A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

0, 1, 1, 6, 1, 8, 1, 27, 10, 10, 1, 39, 1, 14, 12, 108, 1, 55, 1, 51, 16, 16, 1, 162, 14, 20, 75, 75, 1, 71, 1, 405, 18, 22, 18, 240, 1, 26, 22, 216, 1, 103, 1, 87, 95, 32, 1, 621, 22, 91, 24, 111, 1, 350, 20, 324, 28, 34, 1, 318, 1, 40, 135, 1458, 24, 119, 1, 123, 34, 131, 1, 945, 1, 44, 119, 147, 24, 151, 1, 837

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

Formula

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

Mathematica

f1[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := d[s[n]]; Array[a, 100] (* Amiram Eldar, Dec 05 2021 *)

Prog

(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); };
A349905(n) = A003415(A003961(n));

Crossrefs

Cf. A003415, A003961, A026424 (positions of odd terms), A028260 (of even terms), A066829 (parity of a(n)).

Cf. also A343221, A343222, A347964, A347965, A348937, A353571.

Cf. A358760, A358761, A358762, A358763 for indices of terms that of the form 4k+j, for j=0..3, and A358750, A358751, A358752, A358753 for their characteristic functions.

Sequence in context: A021622 A073228 A256853 * A258404 A244692 A248589

Adjacent sequences: A349902 A349903 A349904 * A349906 A349907 A349908

Keyword

nonn

Author

Antti Karttunen, Dec 05 2021

Status

approved