176881493, 445570839299219762020391212081493

Offset: 1

Wolfdieter Lang, Feb 15 2005

For the rationals Zeta(k,n) for k = 1..10 and n = 1..20, see the W. Lang link.

a(n) gives the partial sum, Zeta(6,n), of Euler's (later Riemann's) Zeta(6). Zeta(k,n), k >= 2, is sometimes also called H(k,n) because for k = 1 these would be the harmonic numbers A001008/A002805. However, H(1,n) does not give partial sums of a convergent series.

			The first few fractions are 1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, ... = A103345/A103346. - _Petros Hadjicostas_, May 10 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, Rational Zeta(k,n) and more.

Cf. A013664, A291456. For the denominators, see A103346.

For k=1..5, see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.

s=0; lst={}; Do[s+=n^1/n^7; AppendTo[lst,Numerator[s]],{n,3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 6] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(Sum_{k=1..n} 1/k^6) = numerator(A291456(n)/(n!)^6).
G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).
Zeta(6, n) = (1/945)*Pi^6 - psi(5, n+1)/5!, see eq. 6.4.3 with m = 5, p. 260, of the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 03 2013

A059378 Jordan function J_5(n).

Original entry on oeis.org

1, 31, 242, 992, 3124, 7502, 16806, 31744, 58806, 96844, 161050, 240064, 371292, 520986, 756008, 1015808, 1419856, 1822986, 2476098, 3099008, 4067052, 4992550, 6436342, 7682048, 9762500, 11510052, 14289858, 16671552, 20511148, 23436248, 28629150, 32505856

Offset: 1

N. J. A. Sloane, Jan 28 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

T. D. Noe, Table of n, a(n) for n=1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Wikipedia, Jordan's totient function.

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A069091 - A069095 (J_6 through J_10).

Cf. A013664.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 5)

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 5]; Array[f, 30]
f[p_, e_] := p^(5*e) - p^(5*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^5*moebius(n/d)),","))

{ for (n = 1, 1000, write("b059378.txt", n, " ", sumdiv(n, d, d^5*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009

from sympy import divisors, mobius
def a(n):
    return sum(d**5 * mobius(n // d) for d in divisors(n))
# Indranil Ghosh, Apr 26 2017

a(n) = Sum_{d|n} d^5*mu(n/d). - Benoit Cloitre, Apr 05 2002
Multiplicative with a(p^e) = p^(5e)-p^(5(e-1)).
Dirichlet generating function: zeta(s-5)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = n^5*Product_{distinct primes p dividing n} (1-1/p^5). - Tom Edgar, Jan 09 2015
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ 315*n^6 / (2*Pi^6). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^5 = 1/zeta(6).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^5/(p^5-1)^2) = 1.0379908060... (End)
O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 26*x^n + 66*x^(2*n) + 26*x^(3*n) + x^(4*n))/(1 - x^n)^6 = x + 31*x^2 + 242*x^3 + 992*x^4 + 3124*x^5 + .... - Peter Bala, Jan 31 2022
From Peter Bala, Jan 01 2024: (Start)
a(n) = Sum_{d divides n} d * J_4(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_3(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_2(d) * J_3(n/d) = Sum_{d divides n} d^4 * J_1(d) * J_4(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n), J(3,n) = A059376(n) and J_4(n) = A059377(n).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_4(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_3(gcd(j, k, n)). (End)
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_3(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_4(j) (apply Lehmer, Theorem 1). - Peter Bala, Jan 30 2024

A266553 Decimal expansion of the generalized Glaisher-Kinkelin constant A(6).

Original entry on oeis.org

1, 0, 0, 5, 9, 1, 7, 1, 9, 6, 9, 9, 8, 6, 7, 3, 4, 6, 8, 4, 4, 4, 0, 1, 3, 9, 8, 3, 5, 5, 4, 2, 5, 5, 6, 5, 6, 3, 9, 0, 6, 1, 5, 6, 5, 5, 0, 0, 6, 9, 3, 2, 1, 1, 4, 0, 0, 9, 8, 0, 5, 1, 5, 7, 4, 0, 8, 1, 4, 6, 8, 7, 0, 3, 4, 2, 9, 9, 4, 6, 3, 2, 7, 7, 1, 9, 6, 7, 0, 8, 1, 7, 0, 8, 8, 4, 1, 4, 6, 8, 7, 3, 5, 4, 1, 1, 1, 0, 0, 2, 2, 4, 0, 3

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 6th Bendersky constant.

			1.00591719699867346844401398355425565639061565500693211400980...

G. C. Greubel, Table of n, a(n) for n = 1..2003

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A013664, A013665, A259071, A027641, A027642

Exp[N[(BernoulliB[6]/4)*(Zeta[7]/Zeta[6]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(6) = exp(- zeta'(-6)) = exp((B(6)/4)*(zeta(7)/zeta(6))).
A(6) = exp(6! * Zeta(7) / (2^7 * Pi^6)). - Vaclav Kotesovec, Jan 01 2016

A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Unitary analog of A078429.
Cf. A013664, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), this sequence (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[Divisible[e, 3], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%3, 1, 0));}

Multiplicative with a(p^e) = p^e if e is a multiple of 3, and p^e-1 otherwise.
Dirichlet g.f.: zeta(s-1) * zeta(3*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) - 1/p^(3*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.71190515701599590826... .

A002760 Squares and cubes.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849

Offset: 1

N. J. A. Sloane

nonn

Catalan's Conjecture states that 8 and 9 are the only pair of consecutive numbers in this sequence. The conjecture was established in 2003 by Mihilescu.

