A109797 First of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.

24, 30, 40, 42, 48, 60, 80, 80, 96, 102, 126, 140, 140, 156, 156, 156, 174, 180, 180, 198, 216, 224, 224, 264, 276, 280, 294, 294, 300, 320, 340, 372, 380, 384, 440, 440, 468, 500, 504, 510, 528, 560, 582, 608, 616, 642, 680, 684, 690, 690, 696, 702, 736, 750

Offset: 1

Author

Walter Kehowski, Aug 15 2005

Keywords

• nonn

Comments

Compatible numbers were introduced by Sachs in analogy to amicable numbers, as admirable numbers are analogous to perfect numbers. Some terms have more than one counterpart (A109798), like 80 (two counterparts: 102 and 104) or 156 (3 counterparts: 210, 230 and 234). - Amiram Eldar, Oct 26 2019

Examples

			sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=42.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
• T. Trotter, Admirable Numbers.

Crossrefs

Cf. A109798, A111592.

Programs

Maple

L:=remove(proc(z) isprime(z) end, [$1..10000]): S:=proc(n) map(proc(z) sigma(n) -2*z end, divisors(n) minus {n}) end; CK:=map(proc(z) [z,S(z)] end, L): CL:=[]: for j from 1 to nops(CK)-1 do x:=CK[j,1]; sx:=sigma(x); Sx:=CK[j,2]; for k from j+1 to nops(CK) while CK[k,1]
				

Mathematica

seq = {}; Do[d = Most[Divisors[n]]; s = Total[d]; Do[m = s - 2*d[[k]]; If[m <= 0 || m <= n, Continue[]]; delta = DivisorSigma[1, m] - m - n; If[delta > 0 && EvenQ[delta] && delta/2 < m && Divisible[m, delta/2], AppendTo[seq, n]], {k, Length[d], 1, -1}], {n, 1, 750}]; seq (* Amiram Eldar, Oct 26 2019 *)