A378521 Möbius transform of A048673.

1, 1, 2, 3, 3, 4, 5, 9, 10, 6, 6, 12, 8, 10, 12, 27, 9, 20, 11, 18, 20, 12, 14, 36, 21, 16, 50, 30, 15, 24, 18, 81, 24, 18, 30, 60, 20, 22, 32, 54, 21, 40, 23, 36, 60, 28, 26, 108, 55, 42, 36, 48, 29, 100, 36, 90, 44, 30, 30, 72, 33, 36, 100, 243, 48, 48, 35, 54, 56, 60, 36, 180, 39, 40, 84, 66, 60, 64, 41, 162, 250

Offset: 1

Author

Antti Karttunen, Nov 30 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A000010, A003961, A003972, A008683, A048673, A055034, A072451, A349136.

Programs

PARI

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378521(n) = sumdiv(n,d,moebius(n/d)*A048673(d));

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378521(n) = if(1==n,n,eulerphi(A003961(n))/2);

Formula

a(n) = Sum_{d|n} A008683(n/d)*A048673(d).

a(n) = A072451(A048673(n)).

a(n) = A055034(A003961(n)) = A349136(A003961(n)).

For n > 1, a(n) = A003972(n)/2 = A000010(A003961(n))/2.