A327823 Odd integers m such that every odd integer k with 1 < k < m and gcd(k,m) = 1 is prime.

1, 3, 5, 7, 9, 15, 21, 45, 105

Offset: 1

Author

Bernard Schott, Sep 26 2019

Keywords

• nonn
• full
• fini

Comments

Solomon W. Golomb and Kee-Wai Lau prove in AMM (see link) that the greatest odd integer with this property is 105.

This sequence is inspirated by the other one: integers q such that every integer k with 1 < k < q and gcd(k,q) = 1 is prime, with 2, 3, 4, 6, 8, 12, 18, 24, 30 in A048597 \ {1}.

The terms 1 and 3 are added after recommendations of Amiram Eldar and Michel Marcus.

Examples

			For m = 15 and 1 < k odd < 15, we have gcd(3,15) = 3, gcd(5,15) = 5, gcd(7,15) = 1, gcd(9,15) = 3, gcd(11,15) = 1, gcd(13,15) = 1. So, gcd(k,15) = 1 only if k is prime and 15 is a term.
For m = 63, we have gcd(25,63) = 1 with 25 no prime, so 63 is not a term.
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, number 105, page 118.

Links

• Solomon W. Golomb and Kee-Wai Lau, Problem E3137, American Mathematical Monthly, Vol. 94, No. 9, Nov. 1987, pp. 883-884.

Crossrefs

Cf. A048597.

Programs

Mathematica

aQ[n_] := OddQ[n] && AllTrue[Select[Range[3, n, 2], CoprimeQ[n, #] &], PrimeQ]; Select[Range[10^3], aQ] (* Amiram Eldar, Sep 27 2019 *)

PARI

isok(m) = {if (m % 2, forstep (k=3, m-1, 2, if ((gcd(k, m) == 1) && !isprime(k), return(0));); return(1););} \\ Michel Marcus, Sep 27 2019