A279496 -id:A279496 - cp's OEIS frontend

A300410 Number of centered square numbers dividing n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3

Offset: 1

Ilya Gutkovskiy, Mar 05 2018

nonn

			a(26) = 2 because 26 has 4 divisors {1, 2, 13, 26} among which 2 divisors {1, 13} are centered square numbers.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Square Number.
Index entries for sequences related to centered polygonal numbers.

Cf. A001844, A046951, A228048, A279496, A300409.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
for k from 1 do
  m:= 2*k*(k+1)+1;
  if m > N then break fi;
  r:= [seq(i,i=m..N,m)];
  V[r]:= map(t->t+1, V[r]);
od:
convert(V,list); # Robert Israel, Mar 05 2018

nmax = 100; Rest[CoefficientList[Series[Sum[x^(2 k (k + 1) + 1)/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=0} x^(2*k*(k+1)+1)/(1 - x^(2*k*(k+1)+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A228048 = 1.440659... . - Amiram Eldar, Jan 02 2024

A358543 a(n) is the smallest number with exactly n divisors that are square pyramidal numbers.

Original entry on oeis.org

1, 5, 30, 140, 420, 1540, 4620, 13860, 78540, 157080, 471240, 1141140, 3603600, 3423420, 13693680, 30630600, 58198140, 116396280, 214414200, 428828400, 581981400, 1163962800, 5354228880, 4073869800, 8147739600, 26771144400, 36082846800, 80313433200, 93699005400, 187398010800

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

nonn

Any terms for n > 25 exceed 10^10. - Lucas A. Brown, Dec 24 2022

a(25) <= 8147739600, a(26) <= 26771144400, a(27) <= 36082846800, a(28) <= 80313433200. - Jon E. Schoenfield, Dec 16 2022

			a(3) = 30 because 30 has 3 square pyramidal divisors {1, 5, 30} and this is the smallest such number.

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Square Pyramidal Number.
Index entries for sequences related to divisors of numbers

Cf. A000330, A005179, A130279, A279496, A358542, A358545.

issqpyr(n) = my(m = sqrtnint(3*n, 3)); n==m*(m+1)*(2*m+1)/6; \\ A253903
a(n) = my(k=1); while (sumdiv(k, d, issqpyr(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(15) from Michel Marcus, Nov 21 2022
a(16)-a(20) from Jinyuan Wang, Nov 28 2022
a(21)-a(22) from Lucas A. Brown, Dec 14 2022
a(23)-a(24) from Lucas A. Brown, Dec 18 2022
a(25) from Lucas A. Brown, Dec 22 2022
a(26)-a(30) from Bert Dobbelaere, May 18 2025

A334926 G.f.: Sum_{k>=1} x^(k(2k^2 + 1)/3) / (1 - x^(k(2k^2 + 1)/3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of octahedral numbers (A005900) dividing n.

Eric Weisstein's World of Mathematics, Octahedral Number.

Cf. A005900, A061704, A175577, A279495, A279496, A300410, A334925.

nmax = 100; CoefficientList[Series[Sum[x^(k (2 k^2 + 1)/3)/(1 - x^(k (2 k^2 + 1)/3)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A175577 = 1.278185... . - Amiram Eldar, Jan 02 2024

A334924 G.f.: Sum_{k>=1} x^(k^2(k + 1)/2) / (1 - x^(k^2(k + 1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of pentagonal pyramidal numbers (A002411) dividing n.

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number.

Cf. A002411, A007862, A195055, A279495, A279496, A279497.

nmax = 100; CoefficientList[Series[Sum[x^(k^2 (k + 1)/2)/(1 - x^(k^2 (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/3 - 2 = A195055 - 2 = 1.289868... . - Amiram Eldar, Jan 02 2024

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A300410 Number of centered square numbers dividing n.

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A358543 a(n) is the smallest number with exactly n divisors that are square pyramidal numbers.

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A334926 G.f.: Sum_{k>=1} x^(k*(2*k^2 + 1)/3) / (1 - x^(k*(2*k^2 + 1)/3)).

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A334924 G.f.: Sum_{k>=1} x^(k^2*(k + 1)/2) / (1 - x^(k^2*(k + 1)/2)).

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Mathematica

Formula

A334926 G.f.: Sum_{k>=1} x^(k(2k^2 + 1)/3) / (1 - x^(k(2k^2 + 1)/3)).

A334924 G.f.: Sum_{k>=1} x^(k^2(k + 1)/2) / (1 - x^(k^2(k + 1)/2)).