A279495 -id:A279495 - cp's OEIS frontend

A300409 Number of centered triangular numbers dividing n.

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

Offset: 1

Ilya Gutkovskiy, Mar 05 2018

nonn

			a(20) = 3 because 20 has 6 divisors {1, 2, 4, 5, 10, 20} among which 3 divisors {1, 4, 10} are centered triangular numbers.

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

Cf. A005448, A007862, A279495, A300410, A306324.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
for k from 1 do
  m:= 3*k*(k+1)/2+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^(3 k (k + 1)/2 + 1)/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=0} x^(3*k*(k+1)/2+1)/(1 - x^(3*k*(k+1)/2+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A306324 = 1.5670651... . - Amiram Eldar, Jan 02 2024

A358542 a(n) is the smallest number with exactly n divisors that are tetrahedral numbers.

Original entry on oeis.org

1, 4, 56, 20, 120, 280, 560, 840, 1680, 10920, 9240, 18480, 55440, 120120, 240240, 314160, 628320, 1441440, 2282280, 7225680, 4564560, 9129120, 13693680, 27387360, 54774720, 68468400, 77597520, 136936800, 155195040, 310390080, 465585120, 775975200, 1163962800

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

nonn

			a(3) = 56 because 56 has 3 tetrahedral divisors {1, 4, 56} and this is the smallest such number.

Lucas A. Brown, Table of n, a(n) for n = 1..37
Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index entries for sequences related to divisors of numbers

Cf. A000292, A005179, A130317, A279495, A358543, A358544.

istetrah(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = my(k=1); while (sumdiv(k, d, istetrah(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(20)-a(22) from Michel Marcus, Nov 21 2022
a(23)-a(30) from Jinyuan Wang, Nov 28 2022
a(31) from Martin Ehrenstein, Dec 02 2022
a(32) and a(33) from Lucas A. Brown, Dec 14 2022

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of divisors of n of the form k*(k^2 + 1)/2 (A006003).

Cf. A006003, A007862, A279495, A300409, A334926.

Cf. A001620, A248177.

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (A248177 + A001620) = 1.343731... . - Amiram Eldar, Jan 02 2024

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

A334988 Sum of tetrahedral numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 36, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 61, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 46, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 89, 1, 1, 1, 5, 1, 11

Offset: 1

Ilya Gutkovskiy, May 18 2020

nonn

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A023533, A068980, A185027, A279495, A334987.

nmax = 90; CoefficientList[Series[Sum[Binomial[k + 2, 3] x^Binomial[k + 2, 3]/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

ist(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = sumdiv(n, d, if (ist(d), d)); \\ Michel Marcus, May 19 2020

G.f.: Sum_{k>=1} binomial(k+2,3) * x^binomial(k+2,3) / (1 - x^binomial(k+2,3)).
L.g.f.: log(G(x)), where G(x) is the g.f. for A068980.
a(n) = Sum_{d|n} A023533(d) * d.

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

cp's OEIS Frontend

A300409 Number of centered triangular numbers dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358542 a(n) is the smallest number with exactly n divisors that are tetrahedral numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A334925 G.f.: Sum_{k>=1} x^(k*(k^2 + 1)/2) / (1 - x^(k*(k^2 + 1)/2)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A334988 Sum of tetrahedral numbers dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).

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)).