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

%I #85 Jun 10 2022 13:31:00

%S 1711,1829,2077,2479,3071,3901,5029,6527,6757,7471,7967,8479,10397,

%T 10981,11581,14141,15167,15517,15871,16591,16957,17701,18079,18847,

%U 19631,20837,22927,23791,25567,26941,27877,28829,29797,30287,31279,31781,32287,35941,38117

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

%C The first two terms that are not semiprimes, and their prime factorizations, are:

%C a(62) = 2*185^2 + 29 = 68479 = 31*47*47,

%C a(63) = 2*187^2 + 29 = 69967 = 31*37*61.

%C --

%C 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 +----+-----------------------------------------------+

%C | p | Residues modulo p of 2*k^2 + 29 |

%C +----+-----------------------------------------------+

%C | 2 | 1 |

%C | 3 | 1, 2 |

%C | 5 | 1, 2, 4 |

%C | 7 | 1, 2, 3, 5 |

%C | 11 | 2, 3, 4, 6, 7, 9 |

%C | 13 | 1, 3, 5, 8, 9, 10, 11 |

%C | 17 | 3, 4, 8, 10, 11, 12, 13, 14, 16 |

%C | 19 | 1, 3, 4, 5, 6, 9, 10, 12, 13, 18 |

%C | 23 | 1, 6, 7, 8, 9, 10, 12, 14, 15, 18, 19, 22 |

%C +----+-----------------------------------------------+

%C Idea and table from _Jon E. Schoenfield_.

%C Example of explanation:

%C if k ~ 0 (mod 3) then k^2 ~ 0 (mod 3), so 2*k^2 + 29 ~ 29 (mod 3) ~ 2 (mod 3);

%C 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 --

%C A number of the form 2*k^2 + 29 has the prime 29 as a factor iff k ~ 0 (mod 29).

%H Michael S. Branicky, <a href="/A352949/b352949.txt">Table of n, a(n) for n = 1..10000</a>.

%H Rémi Guillaume, <a href="/A352949/a352949_1.txt">Examples of prime factorizations and prime factor distributions, and proofs</a>.

%F a(n) = 2*(A007642(n))^2 + 29.

%e a(5) = 3071 = 37*83 = 2*39^2 + 29 is composite and of the form 2*k^2 + 29.

%e a(62) = 68479 = 31*47^2 = 2*185^2 + 29 is composite and of the form 2*k^2 + 29.

%t Select[2*Range[150]^2 + 29, CompositeQ] (* _Amiram Eldar_, Apr 15 2022 *)

%o (Python)

%o from sympy import isprime

%o 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 Cf. A007642 for arguments k.

%Y Cf. 2*A353004^2 + 29 = A241554, which is a subsequence, for semiprimes.

%Y Cf. 2*A352800^2 + 29 = A007641 for primes.

%K nonn

%O 1,1

%A _Rémi Guillaume_, Apr 10 2022