A133946 a(n) = Sum phi(k), where the sum is over the integers k which are the "non-isolated divisors" of 2n and phi(k) is the Euler totient function (phi(k) = A000010(k)). A positive divisor k of n is non-isolated if k-1 and/ or k+1 also divides n.

2, 2, 4, 2, 2, 6, 2, 2, 4, 8, 2, 6, 2, 2, 10, 2, 2, 6, 2, 8, 12, 2, 2, 6, 2, 2, 4, 12, 2, 12, 2, 2, 4, 2, 2, 16, 2, 2, 4, 8, 2, 14, 2, 2, 20, 2, 2, 6, 2, 8, 4, 2, 2, 6, 16, 12, 4, 2, 2, 12, 2, 2, 12, 2, 2, 20, 2, 2, 4, 8, 2, 16, 2, 2, 10, 2, 2, 22, 2, 8, 4, 2, 2, 18, 2, 2, 4, 2, 2, 22, 20, 2, 4, 2, 2, 6, 2

Offset

1,1

Comments

No odd integer has any non-isolated divisors.

a(n) = 2n - A133945(2n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..32768

Formula

(PARI) A133946(n) = { n = 2*n; sumdiv(n,d,if((d>1 && !(n%(d-1))) || !(n%(d+1)), eulerphi(d))); }; \\ Antti Karttunen, Jan 15 2025

Mathematica

Table[Plus @@ EulerPhi[Select[Divisors[2n], If[ # > 1, IntegerQ[2n/(# - 1)]] || IntegerQ[2n/(# + 1)] &]], {n, 1, 80}] (* Stefan Steinerberger, Oct 04 2007 *)

Crossrefs

Cf. A133945.

Sequence in context: A321320 A366808 A175462 * A286250 A278244 A278542

Adjacent sequences: A133943 A133944 A133945 * A133947 A133948 A133949

Keyword

nonn,changed

Author

Leroy Quet, Oct 03 2007

Extensions

More terms from Stefan Steinerberger, Oct 04 2007

Extended by Ray Chandler, May 28 2008

Status

approved