A380070 Square pyramidal numbers sp(n) with a zeroless decimal representation such that (product of decimal digits of sp(n)) / n is an integer.

1, 55, 385, 5525, 7714, 9455, 31395, 116795, 3382596, 6597495, 14352975, 38381931, 63866976, 67938794, 151289984, 726434136, 2733212496, 2769972525, 6477521344, 13765386816, 16149163995, 17585156875, 19598599944, 78466472175, 129166736265, 371557934784, 597944938275, 982218617856

Offset

1,2

Comments

I conjecture the sequence has only finitely many members, i.e., for all n, sp(n) < 10^r for a large enough r.

Links

Michel Marcus, Table of n, a(n) for n = 1..60

Example

For n = 28, sp(28) = 7714, product of decimal digits of sp(28) = 7*7*1*4 = 196 and 196 / 28 = 7 so sp(28) = 7714 is in the sequence.

Mathematica

sp[n_] := n*(n + 1)*(2*n + 1)/6; seq[lim_] := Module[{v = Table[Times @@ IntegerDigits[sp[n]], {n, 1, lim}]/Range[lim], ind}, ind = Position[v, _?(# > 0 && IntegerQ[#] &)] // Flatten; sp[ind]]; seq[15000] (* Amiram Eldar, Jan 11 2025 *)

Prog

(PARI) sp(n) = n*(n+1)*(2*n+1)/6; \\ A000330
lista(nn) = my(list=List()); for (n=1, nn, my(x=sp(n), d=digits(x)); if (vecmin(d) && !(vecprod(d) % n), listput(list, x)); ); Vec(list); \\ Michel Marcus, Jan 11 2025

Crossrefs

Cf. A000330, A007954, A052382.

Sequence in context: A063653 A222348 A075740 * A340240 A355511 A129217

Adjacent sequences: A380065 A380067 A380068 * A380071 A380072 A380073

Keyword

nonn,base

Author

Ctibor O. Zizka, Jan 11 2025

Extensions

More terms from Michel Marcus, Jan 11 2025

Status

approved