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A217627 a(n) is the sum of the products of the nonzero digits of the numbers from 1 to n.

%I A217627 #19 Sep 20 2017 18:11:21
%S A217627 1,3,6,10,15,21,28,36,45,46,47,49,52,56,61,67,74,82,91,93,95,99,105,
%T A217627 113,123,135,149,165,183,186,189,195,204,216,231,249,270,294,321,325,
%U A217627 329,337,349,365,385,409,437,469,505,510,515,525,540,560,585,615,650
%N A217627 a(n) is the sum of the products of the nonzero digits of the numbers from 1 to n.
%C A217627 The formula a(10^k) = 46^k can be easily derived from the Multinomial Theorem, inspecting the expansion of (1+1+2+3+...+9)^k, where the second '1's takes the place of '0' (since we are neglecting the zeros in the products). This formula can be generalized as follows:
%C A217627 Let B>1 be the base used for representation. Let D be a subset of {1,2,...,B-1}. Using base B, let A(n) be the sum of the products of the digits in D of the numbers up to n. Then, A(B^k)=(B+S-|D|)^k, where |D| is the cardinality of D and S is the sum of the elements of D. For example, in base 10, with D={1,3,5,7,9}, (i.e., A(n)= sum of the products of the odd digits of the numbers up to n) we have A(k)=(10+(1+3+5+7+9)-5)^k = 30^k.
%H A217627 Giovanni Resta, <a href="/A217627/b217627.txt">Table of n, a(n) for n = 1..10000</a>
%H A217627 Wikipedia, <a href="http://en.wikipedia.org/wiki/Multinomial_theorem">Multinomial theorem</a>
%F A217627 a(10^k) = 46^k.
%e A217627 a(10) = 1+2+3+4+5+6+7+8+9+1 = 46
%t A217627 pp[n_]:=Times@@Select[IntegerDigits[n],#>0 &]; Accumulate[pp /@ Range[100]]
%Y A217627 Cf. A061076 (the same sum, when zeros are taken into account).
%K A217627 nonn,base,easy
%O A217627 1,2
%A A217627 _Giovanni Resta_, Oct 18 2012