A378749 -id:A378749 - cp's OEIS frontend

A378747 a(n) = A048673(n) - A001065(n).

1, 1, 2, 2, 3, 2, 5, 7, 9, 3, 6, 7, 8, 7, 9, 26, 9, 17, 11, 10, 17, 6, 14, 32, 19, 10, 50, 22, 15, 11, 18, 91, 18, 9, 26, 58, 20, 13, 26, 45, 21, 29, 23, 19, 55, 18, 26, 127, 53, 31, 27, 31, 29, 122, 29, 85, 35, 15, 30, 50, 33, 22, 97, 302, 41, 20, 35, 28, 46, 42, 36, 215, 39, 22, 74, 40, 53, 38, 41, 178, 273, 21

Offset: 1

Antti Karttunen, Dec 09 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

Cf. A000203, A001065, A003961, A048673, A337378 (where a(n) > n), A337379 (where a(n) <= n), A378748 (Möbius transform), A378749 (Dirichlet inverse).

Cf. also A286385, A349573.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378747(n) = (A048673(n)-(sigma(n)-n));

a(n) = n + (A003961(n)+1)/2 - A000203(n).

A378750 Dirichlet inverse of A377984, where A377984(n) = 2*sigma(n) - A003961(n).

Original entry on oeis.org

1, -3, -3, 4, -5, 9, -5, 0, 8, 15, -11, -8, -11, 15, 17, 8, -17, -24, -17, -16, 21, 33, -19, 0, 12, 33, 24, -8, -29, -51, -27, 24, 35, 51, 31, 56, -35, 51, 39, 0, -41, -63, -41, -40, -28, 57, -43, 0, 32, -36, 53, -32, -49, -72, 57, 0, 57, 87, -59, 48, -57, 81, -4, 88, 61, -105, -65, -64, 67, -93, -71, 0, -69, 105

Offset: 1

Antti Karttunen, Dec 09 2024

sign

Cf. A000203, A003961, A377984.

Cf. also A378749.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A377984(n) = (2*sigma(n) - A003961(n));
memoA378750 = Map();
A378750(n) = if(1==n,1,my(v); if(mapisdefined(memoA378750,n,&v), v, v = -sumdiv(n,d,if(dA377984(n/d)*A378750(d),0)); mapput(memoA378750,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA377984(n/d) * a(d).

cp's OEIS Frontend

A378747 a(n) = A048673(n) - A001065(n).

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