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

Author

Robert G. Wilson v, Dec 15 2001

Keywords

• nonn

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.

Crossrefs

Intersection of A002808 and A066191.

Cf. A000396, A091570, A383428.

Programs

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 *)

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