A066192 Composite numbers k such that the sum of the odd aliquot parts of k divides k.

4, 8, 12, 16, 24, 32, 48, 56, 64, 96, 112, 120, 128, 192, 224, 240, 256, 384, 448, 480, 512, 528, 672, 768, 896, 960, 992, 1024, 1056, 1344, 1456, 1536, 1792, 1920, 1984, 2048, 2112, 2160, 2208, 2688, 2912, 3072, 3584, 3840, 3968, 4096, 4224, 4320, 4416

Offset

1,1

Comments

From Amiram Eldar, Apr 27 2025: (Start)

If there is an odd term in this sequence it must be an odd perfect number (A000396). If k is an odd term then d = sigma(k)-k divides k. If d < k then sigma(k) = k + d with d being an aliquot divisor of k which is possible only if k is prime and d = 1. Therefore, d = k and k must be an odd perfect number.

This sequence is infinite because if k is a term then 2*k is also a term. The primitive terms are in A383428. (End)

Links

Amiram Eldar, Table of n, a(n) for n = 1..4905 (terms below 10^11; terms 1..1000 from Harry J. Smith)

Index entries for sequences where any odd perfect numbers must occur.

Mathematica

Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; od = 1; k = 2; While[ k <= l, If[ OddQ[ d[ [ k ] ] ], od = od + d[ [ k ] ] ]; k++ ]; If[ !PrimeQ[ n ] && IntegerQ[ n/od ], Print[ n ] ], {n, 2, 10^4} ]
Select[Range[4500], CompositeQ[#] && Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* Amiram Eldar, Apr 27 2025 *)

Prog

(PARI) { n=0; for (m=4, 10^9, if (isprime(m), next); d=divisors(m); s=1; for (i=2, numdiv(m) - 1, if (d[i]%2, s += d[i])); if (s > 0 && m%s == 0, write("b066192.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010
(PARI) isok(k) = if(k == 1, 0, !isprime(k) && !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ Amiram Eldar, Apr 27 2025

Crossrefs

Intersection of A002808 and A066191.

Cf. A000396, A091570, A383428.

Sequence in context: A212502 A071385 A329883 * A097981 A330972 A033833

Adjacent sequences: A066189 A066190 A066191 * A066193 A066194 A066195

Keyword

nonn

Author

Robert G. Wilson v, Dec 15 2001

Status

approved