A153326 Composite numbers k such that k+1+d is prime for all nontrivial divisors d which divide k.
4, 8, 9, 25, 27, 35, 39, 55, 65, 119, 125, 185, 203, 219, 235, 237, 289, 305, 319, 341, 415, 417, 437, 515, 535, 597, 649, 655, 671, 685, 749, 755, 905, 935, 959, 979, 989, 1003, 1043, 1079, 1111, 1119, 1165, 1227, 1247, 1285, 1299, 1315, 1343, 1355, 1465, 1469, 1565, 1649, 1681, 1735, 1739, 1829
Offset: 1
Examples
For k = 8, the nontrivial divisors are 2 and 4 and (8+1) + 2 = 11 and (8+1) + 4 = 13 are both primes. For 35 the nontrivial divisors are 5 and 7. With (35+1) + 5 = 41 and (35+1) + 7 = 43, both sums are primes.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
q[k_] := CompositeQ[k] && AllTrue[Divisors[k][[2 ;; -2]], PrimeQ[k + # + 1] &]; Select[Range[2000], q] (* Amiram Eldar, Aug 05 2024 *)
Formula
{k: k+1+d in A000040 for all 1 < d < k with d|k}.
Extensions
Added 4, replaced 121 by 125, extended, simplified definition, added non-semiprime examples. R. J. Mathar, Jan 06 2009
Comments