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User: Jan Kostanjevec

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A380979 Composites that cause a witness to be added to a set of Fermat witnesses: a(n) is the smallest composite number that is not guaranteed composite using Fermat's Little Theorem by the witness A380978(i) for any i < n.

Original entry on oeis.org

4, 341, 1105, 1729, 29341, 75361, 162401, 252601, 294409, 334153, 399001, 1152271, 1615681, 2508013, 3581761, 3828001, 6189121, 6733693, 10024561, 10267951, 14469841, 17098369, 17236801, 19384289, 23382529, 29111881, 34657141, 53711113, 64377991, 79411201, 79624621
Offset: 1

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Author

Jan Kostanjevec, Feb 10 2025

Keywords

Comments

A380978(n) is defined as the minimal Fermat witness that guarantees the compositeness of a(n). See the Weisstein link for details of the guarantee -- the option that uses a property derived from Fermat's little theorem.
To what extent does this differ from A135720 sorted? - Peter Munn, Mar 12 2025

Examples

			a(1) = 4, since 4 is the smallest composite number and we need to add a witness to the empty set to guarantee its compositeness. 2 is the minimal Fermat witness for the compositeness of 4, so the set of witnesses becomes {2}.
a(2) = 341, since 341 is the smallest composite number that requires a witness other than 2, namely 3.
a(3) = 1105, since 1105 is the smallest composite number that requires a witness other than 2 and 3, namely 5.

Links

Crossrefs

Cf. A001567, A002997, A006945, A098654, A135720, A380978 (new minimal Fermat witness).

Extensions

More terms from Jinyuan Wang, Mar 05 2025

A380978 Sequence of minimal Fermat witnesses for compositeness. a(n) is the least k such that the smallest composite number that is a Fermat pseudoprime to bases {a(i) : 1 <= i < n} is not a Fermat pseudoprime to base k.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 41, 37, 19, 31, 43, 23, 53, 29, 101, 61, 109, 71, 67, 73, 113, 151, 89, 97, 211, 191, 157, 163, 193, 139, 281, 107, 103, 181, 47, 127, 271, 131, 307, 59, 257, 229, 331, 337, 199, 241, 461, 239, 617, 367, 263, 401, 251, 149, 421, 137, 277
Offset: 1

Views

Author

Jan Kostanjevec, Feb 10 2025

Keywords

Examples

			For n = 1, a(1) = 2, since 2 is the first Fermat witness, proving the compositeness of 4.
For n = 2, a(2) = 3, since 3 is the next required Fermat witness, proving the compositeness of 341 (all previous composites are witnessed by 2).
For n = 3, a(3) = 5, since 5 is the next required Fermat witness, proving the compositeness of  1105 (all previous composites are witnessed by 2 and 3).

Links

Crossrefs

Cf. A001567, A089105, A321790, A380979.

Formula

a(1) = 2, otherwise a(n) = A321790(k), where k is such that A001567(k) = A380979(n). - Peter Munn, Mar 12 2025

Extensions

More terms from Jinyuan Wang, Mar 05 2025

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