A348937 - OEIS

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

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OFFSET

1,2

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

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

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

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 *)

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); };

A348937(n) = (A003961(n) - A003415(n));

CROSSREFS

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

Sequence in context: A077312 A325558 A140779 * A117890 A108853 A257085

Adjacent sequences: A348934 A348935 A348936 * A348938 A348939 A348940

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 06 2021

STATUS

approved