A038510 Composite numbers with smallest prime factor >= 7.

49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 343, 361, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 529, 533, 539, 551, 553, 559, 581, 583

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

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

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

Composite numbers not divisible by 2, 3 or 5. - Lekraj Beedassy, Jun 30 2004

Composite numbers k such that k^4 mod 30 = 1. - Gary Detlefs, Dec 09 2012

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

"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

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

References

• J. H. Silverman, A Friendly Introduction to Number Theory, 2nd Edn. "Appendix A: Factorization of Small Composite Integers", Prentice Hall NY 2001.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A Curious Cubic Identity and Self-similar Sums of Squares, 2006, pp. 73-74.

Crossrefs

Cf. A001511, A051064, A055457, A038509, A071904.

Cf. A070884, A038511.

Intersection of A002808 and A007775.

Programs

Maple

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

Mathematica

Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 7 &] (* T. D. Noe, Mar 16 2013 *)

PARI

is(n)=gcd(n,30)==1 && !isprime(n) \\ Charles R Greathouse IV, Dec 09 2012

Formula

a(n) ~ 3.75n. - Charles R Greathouse IV, Dec 09 2012

Extensions

Corrected by Ralf Stephan, Apr 04 2003