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 A278921 Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1 and (q-p)^(q+1) + 1. 2
 10, 15, 65, 221, 493, 671, 1147, 1219, 3439, 5069, 12209, 14893, 20737, 24503, 30083, 49813, 61937, 77507, 91277, 97297, 100337, 102719, 109283, 109783, 113521, 132427, 144301, 178991, 204851, 244523, 245041, 246559, 257149, 258749, 312167, 339497, 397219, 433091, 434617, 461893, 465763 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS q is always a Pythagorean prime (A002144). Semiprimes of the form p*q where p < q such that q divides p^(q+1) + k and (q-p)^(q+1) + k: k = 1: (this sequence); k = 2: 6, 33, 119, 247, 451, ... k = 3: 14, 35, 91, 341, ... k = 4: 39, 145, 371, ... For every positive odd number q (whether prime or not), every integer p in 0..q, and every integer k, if q divides p^(q+1) + k, then it necessarily follows that q also divides (q-p)^(q+1) + k; thus, this sequence could be more simply defined as "Semiprimes of the form p*q where p < q such that q divides p^(q+1) + 1." - Jon E. Schoenfield, Dec 07 2016 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 MATHEMATICA Take[#, 41] &@ Union@ Flatten@ Table[Function[q, q Select[Prime@ Range@ n, Function[p, And[Divisible[p^(q + 1) + 1, q], Divisible[(q - p)^(q + 1) + 1, q]]]]]@ Prime@ n, {n, 600}] (* Michael De Vlieger, Dec 02 2016 *) PROG (PARI) list(lim)=my(v=List()); forprime(q=5, lim\2, if(q%4>2, next); forprime(p=2, min(lim\q, q-2), if(Mod(p, q)^(q+1)==-1 && Mod(q-p, q)^(q+1)==-1, listput(v, p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 02 2016 CROSSREFS Cf. A001358, A002144, A006881, A279024. Sequence in context: A092192 A119039 A047189 * A035407 A020139 A056522 Adjacent sequences: A278918 A278919 A278920 * A278922 A278923 A278924 KEYWORD nonn AUTHOR Juri-Stepan Gerasimov, Dec 01 2016 STATUS approved

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Last modified March 26 22:49 EDT 2023. Contains 361553 sequences. (Running on oeis4.)