A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

1, 1, 1, 3, 3, 1, 5, 9, 5, 3, 3, 3, 1, 5, 3, 27, 9, 5, 11, 9, 5, 3, 7, 9, 21, 1, 25, 15, 15, 3, 9, 81, 3, 9, 15, 15, 5, 11, 1, 27, 21, 5, 23, 9, 15, 7, 13, 27, 55, 21, 9, 3, 29, 25, 9, 45, 11, 15, 15, 9, 33, 9, 25, 243, 3, 3, 35, 27, 7, 15, 9, 45, 39, 5, 21, 33, 15, 1, 41, 81, 125, 21, 11, 15, 27, 23, 15, 27, 3, 15

Offset: 1

Author

Antti Karttunen, Dec 29 2020

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000010, A000265, A003961, A003972, A005117, A053575, A151800, A339903, A340075.

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A339904(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,my(q=nextprime(1+f[i,1])); A000265(q-1)*(q^(f[i,2]-1))));

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339903(n) = A000265(eulerphi(A003961(n)));

Formula

Multiplicative with a(p^e) = A000265(q-1) * q^(e-1), where q = A151800(p), the next prime larger than p.

a(n) = A000265(A003972(n)) = A053575(A003961(n)) = A000265(A000010(A003961(n))).

For all squarefree numbers k, a(k) = A339903(k).