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A109797
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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.
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2
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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
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1,1
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COMMENTS
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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
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LINKS
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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]
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EXAMPLE
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sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=42.
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A109798, A111592.
Sequence in context: A284174 A292982 A334972 * A129656 A334974 A048945
Adjacent sequences: A109794 A109795 A109796 * A109798 A109799 A109800
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KEYWORD
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nonn
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AUTHOR
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Walter Kehowski, Aug 15 2005
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STATUS
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approved
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