cp's OEIS Frontend

A352949 Composite numbers of the form 2*k^2 + 29.

%I A352949 #85 Jun 10 2022 13:31:00
%S A352949 1711,1829,2077,2479,3071,3901,5029,6527,6757,7471,7967,8479,10397,
%T A352949 10981,11581,14141,15167,15517,15871,16591,16957,17701,18079,18847,
%U A352949 19631,20837,22927,23791,25567,26941,27877,28829,29797,30287,31279,31781,32287,35941,38117
%N A352949 Composite numbers of the form 2*k^2 + 29.
%C A352949 The first two terms that are not semiprimes, and their prime factorizations, are:
%C A352949   a(62) = 2*185^2 + 29 = 68479 = 31*47*47,
%C A352949   a(63) = 2*187^2 + 29 = 69967 = 31*37*61.
%C A352949 --
%C A352949 No number of the form 2^k*2 + 29 has any prime factor < 29, as can be proved by showing that 2*k^2 + 29 (mod p) takes only nonzero values for all primes p < 29:
%C A352949   +----+-----------------------------------------------+
%C A352949   |  p | Residues modulo p of 2*k^2 + 29               |
%C A352949   +----+-----------------------------------------------+
%C A352949   |  2 | 1                                             |
%C A352949   |  3 | 1,  2                                         |
%C A352949   |  5 | 1,  2,  4                                     |
%C A352949   |  7 | 1,  2,  3,  5                                 |
%C A352949   | 11 | 2,  3,  4,  6,  7,  9                         |
%C A352949   | 13 | 1,  3,  5,  8,  9, 10, 11                     |
%C A352949   | 17 | 3,  4,  8, 10, 11, 12, 13, 14, 16             |
%C A352949   | 19 | 1,  3,  4,  5,  6,  9, 10, 12, 13, 18         |
%C A352949   | 23 | 1,  6,  7,  8,  9, 10, 12, 14, 15, 18, 19, 22 |
%C A352949   +----+-----------------------------------------------+
%C A352949 Idea and table from _Jon E. Schoenfield_.
%C A352949 Example of explanation:
%C A352949 if k ~ 0 (mod 3) then k^2 ~ 0 (mod 3), so 2*k^2 + 29 ~ 29 (mod 3) ~ 2 (mod 3);
%C A352949 if k ~ 1 (mod 3) or if k ~ 2 (mod 3) ~ -1 (mod 3), then k^2 ~ 1 (mod 3), so 2*k^2 + 29 ~ 31 (mod 3) ~ 1 (mod 3).
%C A352949 --
%C A352949 A number of the form 2*k^2 + 29 has the prime 29 as a factor iff k ~ 0 (mod 29).
%H A352949 Michael S. Branicky, <a href="/A352949/b352949.txt">Table of n, a(n) for n = 1..10000</a>.
%H A352949 Rémi Guillaume, <a href="/A352949/a352949_1.txt">Examples of prime factorizations and prime factor distributions, and proofs</a>.
%F A352949 a(n) = 2*(A007642(n))^2 + 29.
%e A352949 a(5) = 3071 = 37*83 = 2*39^2 + 29 is composite and of the form 2*k^2 + 29.
%e A352949 a(62) = 68479 = 31*47^2 = 2*185^2 + 29 is composite and of the form 2*k^2 + 29.
%t A352949 Select[2*Range[150]^2 + 29, CompositeQ] (* _Amiram Eldar_, Apr 15 2022 *)
%o A352949 (Python)
%o A352949 from sympy import isprime
%o A352949 print([m for m in (2*k**2+29 for k in range(140)) if not isprime(m)]) # _Michael S. Branicky_, Apr 15 2022
%Y A352949 Cf. A007642 for arguments k.
%Y A352949 Cf. 2*A353004^2 + 29 = A241554, which is a subsequence, for semiprimes.
%Y A352949 Cf. 2*A352800^2 + 29 = A007641 for primes.
%K A352949 nonn
%O A352949 1,1
%A A352949 _Rémi Guillaume_, Apr 10 2022