A255384 a(n) = sopfr(n)^2 - 2n, where sopfr(n) is the sum of the prime factors of n with multiplicity.

-2, 0, 3, 8, 15, 13, 35, 20, 18, 29, 99, 25, 143, 53, 34, 32, 255, 28, 323, 41, 58, 125, 483, 33, 50, 173, 27, 65, 783, 40, 899, 36, 130, 293, 74, 28, 1295, 365, 178, 41, 1599, 60, 1763, 137, 31, 533, 2115, 25, 98, 44, 298, 185, 2703, 13, 146, 57, 370, 845

Offset: 1

Author

Wesley Ivan Hurt, May 05 2015

Keywords

• sign

Comments

If n is prime, then a(n) = n*(n-2). If n is semiprime, then a(n) gives the sum of the squares of the prime factors of n (with multiplicity).

a(n) is negative for n = 1, 81, 90, 96, 100, 108, 120, 125, 126, 128, 135, .... - Charles R Greathouse IV, May 06 2015

Examples

			a(6) = sopfr(6)^2 - 2(6) = (2+3)^2 - 12 = 25 - 12 = 13.
a(8) = sopfr(8)^2 - 2(8) = (2+2+2)^2 - 16 = 36 - 16 = 20.
		

Crossrefs

Cf. A074373 (sopfr^2), A001414 (sopfr).

Programs

Mathematica

sopfr[n_] := Plus @@ Times @@@ FactorInteger@n; f[1] = 0; Table[sopfr[n]^2 - 2 n, {n, 100}]

PARI

sopfr(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]*f[i,2])
a(n)=sopfr(n)^2 - 2*n \\ Charles R Greathouse IV, May 06 2015

Formula

a(n) = A074373(n) - A005843(n).