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.

1, 2, 4, 5, 6, 10, 10, 15, 19, 14, 12, 29, 16, 24, 27, 49, 18, 54, 22, 39, 45, 26, 28, 91, 39, 36, 98, 67, 30, 74, 36, 163, 51, 38, 65, 165, 40, 48, 69, 121, 42, 124, 46, 69, 136, 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

Offset: 1

Author

Antti Karttunen, Nov 06 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A003961, A168036, A286385, A336853, A343224, A347964, A347965.

Programs

Mathematica

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 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348937(n) = (A003961(n) - A003415(n));

Formula

a(n) = A003961(n) - A003415(n).

a(n) = A336853(n) - A168036(n).

a(n) = A286385(n) + A343224(n).