A330136 -id:A330136 - cp's OEIS frontend

A330137 Numbers m such that 1 < gcd(m, 30) < m and m does not divide 30^e for e >= 0.

14, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 82, 84, 85, 86, 87, 88, 92, 93, 94, 95, 98, 99, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 122, 123, 124, 126, 129, 130, 132, 134

Offset: 1

Michael De Vlieger, Dec 02 2019

Numbers m that are neither 5-smooth nor reduced residues mod 30. Such numbers m have at least 1 prime factor p <= 5 and at least 1 prime factor q > 5.
Complement of the union of A007775 and A051037.
Analogous to A105115 for A120944(2) = 10. This sequence applies to the second primorial in A120944, i.e., 30 = A002110(2).

			14 is in the sequence since gcd(14, 30) = 2 and 14 does not divide 30^e with integer e >= 0.
15 is not in the sequence since 15 | 30.
16 is not in the sequence since 16 | 30^4.
17 is not in the sequence since 17 is coprime to 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A007775, A051037, A105115, A120944, A306999, A307589, A316991, A316992, A330136.

With[{nn = 135, k = 30}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A376158 Numbers k having two prime divisors p < q such that p! <= k <= q!.

Original entry on oeis.org

6, 10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 56, 57, 58, 60, 62, 63, 66, 68, 69, 70, 74, 75, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 129, 130, 132, 134, 136, 138, 140, 141, 142, 145

Offset: 1

Mike Jones, Sep 12 2024

nonn

Different from A330136.

			40 is in the list because 40 has at least 2 distinct prime divisors, and the smallest prime divisor of 40 is 2 and the largest prime divisor of 40 is 5, and 2! <= 40 <= 5! because 2! = 2 and 5! = 120.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A020639, A006530, A330136.

q:= n-> (s-> nops(s)>1 and min(s)!<=n and n<=max(s)!)(numtheory[factorset](n)):
select(q, [$2..150])[];  # Alois P. Heinz, Sep 20 2024

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && k >= p[[1]]! && k <= p[[-1]]!]; Select[Range[125], q] (* Amiram Eldar, Sep 20 2024 *)

cp's OEIS Frontend

A330137 Numbers m such that 1 < gcd(m, 30) < m and m does not divide 30^e for e >= 0.

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