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%I #24 Mar 21 2021 11:57:39
%S 1,2,4,6,8,10,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,70,
%T 72,78,80,84,88,90,96,100,104,108,112,120,126,128,132,140,144,150,156,
%U 160,162,168,176,180,192,196,198,200,204,208,210,216,220,224,228,234,240
%N Numbers k such that every prime up to sigma(k) is a sum of divisors of k.
%C Sequence is infinite as sigma (2^n) = 2^(n+1)-1 and a(2^n) = pi(2^(n+1)-1).
%C Does this sequence include any non-members of A005153 other than 10, 70 and 836? - _Franklin T. Adams-Watters_, Apr 28 2006
%C The answer to the previous comment is yes, this sequence has many terms that are not in A005153. See A174434. - _T. D. Noe_, Mar 19 2010
%H Amiram Eldar, <a href="/A093891/b093891.txt">Table of n, a(n) for n = 1..10000</a>
%e 4 is a member as sigma(4) = 7 and all the primes up to 7 are a partial sum of divisors of 4, since divisors of 4 are 1, 2 and 4 and because primes arising are 2, 3 = 1+2, 5 = 1+4 and 7 = 1+2+4.
%t Select[Range[240], SubsetQ[Total /@ Rest@ Subsets@ Divisors[#], Prime@ Range@ PrimePi@ DivisorSigma[1, #]] &] (* _Michael De Vlieger_, Mar 19 2021 *)
%o (PARI) isok(m) = {my(d=divisors(m), vp = primes(primepi(sigma(m)))); for (i=1, 2^#d - 1, my(b = Vecrev(binary(i)), x = sum(k=1, #b, b[k]*d[k])); if (vecsearch(vp, x), vp = setminus(vp, Set(x))); if (#vp == 0, return (1)););} \\ _Michel Marcus_, Mar 19 2021
%Y Cf. A093890, A093892.
%Y Cf. A005153.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Apr 23 2004
%E More terms from _Franklin T. Adams-Watters_, Apr 28 2006
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