A316907 -id:A316907 - cp's OEIS frontend

A316940 Smallest "anti-Carmichael pseudoprime" to base n.

35, 7957, 16531, 1247, 17767, 35, 817, 2501, 697, 4141, 2257, 143, 9577, 2257, 4187, 1247, 3991, 221, 7957, 2059, 55, 161, 1027, 115, 403, 475, 247, 4553, 35, 247, 6289, 697, 1853, 35, 1247, 35, 589, 221, 95, 533, 35, 559, 77, 215, 253, 235, 221, 329, 247, 119

Offset: 1

Thomas Ordowski, Jul 17 2018

nonn

a(n) is the smallest k such that n^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

All listed terms are semiprime and squarefree, except a(26) = 475 = 5^2*19.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A121707 (probably "anti-Carmichael numbers": n such that p-1 does not divide n-1 for every prime p dividing n).

Cf. A316907 ("anti-Carmichael pseudoprimes" to base 2).

Table[Block[{k = 2}, While[Nand[PowerMod[n, k - 1, k] == 1, AllTrue[FactorInteger[k][[All, 1]] - 1, Mod[k - 1, #] != 0 &]], k++]; k], {n, 50}] (* Michael De Vlieger, Jul 20 2018 *)

isok(k, n) = {if (!isprime(k) && Mod(n, k)^(k-1) == 1, f = factor(k)[,1]; for (j=1, #f~, if (!((k-1) % (f[j]-1)), return (0));); return (1);); return (0);}
a(n) = {my(k=2); while(!isok(k, n), k++); k;} \\ Michel Marcus, Jul 17 2018

More terms from Michel Marcus, Jul 17 2018

A300762 Numbers k > 1 such that 2^k == 2 (mod k) and gcd(k, 3^k - 3) = 1.

Original entry on oeis.org

35333, 42799, 49981, 60787, 150851, 162193, 164737, 241001, 253241, 256999, 280601, 452051, 481573, 556169, 617093, 665333, 722201, 838861, 1016801, 1252697, 1507963, 1534541, 1678541, 1826203, 2134277, 2269093, 2304167, 2313697, 2537641, 2617451, 2811271

Offset: 1

Thomas Ordowski, Aug 15 2018

nonn

Numbers k > 1 such that 2^(k-1) == 1 (mod k) and gcd(k, 3^(k-1)-1) = 1.

Are there infinitely many such "anti-Carmichael pseudoprimes (2,3)"?

Subsequence of A001567 and of A316907 and probably of A121707.

Select[Range[2 10^6], PowerMod[2, #, #] == 2 && GCD[#, # + PowerMod[3, #, #] - 3] == 1 &] (* Giovanni Resta, Aug 18 2018 *)

isok(k) = (k != 1) && (Mod(2, k)^k == Mod(2, k)) && (gcd(k, 3^k - 3) == 1); \\ Michel Marcus, Aug 15 2018

More terms from Michel Marcus, Aug 15 2018

More terms from Giovanni Resta, Aug 18 2018

A316908 a(n) is the smallest k with n prime factors such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

Original entry on oeis.org

7957, 617093, 134564501, 384266404601, 8748670222601, 6105991025919737, 901196605940857381

Offset: 2

Thomas Ordowski, Jul 16 2018

Conjecture: a(n) > A006931(n) for every n > 2.
a(6)-a(8) derived from Feitsma's tables of pseudoprimes. a(9) > 2^64. - Giovanni Resta, Jul 19 2018
From Daniel Suteu, Jun 08 2020: (Start)
a(9)  <= 521957994426556057126261,
a(10) <= 1315856103949347820015303981,
a(11) <= 6357507186189933506573017225316941,
a(12) <= 77822245466150976053960303855104674781. (End)

Jan Feitsma and William Galway, Tables of pseudoprimes and related data

Cf. A001567, A121707, A316907.

More terms from Michel Marcus, Jul 16 2018

a(6)-a(8) from Giovanni Resta, Jul 19 2018

cp's OEIS Frontend

A316940 Smallest "anti-Carmichael pseudoprime" to base n.

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A300762 Numbers k > 1 such that 2^k == 2 (mod k) and gcd(k, 3^k - 3) = 1.

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Mathematica

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Extensions

A316908 a(n) is the smallest k with n prime factors such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

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Author

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Extensions