A277794 Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.

4, 21, 85, 129, 201, 237, 324, 325, 517, 549, 669, 837, 865, 1081, 1137, 1161, 1165, 1309, 1389, 1677, 1765, 2169, 2233, 2304, 2305, 2469, 2709, 2737, 2761, 3265, 3297, 3745, 3961, 4161, 4285, 4693, 4705, 4741, 4989, 5061, 5221, 5349, 5817, 5949, 6249, 6457, 6517, 6685, 6789, 6813, 6853, 6921

Offset

1,1

Comments

Intersection of A037020 and A200981.

Numbers k such that A000005(A001065(k)) = A000005(A024816(k)) = 2 or A000005(A000203(k) - k) = A000005(A000217(k) - A000203(k)) = 2.

All terms are composite (A002808).

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sums of divisors

Example

21 is in the sequence because 21 has three proper divisors {1, 3, 7}, and therefore seventeen non-divisors {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, so the sum of proper divisors is 1 + 3 + 7 = 11 (which is prime) and the sum of non-divisors is 2 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 199 (which is also prime).
22 is not in the sequence because its three proper divisors {1, 2, 11} add up to 14, which is composite.

Maple

f:= proc(n) local t; t:= numtheory:-sigma(n) - n; isprime(t) and isprime(n*(n-1)/2 - t) end proc:
select(f, [$1..10^4]); # Robert Israel, Nov 10 2016

Mathematica

Select[Range[7000], DivisorSigma[0, #1 ((#1 + 1)/2) - DivisorSigma[1, #1]] == 2 && DivisorSigma[0, DivisorSigma[1, #1] - #1] == 2 & ]

Crossrefs

Cf. A000005, A000203, A000217, A002808, A037020, A200981.

Sequence in context: A163697 A134424 A348621 * A292126 A174797 A328345

Adjacent sequences: A277791 A277792 A277793 * A277795 A277796 A277797

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Oct 31 2016

Status

approved