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%I #70 Oct 22 2021 09:37:46
%S 49,77,91,119,121,133,143,161,169,187,203,209,217,221,247,253,259,287,
%T 289,299,301,319,323,329,341,343,361,371,377,391,403,407,413,427,437,
%U 451,469,473,481,493,497,511,517,527,529,533,539,551,553,559,581,583
%N Composite numbers with smallest prime factor >= 7.
%C Let A = set of numbers of form 6n + 1, B = numbers of form 6n - 1. Eliminating numbers of form 25 + 30s from A and those of form 35 + 30s from B we obtain sets A* and B*. Removing all terms of the sequence from the union of A* and B*, only prime numbers remain. - Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002
%C Divide n by a*b*c where a = 2^(A001511(n)-1), b = 3^(A051064(n)-1) and c = 5^(A055457(n) -1). Then the resulting sequence includes only primes and a(n). - _Alford Arnold_, Sep 08 2003
%C Composite numbers not divisible by 2, 3 or 5. - _Lekraj Beedassy_, Jun 30 2004
%C Composite numbers k such that k^4 mod 30 = 1. - _Gary Detlefs_, Dec 09 2012
%C Composite numbers congruent to 1, 7, 11, 13, -13, -11, -7, -1 (mod 30). Since asymptotically, 100% of integers are composite, we have a(n)/n ~ 30/phi(30) = 30/8 = 3.75. - _Daniel Forgues_, Mar 16 2013
%C "John [Conway] recommends the more refined partition [of the positive numbers]: 1, prime, trivially composite, or nontrivially composite. Here, a composite integer is trivially composite if it is divisible by 2, 3, or 5." See link to (van der Poorten, Thomsen, and Wiebe; 2006) pp. 73-74. - _Daniel Forgues_, Jan 30 2015, Feb 04 2015
%C For the eight congruences coprime to 30, we can use one byte to encode the "primality/non-primality (unit or composite)" for each [30*n, 30*(n+1)[, n >= 0, closed-open interval, either as little endian binary sequence {01111111, 11111011, 11110111, 01111110, ...}, or as big endian binary sequence {11111110, 11011111, 11101111, 01111110, ...}, which we may then express in base 10. - _Daniel Forgues_, Feb 05 2015
%D J. H. Silverman, A Friendly Introduction to Number Theory, 2nd Edn. "Appendix A: Factorization of Small Composite Integers", Prentice Hall NY 2001.
%H T. D. Noe, <a href="/A038510/b038510.txt">Table of n, a(n) for n = 1..10000</a>
%H Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, <a href="https://www.researchgate.net/profile/Michael_Kleber/publication/225591088_A_curious_cubic_identity_and_self-similar_sums_of_squares/links/55c635f308aeca747d633576.pdf">A Curious Cubic Identity and Self-similar Sums of Squares</a>, 2006, pp. 73-74.
%F a(n) ~ 3.75n. - _Charles R Greathouse IV_, Dec 09 2012
%p for n from 1 to 583 do if n^4 mod 30 = 1 and not isprime(n) then print(n)fi od; # _Gary Detlefs_, Dec 09 2012
%t Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 7 &] (* _T. D. Noe_, Mar 16 2013 *)
%o (PARI) is(n)=gcd(n,30)==1 && !isprime(n) \\ _Charles R Greathouse IV_, Dec 09 2012
%Y Cf. A001511, A051064, A055457, A038509, A071904.
%Y Cf. A070884, A038511.
%Y Intersection of A002808 and A007775.
%K nonn
%O 1,1
%A _Jeff Burch_
%E Corrected by _Ralf Stephan_, Apr 04 2003
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