A066191 Numbers k such that the sum of the odd aliquot parts of k divides k.
2, 3, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 23, 24, 29, 31, 32, 37, 41, 43, 47, 48, 53, 56, 59, 61, 64, 67, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 120, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193, 197, 199, 211, 223
Offset: 1
Examples
12 is in the sequence because the odd aliquot parts of 12 are {1,3} and their sum divides 12.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Programs
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Maple
with(numtheory):soa:=proc(n) local div,s,j: div:=convert(divisors(n),list): s:=0: for j from 1 to nops(div)-1 do if div[j] mod 2=1 then s:=s+div[j] else s:=s: fi: od: end: p:=proc(n) if type(n/soa(n),integer)=true then n else fi end: seq(p(n),n=1..240); # Emeric Deutsch, Feb 26 2005
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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[ IntegerQ[ n/od ], Print[ n ] ], {n, 2, 200} ] Select[Range[2, 225], Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* Amiram Eldar, Apr 27 2025 *) -
PARI
{ n=0; for (m=2, 10^9, 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("b066191.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010 -
PARI
isok(n) = !(n % sumdiv(n, d, d*(d%2)*(d!=n))); \\ Michel Marcus, Apr 06 2015
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PARI
isok(k) = if(k == 1, 0, !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ Amiram Eldar, Apr 27 2025
Extensions
More terms from Emeric Deutsch, Feb 26 2005