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A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

%I A348937 #17 Dec 06 2021 11:03:01
%S A348937 1,2,4,5,6,10,10,15,19,14,12,29,16,24,27,49,18,54,22,39,45,26,28,91,
%T A348937 39,36,98,67,30,74,36,163,51,38,65,165,40,48,69,121,42,124,46,69,136,
%U A348937 62,52,293,107,102,75,97,58,294,75,205,93,62,60,223,66,78,224,537,101,134,70,99,119,172,72,519,78,84,190,127
%N A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.
%H A348937 Antti Karttunen, <a href="/A348937/b348937.txt">Table of n, a(n) for n = 1..20000</a>
%H A348937 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A348937 a(n) = A003961(n) - A003415(n).
%F A348937 a(n) = A336853(n) - A168036(n).
%F A348937 a(n) = A286385(n) + A343224(n).
%t A348937 f1[p_, e_] := e/p; f2[p_, e_] := NextPrime[p]^e; a[n_] := Times @@ f2 @@@ (f = FactorInteger[n]) - n * Plus @@ f1 @@@ f; Array[a, 100] (* _Amiram Eldar_, Nov 06 2021 *)
%o A348937 (PARI)
%o A348937 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A348937 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A348937 A348937(n) = (A003961(n) - A003415(n));
%Y A348937 Cf. A003415, A003961, A168036, A286385, A336853, A343224, A347964, A347965.
%K A348937 nonn
%O A348937 1,2
%A A348937 _Antti Karttunen_, Nov 06 2021