A316991 -id:A316991 - cp's OEIS frontend

A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

6, 10, 12, 18, 20, 21, 24, 30, 33, 35, 36, 39, 40, 42, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 78, 80, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 126, 129, 130, 132, 138, 140, 141, 144, 145, 147, 150

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003593 and A229829.
Analogous to A081062 and A105115 that apply to A120944(1)=6 and A120944(2)=10, respectively.
This sequence applies to term A120944(4)=15.

			6 is in the sequence since gcd(6, 15) = 3 and 6 does not divide 15^e with integer e >= 0.
2 and 4 are not in the sequence since they are coprime to 15.
3 and 5 are not in the sequence since they are divisors of 15.
9 is not in the sequence since 9 | 15^2.

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

Cf. A003593, A081062, A105115, A120944, A229829, A316991.

With[{nn = 150, k = 15}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A330136 Numbers m such that 1 < gcd(m, 6) < m and m does not divide 6^e for e >= 0.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Dec 02 2019

Numbers m that are neither 3-smooth nor reduced residues mod 6. Such numbers m have at least 1 prime factor p <= 3 and at least 1 prime factor q > 3.
Complement of the union of A003586 and A007310. Analogous to A105115 for A120944(2) = 10. This sequence applies to A120944(1) = 6 = A002110(1).
The only composite n in A024619 for which n < A096014(n) is 6. Let n be a composite that is not a prime power (i.e., in A024619), let p = lpf(n) = A020639(n), and let q = A053669(n) be the smallest prime that does not divide n. We observe that A096014(n) = A020639(n) * A053669(n) = pq. Such n with n < pq must minimize one factor while maximizing the other. The prime p is minimum when n is even, and q is greatest when n is the product p_k# of the smallest k primes, i.e., when n is in A002110. Alternatively, q is minimum when n is odd, however, n > 2p since n is the product of at least two distinct odd primes. Since p_k# greatly increases as k increments, while A053669(p_k#) = p_(k + 1), and observing that A096014(30) = 2 * 7 = 14, the only composite n in A024619 such that n < pq is 6.

			All m < 10 are not in the sequence since they either divide 6^e with integer e >= 0 or are coprime to 6.
10 is in the sequence since gcd(6, 10) = 2 and 10 does not divide 6^e with integer e >= 0.
11 is not in the sequence since 11 is coprime to 6.
12 is not in the sequence since 12 | 6^2.

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

Cf. A002110, A003586, A007310, A020639, A024619, A053669, A096014, A105115, A120944, A306999, A307589, A316991, A316992, A330137.

With[{nn = 123, k = 6}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

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

Original entry on oeis.org

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] &]]

cp's OEIS Frontend

A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A330136 Numbers m such that 1 < gcd(m, 6) < m and m does not divide 6^e for e >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica