

A038510


Composite numbers with smallest prime factor >= 7.


2



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
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OFFSET

1,1


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
Composite numbers such that the denominator of (n2)*binomial(2n, 4)/binomial(n, 4) is n  3.  Gary Detlefs, May 19 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. 7374.  Daniel Forgues, Jan 30 2015, Feb 04 2015
For the eight congruences coprime to 30, we can use one byte to encode the "primality/nonprimality (unit or composite)" for each [30*n, 30*(n+1)[, n >= 0, closedopen 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 Selfsimilar Sums of Squares, 2006, pp. 7374.


FORMULA

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


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 *)


PROG

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


CROSSREFS

Cf. A001511, A051064, A055457, A038509, A071904.
Cf. A070884, A038511.
Sequence in context: A112074 A112057 A260571 * A063163 A103216 A036307
Adjacent sequences: A038507 A038508 A038509 * A038511 A038512 A038513


KEYWORD

nonn


AUTHOR

Jeff Burch


EXTENSIONS

Corrected by Ralf Stephan, Apr 04 2003


STATUS

approved



