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%I #39 Aug 06 2021 15:21:53
%S 5,15,36,74,141,250,426,696,1106,1711,2593,3852,5635,8118,11548,16231,
%T 22577,31092,42447,57464,77213,103009,136529,179830,235514,306751,
%U 397506,512607,658030,841020,1070490,1357195,1714274,2157539,2706174,3383187,4216358
%N Number of distinct factorizations of 105*2^n.
%H Alois P. Heinz, <a href="/A093802/b093802.txt">Table of n, a(n) for n = 0..10000</a>
%e 105*A000079 is 105, 210, 420, 840, 1680, 3360, ... and there are 15 distinct factorizations of 210 so a(1) = 15.
%e a(0) = 5: 105*2^0 = 105 = 3*5*7 = 3*35 = 5*21 = 7*15. - _Alois P. Heinz_, May 26 2013
%p with(numtheory):
%p b:= proc(n, k) option remember;
%p `if`(n>k, 0, 1) +`if`(isprime(n), 0,
%p add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
%p end:
%p a:= n-> b((105*2^n)$2):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, May 26 2013
%t b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0,
%t Sum[If[d > k, 0, b[n/d, d]], {d, Divisors[n][[2;;-2]]}]];
%t a[n_] := b[105*2^n, 105*2^n];
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 15 2021, after _Alois P. Heinz_ *)
%Y Similar sequences: 45*A000079 => A002763, [1, 3, 9, 27, 81, 243...]*A000079 => A054225, 1*A002110 => A000110, 2*A002110 => A035098, A000142 => A076716.
%Y Cf. A001055, A126442, A129306, A346823.
%Y Column k=3 of A346426.
%K easy,nonn
%O 0,1
%A _Alford Arnold_, May 19 2004
%E 2 more terms from _Alford Arnold_, Aug 29 2007
%E Corrected offset and extended beyond a(7) by _Alois P. Heinz_, May 26 2013
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