A161918 Numbers n such that the sum of the divisors minus the sum of the prime factors (counted with multiplicity) is equal to n+1.

6, 8, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 23 2009

Keywords

• easy
• nonn

Comments

Equals A006881 union {8}. - Franklin T. Adams-Watters, Jun 26 2009

Examples

			n=21: Sum_divisors (1,3,7,21) = 32; Sum_prime_factors (3,7) = 10 -> 32-10 = 22. n=55: Sum_divisors (1,5,11,55) = 72; Sum_prime_factors (5,11) = 16 -> 72-16 = 56.
		

Links

• Jayanta Basu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A161917, A006881, A151797, A030229.

Programs

Maple

with(numtheory); P:=proc(i) local b,c,j,s,n; for n from 2 by 1 to i do b:=(convert(ifactors(n),`+`)-1); c:=nops(b); j:=0; s:=0; for j from c by -1 to 1 do s:=s+convert(b[j],`*`); od; if n=sigma(n)-s-1 then print(n); fi; od; end: P(500);

Mathematica

Select[Range[203], DivisorSigma[1, #] - Total[Times @@@ FactorInteger[#]] == # + 1 &] (* Jayanta Basu, Aug 11 2013 *)

PARI

\\ from M. F. Hasler
isA161918(n)={ n+1 == sigma(n)-(n=factor(n))[,1]~*n[,2] }
for(n=1,500, isA161918(n)&print1(n","))

Extensions

Edited by N. J. A. Sloane, Jun 27 2009 incorporating suggestions from R. J. Mathar, M. F. Hasler, Benoit Jubin and others.