A109490 -id:A109490 - cp's OEIS frontend

A109489 Value of Product[k/sd(k,2),k=1..n], where sd(k,b) is the sum of the digits of k represented in base b.

1, 2, 3, 12, 30, 90, 210, 1680, 7560, 37800, 138600, 831600, 3603600, 16816800, 63063000, 1009008000, 8576568000, 77189112000, 488864376000, 4888643760000, 34220506320000, 250950379680000, 1442964683160000, 17315576197920000

Offset: 1

John W. Layman, Jun 29 2005

It appears that Product[k/sd(k,b),k=1..n] is an integer for all integers n>0 and b>1. Is this known or easy to prove?

It is not true! The product is not an integer for b=2 and n=422 (it has a denominator of 5). B-file contains all terms before that. - Robert Israel, Jan 21 2018

			The base 2 representations of 1,2,3,4 are 1,10,11,100 so a(4)=(1/1)(2/1)(3/2)(4/1)=12.

Robert Israel, Table of n, a(n) for n = 1..421

Cf. A003601, A109490, A109491, A109492.

P:= 1: A[1]:= P:
for n from 2 to 100 do
  P:= P*n/convert(convert(n,base,2),`+`);
  A[n]:= P;
od:
seq(A[i],i=1..100); # Robert Israel, Jan 21 2018

a(n) = prod(k=1, n, k/hammingweight(k)); \\ Michel Marcus, Jul 10 2014

cp's OEIS Frontend

A109489 Value of Product[k/sd(k,2),k=1..n], where sd(k,b) is the sum of the digits of k represented in base b.

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