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A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

%I A293447 #15 Feb 08 2020 20:44:22
%S A293447 0,1,3,2,7,4,15,3,6,8,31,5,63,16,10,4,127,7,255,9,18,32,511,6,14,64,9,
%T A293447 17,1023,11,2047,5,34,128,22,8,4095,256,66,10,8191,19,16383,33,13,512,
%U A293447 32767,7,30,15,130,65,65535,10,38,18,258,1024,131071,12,262143,2048,21,6,70,35,524287,129,514,23,1048575,9,2097151,4096,17,257,46,67,4194303,11,12
%N A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.
%C A293447 Original, equal definition: totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).
%H A293447 Antti Karttunen, <a href="/A293447/b293447.txt">Table of n, a(n) for n = 1..8192</a>
%H A293447 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A293447 Totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).
%F A293447 a(1) = 0, and for n > 1, a(n) = A005187(A087207(n)) + a(A003557(n)).
%F A293447 Other identities:
%F A293447 For all n >= 1, a(A293442(n)) = A046645(n).
%F A293447 For all n >= 2 and all k >= 0, a(n^k) = k*a(n).
%F A293447 For all n >= 1, a(n) >= A048675(n) >= A331740(n) >= A331591(n).
%o A293447 (PARI)
%o A293447 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; }; \\ This function from _Charles R Greathouse IV_
%o A293447 A293447(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2] * A005187(2^(primepi(f[k, 1])-1))); }
%o A293447 (Scheme)
%o A293447 (define (A293447 n) (cond ((= 1 n) 0) (else (+ (A005187 (A087207 n)) (A293447 (A003557 n))))))
%o A293447 ;; Alternatively:
%o A293447 (define (A293447 n) (if (= 1 n) 0 (+ (A005187 (A000079 (+ -1 (A061395 n)))) (A293447 (/ n (A006530 n))))))
%Y A293447 Cf. A000225, A000720, A003557, A005187, A087207.
%Y A293447 Cf. also A046645, A048675, A293442, A331591, A331740.
%K A293447 nonn
%O A293447 1,3
%A A293447 _Antti Karttunen_, Nov 09 2017
%E A293447 Definition simplified by _Antti Karttunen_, Feb 05 2020