Subsequence of A022549. - Reinhard Zumkeller, Jul 17 2010

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 236.

Michael De Vlieger, Table of n, a(n) for n = 1..10443 (first 1000 terms from Zak Seidov)
Yuri F. Bilu, Catalan's Conjecture (After Mihilescu), Astérisque, No. 294, 1-26, 2004.
Yuri F. Bilu, Catalan Without Logarithmic Forms (after Bugeaud, Hanrot and Mihailescu), J. Théor. Nombres Bordeaux 17, 69-85, 2005.
David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 4.
Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
Preda Mihǎilescu, A Class Number Free Criterion for Catalan's Conjecture, J. Number Th. 99 225-231, 2003.
Preda Mihǎilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. MR 2076124.
Paulo Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), p. 1-11.
Paulo Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.

Cf. A131799; union of A000290 and A000578.
First differences in A075052. [From Zak Seidov, May 10 2010]
Cf. A002117, A013661, A013664.

[n: n in [0..1600] | IsIntegral(n^(1/3)) or IsIntegral(n^(1/2))]; // Bruno Berselli, Feb 09 2016

nMax=2000;Union[Range[0,nMax^(1/2)]^2,Range[0,nMax^(1/3)]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
nxt[n_] := Min[ Floor[1 + Sqrt[n]]^2, Floor[1 + n^(1/3)]^3]; NestList[ nxt, 0, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isok(n) = issquare(n) || ispower(n, 3); \\ Michel Marcus, Mar 29 2016

from math import isqrt
from sympy import integer_nthroot
def A002760(n):
    def f(x): return n-1+x+integer_nthroot(x,6)[0]-integer_nthroot(x,3)[0]-isqrt(x)
    m, k = n-1, f(n-1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 09 2024

Sum_{n>=2} 1/a(n) = zeta(2) + zeta(3) - zeta(6). - Amiram Eldar, Dec 19 2020

A157289 Decimal expansion of Zeta(3)/Zeta(6).

Original entry on oeis.org

1, 1, 8, 1, 5, 6, 4, 9, 4, 9, 0, 1, 0, 2, 5, 6, 9, 1, 2, 5, 6, 9, 3, 9, 9, 7, 3, 4, 1, 6, 0, 4, 5, 4, 2, 6, 0, 5, 4, 7, 0, 2, 3, 2, 6, 0, 7, 6, 8, 6, 8, 2, 6, 1, 0, 2, 8, 3, 0, 4, 3, 1, 4, 8, 8, 7, 7, 2, 0, 5, 4, 2, 1, 1, 1, 0, 3, 1, 8, 8, 3, 9, 9, 0, 0, 2, 9, 9, 4, 8, 7, 1, 1, 8, 4, 4, 4, 9, 2, 7, 0, 1, 1, 4, 8

Offset: 1

R. J. Mathar, Feb 26 2009

The Product_{p = primes = A000040} (1+1/p^3), the cubic analog to A082020.

			1.181564949010256912569399734... = (1+1/2^3)*(1+1/3^3)*(1+1/5^3)*(1+1/7^3)*...

Cf. A002117, A005117, A013664, A030078, A062838, A082020.

Maple
```
evalf(Zeta(3)/Zeta(6)) ;
```

RealDigits[Zeta[3]/Zeta[6],10,120][[1]] (* Harvey P. Dale, Jul 23 2016 *)

Equals A002117/A013664 = Product_{i} (1+1/A030078(i)).

Equals Sum_{k>=1} 1/A062838(k) = Sum_{k>=1} 1/A005117(k)^3. - Amiram Eldar, May 22 2020

A008881 a(n) = Product_{j=0..5} floor((n+j)/6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 96, 144, 216, 324, 486, 729, 972, 1296, 1728, 2304, 3072, 4096, 5120, 6400, 8000, 10000, 12500, 15625, 18750, 22500, 27000, 32400, 38880, 46656, 54432, 63504, 74088, 86436, 100842, 117649, 134456, 153664, 175616, 200704

Offset: 0

N. J. A. Sloane

For n >= 6, a(n) is the maximal product of 6 positive integers with sum n. - Wesley Ivan Hurt, Jun 29 2022

The maximal product of k positive variables when their sum is equal to s is obtained when each term = s/k; hence, a(6m) = m^6 (A001014). - Bernard Schott, Jul 28 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,5,-10,5,0,0,0,-10,20,-10,0,0,0,10,-20,10,0,0,0,-5,10,-5,0,0,0,1,-2,1).

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), this sequence (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A001014 (6th power), A008588 (multiples of 6), A013664.

List([0..50], n-> Product([0..5], j-> Int((n+j)/6))); # G. C. Greubel, Sep 13 2019

[(&*[Floor((n+j)/6): j in [0..5]]): n in [0..50]]; // G. C. Greubel, Sep 13 2019

seq( mul( floor((n+i)/6), i=0..5 ), n=0..80);

Product[Floor[(Range[51]+j-2)/6], {j,6}] (* G. C. Greubel, Sep 13 2019 *)

vector(50, n, prod(j=0,5, (n+j)\6) ) \\ G. C. Greubel, Sep 13 2019

[product(floor((n+j)/6) for j in (0..5)) for n in (0..50)] # G. C. Greubel, Sep 13 2019

Sum_{n>=6} 1/a(n) = 1 + zeta(6). - Amiram Eldar, Jan 10 2023

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

			a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
Cf. A001160, A013664, A131472.

terms = 30;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

a(n) = A145095(n)/504 for n > 0.
G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356193 at n=16.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A013664, A268335, A335988.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A335988.
a(n) = A356191(n) iff n is a powerful number (A001694).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022

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