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A378176 Sum over all multiplicative partitions mu of n (with factors > 1) of the encoding as Product_{j in mu} prime(j).

%I A378176 #17 Nov 20 2024 05:20:09
%S A378176 1,3,5,16,11,28,17,67,48,62,31,156,41,94,102,303,59,270,67,334,158,
%T A378176 172,83,743,218,224,343,508,109,707,127,1173,292,316,336,1651,157,364,
%U A378176 372,1587,179,1091,191,926,960,448,211,3468,516,1202,528,1198,241,2209
%N A378176 Sum over all multiplicative partitions mu of n (with factors > 1) of the encoding as Product_{j in mu} prime(j).
%H A378176 Alois P. Heinz, <a href="/A378176/b378176.txt">Table of n, a(n) for n = 1..10000</a>
%F A378176 a(prime(n)) = a(A000040(n)) = A006450(n).
%e A378176 The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} = {19, 21, 27}; the sum gives a(8) = 67.
%p A378176 b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
%p A378176       ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
%p A378176     end:
%p A378176 a:= n-> add(i, i=b(n)):
%p A378176 seq(a(n), n=1..54);
%Y A378176 Row sums of A378175.
%Y A378176 Cf. A000040, A001055, A006450, A145519, A377853.
%K A378176 nonn
%O A378176 1,2
%A A378176 _Alois P. Heinz_, Nov 18 2024