A350735 Semiprimes p*q with p <= q such that Sum_{primes r <= p} (q mod r) = q.

143, 169, 209, 1943, 8413, 11773, 288727, 292421, 544987, 1519381, 1798397, 3245527, 3506509, 4528499, 7043693, 9682711, 10476493, 11670493, 12603709, 16051433, 21499519, 21916327, 64595353, 68086903, 75022813, 81430093, 90537803, 134473993, 136693819, 146316323

Offset

1,1

Links

Table of n, a(n) for n=1..30.

Example

a(3) = 209 is a term because 209 = 11*19 with Sum_{primes r <= 11} (19 mod r) = (19 mod 2) + (19 mod 3) + (19 mod 5) + (19 mod 7) + (19 mod 11) = 1+1+4+5+8 = 19.

Maple

filter:= proc(n) local p, q, r, i;
  if numtheory:-bigomega(n) <> 2 then return false fi;
  p, q:= (min, max)(numtheory:-factorset(n));
  q = add(q mod r, r = select(isprime, [2, seq(i, i=3..p, 2)]))
end proc:
select(filter, [seq(i, i=9..600000, 2)]);

Mathematica

seqQ[n_] := Module[{f = FactorInteger[n], p = 0, q}, If[f[[;; , 2]] == {1, 1}, p = f[[1, 1]]; q = f[[2, 1]]]; If[f[[;; , 2]] == {2}, p = q = f[[1, 1]]]; p > 0 && Sum[Mod[q, r], {r, Select[Range[p], PrimeQ]}] == q]; Select[Range[600000], seqQ] (* Amiram Eldar, Jan 13 2022 *)

Crossrefs

Sequence in context: A160781 A217141 A097435 * A073954 A227868 A353059

Adjacent sequences: A350732 A350733 A350734 * A350736 A350737 A350738

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Jan 12 2022

Extensions

a(10)-a(22) from Amiram Eldar, Jan 13 2022

a(23)-a(30) from Daniel Suteu, May 12 2022

Status

approved