

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
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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. 293295.
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)=962=94 and sigma(52)2(2)=984=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: A068544 A284174 A292982 * A129656 A048945 A111398
Adjacent sequences: A109794 A109795 A109796 * A109798 A109799 A109800


KEYWORD

nonn


AUTHOR

Walter Kehowski, Aug 15 2005


STATUS

approved



