A140826 Arithmetic nondivisor means.

3, 4, 5, 7, 11, 13, 17, 18, 19, 20, 23, 24, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Author

Ctibor O. Zizka, Jul 17 2008

Keywords

• easy
• nonn

Comments

Numbers n such that A024816(n)/(n-A000005(n)) is an integer.

Numbers n such that A231167(n) = 0. - Jaroslav Krizek, Nov 07 2013

Union of odd primes (A065091) and composites from A230605 (4, 18, 20, 24, 432, 588...). - Jaroslav Krizek, Nov 07 2013

Examples

			n=18: numbers less than n which do not divide n are 4,5,7,8,10,11,12,13,14,15,16,17.
antisigma_1(18) = 4+5+7+8+10+11+12+13+14+15+16+17 = 132.
antisigma_0(18) = 12.
132/12 = 11 which is an integer so n=18 belongs to the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A003601, A024816, A049820, A000203.

Programs

Maple

A024816 := proc(n) n*(n+1)/2-numtheory[sigma](n) ; end:
isA140826 := proc(n) if A024816(n) mod ( n-A000005(n)) = 0 then true; else false; fi; end:
for n from 3 to 400 do if isA140826(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Dec 13 2008

Mathematica

Select[Range[3, 300], IntegerQ[(#^2 + # - 2 DivisorSigma[1, #])/(2# - 2 DivisorSigma[0, #])]&] (* Jean-François Alcover, May 11 2023 *)

PARI

isok(n) = (nnd = n - numdiv(n)) && !((n*(n+1)/2-sigma(n)) % nnd); \\ Michel Marcus, Nov 09 2013

Formula

Numbers n such that (n*n+n-2*A000203(n))/(2*n-2*A000005(n)) is an integer.

Extensions

Inserted 20 and extended by R. J. Mathar, Dec 13 2008