cp's OEIS Frontend

A358704 Numbers m such that the sum of the prime divisors and the sum of the nonprime divisors of m^2+1 are both prime.

%I A358704 #9 Dec 19 2022 09:45:53
%S A358704 3,9,172,309,327,392,473,483,557,578,633,657,693,699,747,767,819,820,
%T A358704 829,909,911,1007,1013,1028,1030,1057,1084,1141,1157,1186,1252,1308,
%U A358704 1311,1382,1577,1585,1620,1682,1721,1722,1727,1749,1841,1849,1874,1972,2019,2134
%N A358704 Numbers m such that the sum of the prime divisors and the sum of the nonprime divisors of m^2+1 are both prime.
%C A358704 The primes of the sequence are 3, 557, 829, 911, 1013, 1721, ...
%C A358704 The corresponding pairs (p, q) = (sum of prime divisors, sum of nonprime divisors) are (7, 11), (43, 83), (163, 36293), ...
%C A358704 There is a subsequence {b(n)} = {3, 9, 309, 699, 819, ...} such that q/p < 2 (see the following table).
%C A358704 Conjecture: when b(n) tends to infinity, q/p tends to 2.
%C A358704 +------+----------+----------+-------------+
%C A358704 | b(n) |        p |        q |      q/p    |
%C A358704 +------+----------+----------+-------------+
%C A358704 |    3 |        7 |       11 | 1.571428571 |
%C A358704 |    9 |       43 |       83 | 1.930232558 |
%C A358704 |  309 |    47743 |    95483 | 1.999937164 |
%C A358704 |  699 |   244303 |   488603 | 1.999987720 |
%C A358704 |  819 |   335383 |   670763 | 1.999991055 |
%C A358704 |  909 |   413143 |   826283 | 1.999992739 |
%C A358704 | 1311 |   859363 |  1718723 | 1.999996509 |
%C A358704 | 1749 |  1529503 |  3059003 | 1.999998039 |
%C A358704 | 3201 |  5123203 | 10246403 | 1.999999414 |
%C A358704 | 4809 | 11563243 | 23126483 | 1.999999741 |
%C A358704 ............................................
%e A358704 3 is in the sequence because the divisors of 3^2 + 1 = 10 are {1, 2, 5, 10} and 1 + 10 = 11 and 2 + 5 = 7 are prime numbers.
%t A358704 f[n_]:=Plus@@Select[Divisors[n^2+1], !PrimeQ[#]&]; g[n_]:=Plus@@First/@FactorInteger[n^2+1]; Select[Range[2200], PrimeQ[f[#]&&PrimeQ[g[#]]]&]
%o A358704 (PARI) isok(m) = my(f=factor(m^2+1), sp=vecsum(f[, 1])); isprime(sp) && isprime(sigma(f)-sp); \\ _Michel Marcus_, Nov 28 2022
%Y A358704 Cf. A193462, A194039, A194594.
%K A358704 nonn
%O A358704 1,1
%A A358704 _Michel Lagneau_, Nov 27 2022