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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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. [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - N. J. A. Sloane, Mar 29 2018]

EXAMPLE

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

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]<sx do y:=CK[k, 1]; if x+y in Sx intersect CK[k, 2] then CL:=[op(CL), [x, y, x+y]] fi od od;

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 *)

CROSSREFS

Cf. A109798, A111592.

Sequence in context: A284174 A292982 A334972 * A129656 A334974 A048945

Adjacent sequences:  A109794 A109795 A109796 * A109798 A109799 A109800

KEYWORD

nonn

AUTHOR

Walter Kehowski, Aug 15 2005

STATUS

approved

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Last modified February 26 11:15 EST 2021. Contains 341631 sequences. (Running on oeis4.)