A181780 -id:A181780 - cp's OEIS frontend

A348788 Values of A347113(k) for k in A348787.

1, 12, 15, 18, 24, 30, 27, 32, 36, 40, 42, 44, 48, 56, 50, 54, 60, 63, 66, 64, 70, 65, 68, 72, 69, 75, 78, 76, 84, 80, 87, 77, 81, 88, 91, 90, 98, 93, 92, 96, 100, 105, 102, 99, 104, 108, 110, 114, 120, 132, 112, 115, 124, 130, 119, 123, 126, 125, 128, 135, 138, 129, 136, 140, 144, 150, 141, 147, 152, 156, 148, 153, 160, 154, 162, 155

Offset: 1

N. J. A. Sloane, Nov 20 2021

nonn

By definition, the points (A348787(k), a(k)) form the main diagonal of A347113.
It appears that apart from a very small number of exceptions this sequence consists of the numbers that are neither primes nor twice primes (A264828).
The known exceptions (based on the first 20000 terms) are: (a) the three primes A347113(1423) = 1327, A347113(10686) = 9967, and A347113(83051) = 77647 that are unusually close to the line y=x, and (b) the 27 terms [8, 9, 16, 20, 21, 25, 28, 33, 35, 39, 45, 49, 51, 52, 55, 57, 85, 95, 111, 116, 117, 121, 133, 143, 145, 169, 187] which are in A264828 but are not in the present sequence.
This list of 27 exceptions is surprisingly similar to A181780, but this may be just a coincidence.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A264828, A347113, A348787, A181780.

A227180 Composite numbers n such that b^(n-1) == 1 (mod n) implies b == -1 or +1 (mod n).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138, 140, 142, 144, 146, 150, 152, 156, 158, 160, 162, 164, 166, 168, 170

Offset: 1

Emmanuel Vantieghem, Jul 03 2013

nonn

The sequence is the union of A111305 with {3^k | k > 1}.

The composite numbers not in this sequence are the Fermat pseudoprimes A181780.

Cf. A111305, A181780, A209211.

FQ[k_]:= Block[{},GCD[EulerPhi[k],k-1]==1||IntegerQ[Log[3,k]]];Select[Range[4,170],FQ]

is(n)=for(b=2, n-2, if(Mod(b, n)^(n-1)==1, return(0))); !isprime(n) \\ Charles R Greathouse IV, Dec 22 2016

A254139 a(n) = smallest composite c for which there exist exactly n bases b with b < c such that b^(c-1) == 1 (mod c), i.e., smallest composite c which is a Fermat pseudoprime to exactly n bases less than c.

Original entry on oeis.org

9, 28, 15, 66, 49, 232, 45, 190, 121, 276, 169, 1106

Offset: 1

Felix Fröhlich, Jan 26 2015

a(13) > 150000.

			With c = 49: there are exactly five bases b with b < 49 such that 49 is a Fermat pseudoprime, namely 18, 19, 30, 31 and 48. Since 49 is the smallest composite having exactly five such bases, a(5) = 49.

Cf. A001567, A141768, A181780.

for(n=1, 20, forcomposite(c=3, , b=2; i=0; while(b < c, if(Mod(b, c)^(c-1)==1, i++); b++); if(i==n, print1(c, ", "); break({1}))))

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A348788 Values of A347113(k) for k in A348787.

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A227180 Composite numbers n such that b^(n-1) == 1 (mod n) implies b == -1 or +1 (mod n).

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A254139 a(n) = smallest composite c for which there exist exactly n bases b with b < c such that b^(c-1) == 1 (mod c), i.e., smallest composite c which is a Fermat pseudoprime to exactly n bases less than c.

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