A086469 Sum of the distinct (smallest) prime signature divisors of n. In case of two or more divisors with the same prime signature the smallest is considered to evaluate the sum. Let this function be defined as psigma(n).

1, 3, 4, 7, 6, 9, 8, 15, 13, 13, 12, 25, 14, 17, 19, 31, 18, 36, 20, 37, 25, 25, 24, 57, 31, 29, 40, 49, 30, 39, 32, 63, 37, 37, 41, 61, 38, 41, 43, 85, 42, 51, 44, 73, 73, 49, 48, 121, 57, 88, 55, 85, 54, 117, 61, 113, 61, 61, 60, 115, 62, 65, 97, 127, 71, 75, 68, 109, 73, 83, 72

Offset

1,2

Comments

Define n as a 'psigma perfect number' if psigma(n) = 2n. 18 is a psigma perfect number. The p sigma divisors are 1,2,6,9 and 18 and the sum = 36. Conjecture: 18 is the only psigma perfect number.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

a(30) = 1 + 2 + 6 + 30 = 39. The divisors 3, 5, 10 and 15 are not considered for the sum as 3 and 5 have the same prime signature as 2 and also 10 and 15 have the same prime signature as 6.

Mathematica

a[n_] := Module[{d = Rest[Divisors[n]]}, 1 + Total@DeleteDuplicatesBy[{#, Sort[FactorInteger[#][[;; , 2]]]} & /@ d, Last][[;; , 1]]]; Array[a, 71] (* Amiram Eldar, Jul 20 2019 *)

Crossrefs

Cf. A086470.

Sequence in context: A351923 A341944 A332994 * A087030 A175187 A332993

Adjacent sequences: A086466 A086467 A086468 * A086470 A086471 A086472

Keyword

nonn

Author

Amarnath Murthy, Jul 21 2003

Extensions

More terms from David Wasserman, Mar 07 2005

Status

approved