A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

0, 0, 0, 1, -1, 2, -1, 6, 4, 1, -4, 11, -4, 3, 3, 25, -7, 20, -7, 12, 7, -2, -8, 44, 0, 0, 36, 22, -13, 23, -12, 90, 0, -5, 4, 77, -16, -3, 4, 55, -19, 41, -19, 15, 43, -2, -20, 155, 12, 24, -3, 25, -23, 134, -9, 93, 1, -11, -28, 98, -27, -6, 75, 301, -5, 32, -31, 18, 4, 46, -34, 266, -33, -12, 48, 28, -5, 50, -37

Offset: 1

Author

Antti Karttunen, Nov 23 2021

Keywords

• sign
• look

Links

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

Crossrefs

Cf. A003961, A048673, A064216, A349571.

Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

PARI

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A349573(n) = (A048673(n)-n);

Formula

a(n) = A048673(n) - n.

a(n) = Sum_{d|n, dA349571(n/d).