A007862 -id:A007862 - cp's OEIS frontend

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

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

A357842 a(n) is the smallest number k for which k and the arithmetic derivative k' (A003415) have exactly n triangular divisors (A000217).

Original entry on oeis.org

2, 27, 18, 72, 612, 1764, 756, 8100, 27000, 97200, 66528, 175500, 93600, 280800, 1731600, 661500, 680400, 3704400, 34177500, 11107800, 16581600, 20065500, 108486000, 102910500, 108353700, 181912500, 314874000, 462672000, 4408236000, 229975200, 2297786400, 672348600, 925041600, 1344697200, 158230800

Offset: 1

Marius A. Burtea, Oct 20 2022

nonn

			2 has only the divisor 1 = A000217(1) and 2' = 1 = A000217(1), so a(1) = 2.
27 and 27' = 27 have the divisors 1 = A000217(1), 3 = A000217(2) triangular numbers, so a(2) = 27.

Cf. A000217, A003415, A007862, A130317.

tr:=func; f:=func;  a:=[]; for n in [1..30] do k:=2 ; while tr(k) ne n or tr(Floor(f(k))) ne n do k:=k+1; end while; Append(~a,k); end for; a;

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); tridiv[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[8*# + 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 2, i}, While[c < len && n < nmax, i = tridiv[n]; If[i <= len && s[[i]] == 0 && tridiv[d[n]] == i, c++; s[[i]] = n]; n++]; s]; seq[10, 10^6] (* Amiram Eldar, Oct 21 2022 *)

f(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ A007862
ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = my(k=2); while((f(k)!=n) || (f(ad(k))!=n), k++); k; \\ Michel Marcus, Oct 23 2022

cp's OEIS Frontend

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

A357842 a(n) is the smallest number k for which k and the arithmetic derivative k' (A003415) have exactly n triangular divisors (A000217).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

cp's OEIS Frontend

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

A357842 a(n) is the smallest number k for which k and the arithmetic derivative k' (A003415) have exactly n triangular divisors (A000217).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

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