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A376409 a(n) = A048675(A376408(n)); Partial sums of A376408.

%I A376409 #7 Nov 08 2024 17:58:34
%S A376409 0,1,3,9,99,353529,274407373885532679,
%T A376409 2443417474326613595267894539584266773823049253134356679026035220285823429
%N A376409 a(n) = A048675(A376408(n)); Partial sums of A376408.
%C A376409 a(8) has 407 digits, a(9) has 2804 digits.
%C A376409 By induction, it is easy to see that formula a(n) = A048675(A376408(n)) implies that from the second term onward, this sequence gives the partial sums of A376408, as A048675 is fully additive.
%H A376409 Antti Karttunen, <a href="/A376409/b376409.txt">Table of n, a(n) for n = 0..8</a>
%H A376409 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A376409 a(0) = 0; and for n >= 1, a(n) = a(n-1) + A376408(n-1) = Sum_{i=0..n-1} A376408(i).
%o A376409 (PARI)
%o A376409 A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
%o A376409 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A376409 A376408(n) = if(!n,1,my(x=A376408(n-1)); x*A019565(x));
%o A376409 A376409(n) = A048675(A376408(n));
%Y A376409 Cf. A019565, A048675, A376407, A376408.
%Y A376409 Cf. also A376401 (an analogous sequence for A276075).
%K A376409 nonn
%O A376409 0,3
%A A376409 _Antti Karttunen_, Nov 04 2024