A380070 - OEIS

%I #17 Jan 12 2025 09:13:16

%S 1,55,385,5525,7714,9455,31395,116795,3382596,6597495,14352975,

%T 38381931,63866976,67938794,151289984,726434136,2733212496,2769972525,

%U 6477521344,13765386816,16149163995,17585156875,19598599944,78466472175,129166736265,371557934784,597944938275,982218617856

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

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

%H Michel Marcus, <a href="/A380070/b380070.txt">Table of n, a(n) for n = 1..60</a>

%e 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.

%t 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 *)

%o (PARI) sp(n) = n*(n+1)*(2*n+1)/6; \\ A000330

%o 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

%Y Cf. A000330, A007954, A052382.

%K nonn,base

%O 1,2

%A _Ctibor O. Zizka_, Jan 11 2025

%E More terms from _Michel Marcus_, Jan 11 2025