A153326 - OEIS

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

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OFFSET

1,1

COMMENTS

4 and 8 are the only even numbers.

Numbers in the sequence which are not semiprimes: 8, 27, 125, 935, 1859, 2849, etc. - R. J. Mathar, Jan 06 2009

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

{k: k+1+d in A000040 for all 1 < d < k with d|k}.

EXAMPLE

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.

MATHEMATICA

q[k_] := CompositeQ[k] && AllTrue[Divisors[k][[2 ;; -2]], PrimeQ[k + # + 1] &]; Select[Range[2000], q] (* Amiram Eldar, Aug 05 2024 *)

CROSSREFS

Cf. A000040, A120806.

Sequence in context: A053810 A076702 A051761 * A368959 A338325 A168363

Adjacent sequences: A153323 A153324 A153325 * A153327 A153328 A153329

KEYWORD

easy,nonn

AUTHOR

J. M. Bergot, Dec 23 2008

EXTENSIONS

Added 4, replaced 121 by 125, extended, simplified definition, added non-semiprime examples. R. J. Mathar, Jan 06 2009

STATUS

approved