A048597 -id:A048597 - cp's OEIS frontend

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

Bernard Schott, Sep 26 2019

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.

			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.

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

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

Cf. A048597.

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

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

A332839 Irregular triangle whose n-th row lists the integers x such that the number of nonprimes (i.e., 1 and composites) in the reduced residue set (RSS(x)) of x equals n, or 0 if there are no such x.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 18, 24, 30, 5, 10, 14, 20, 42, 60, 7, 9, 16, 36, 48, 90, 15, 22, 54, 84, 26, 28, 66, 120, 11, 21, 32, 40, 72, 78, 210, 13, 34, 50, 38, 44, 70, 150, 102, 114, 126, 17, 27, 46, 56, 96, 108, 180, 19, 33, 52, 132, 25, 45, 80, 168, 0, 23, 39, 58, 62, 110, 138

Offset: 1

Michel Marcus, Feb 26 2020

			Triangle begins:
1, 2, 3, 4, 6, 8, 12, 18, 24, 30;
5, 10, 14, 20, 42, 60;
7, 9, 16, 36, 48, 90;
15, 22, 54, 84;
26, 28, 66, 120;
11, 21, 32, 40, 72, 78, 210;
...

Abhijit A J, A. Satyanarayana Reddy, Number of non-primes in the set of units modulo n, arXiv:1907.09908 [math.GM], 2019. See p. 3.

Cf. A048597 (1st row), A072022 (least x), A074915 (largest x), A076366 (row lengths).

t = Select[ Table[{ EulerPhi[n] - PrimePi[n] + PrimeNu[n], n}, {n, 2000}], #[[1]] <= 100 &]; c = Complement[Range[100], First /@ t]; Last /@ (Sort@ Join[ Transpose[{c, 0 c}], t]) (* Giovanni Resta, Feb 26 2020 *)

cp's OEIS Frontend

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

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