A210454 -id:A210454 - cp's OEIS frontend

A210461 Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.

91, 7381, 597871, 3922632451, 317733228541, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset: 1

Bruno Berselli, Jan 22 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

This is the case a=3 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

			91 is in the sequence because 91=((3^3-1)/2)*((3^3+1)/4), even if p=3 divides 3*(3^2-1), and 3^90 = (91*8+1)^15 == 1 (mod 91).
7381 is in the sequence because 7381=((3^5-1)/2)*((3^5+1)/4) and 3^7380 = (7381*472400+1)^369 == 1 (mod 7381).

Michele Cipolla, Sui numeri composti P che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica 9 (1904), p. 139-160.

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013.
Y. Hamahata and Y. Kokubun, Cipolla Pseudoprimes, Journal of Integer Sequences, Vol. 10 (2007).

Cf. A005935, A002452, A065091, A210454, A217853.

a210461 = (`div` 8) . (subtract 1) . (9 ^) . a065091
-- Reinhard Zumkeller, Jan 22 2013

Magma
```
[(9^NthPrime(n)-1)/8: n in [2..12]];
```

P:=proc(q)local n;
for n from 2 to q do print((9^ithprime(n)-1)/8);
od; end: P(100); # Paolo P. Lava, Oct 11 2013

Mathematica
```
(9^# - 1)/8 & /@ Prime[Range[2, 12]]
```

Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
makelist((9^Prime(n)-1)/8, n, 2, 12);

A216170 Fermat pseudoprimes to base 2 of the form (n^2 + 2*n)/3.

Original entry on oeis.org

341, 645, 2465, 2821, 4033, 5461, 8321, 15841, 25761, 31621, 68101, 83333, 162401, 219781, 282133, 348161, 530881, 587861, 653333, 710533, 722261, 997633, 1053761, 1082401, 1193221, 1246785, 1333333, 1357441, 1398101, 1489665, 1584133, 1690501, 1735841

Offset: 1

Marius Coman, Sep 03 2012

nonn

The corresponding values of n: 31, 43, 85, 91, 109, 127, 157, 217, 277, 307, 451, 499, 697, 811, 919, 1021, 1261, 1327, 1399, 1459, 1471, 1729, 1777, 1801, 1891, 1933, 1999, 2017, 2047, 2113, 2177, 2251.
The formula can be generalized this way: Fermat pseudoprimes to base 2 of the form (n^m + m*n)/(m+1).
For m = 3, the formula becomes (n^3 + 3*n)/4, from which the Poulet numbers 341, 1729, 188461, 228241, and 1082809 (for n = 11, 19, 91, 97, and 163, respectively) were obtained.
Conjecture: For any m natural, m > 1, there exists a series with infinitely many Fermat pseudoprimes to base 2, P, formed this way: P = (n^m + m*n)/(m+1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Poulet Number

Cf. A001567, A210454 (subsequence).

t = Select[Table[n (n + 2)/3, {n, 4, 10000}], IntegerQ]; Select[t, PowerMod[2, # - 1, #] == 1 &] (* T. D. Noe, Sep 03 2012 *)

list(lim)=my(v=List(),t); lim\=1; forstep(n=31,sqrtint(3*lim+1)-1,[2,1], t=t=n*(n+2)/3; if(Mod(2,t)^t==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

A293626 Numbers of the form (2^(2p) + 1)/5, where p is a prime > 5.

Original entry on oeis.org

3277, 838861, 13421773, 3435973837, 54975581389, 14073748835533, 57646075230342349, 922337203685477581, 3777893186295716170957, 967140655691703339764941, 15474250491067253436239053, 3961408125713216879677197517, 16225927682921336339157801028813

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Rotkiewicz proved that all the terms in this sequence are Fermat pseudoprimes to base 2 (A001567).

			3277 = (2^(2*7) + 1)/5 is the first term, corresponding to the prime p = 7.

Andrzej Rotkiewicz, Sur les formules donnant des nombres pseudopremiers, Colloquium Mathematicae, Vol. 12, No. 1 (1964), pp. 69-72.

Cf. A001567, A210454.

p = Select[Range[7,60], PrimeQ]; (2^(2p) + 1)/5

A371757 Numbers of the form (4^(p^(k+1)) + 1)/(4^(p^k) + 1), where k >= 1 and p is an odd prime other than 5.

Original entry on oeis.org

4033, 68719214593, 19341632594266545643831297, 324518553658426708768757511094273, 1684996264962499703367587717863072443065045481313942556034056847361

Offset: 1

Amiram Eldar, Apr 05 2024

nonn

Mąkowski and Rotkiewicz (1969) proved that all the terms are Fermat pseudoprimes to base 2 (A001567).

The next term has 94 digits and is too large to include in the data section.

Amiram Eldar, Table of n, a(n) for n = 1..18
Andrzej Mąkowski and Andrzej Rotkiewicz, On pseudoprime numbers of special form, Colloquium Mathematicum, Vol. 2, No. 20 (1969), pp. 269-271; alternative link.
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Subsequence of A001567.

Cf. A210454, A293626.

f[p_, k_] := (4^(p^(k + 1)) + 1)/(4^(p^k) + 1);
seq[max_] := Module[{s = {}, p = 3, f1, k, addFlag = True}, While[addFlag, If[p == 5, Continue[]]; k = 1; addFlag = False; While[(f1 = f[p, k]) < max, AppendTo[s, f1]; addFlag = True; k++]; p = NextPrime[p, If[p == 3, 2, 1]]]; Sort[s]]; seq[10^80]

A243292 Odd numbers n such that (4^n - 1)/3 is a base-2 pseudoprime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Arkadiusz Wesolowski, Jun 02 2014

nonn

Odd numbers not divisible by 3 that are primes or pseudoprimes to base 4.

Wikipedia, Fermat pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A000040, A001567, A020136, A210454.

[n: n in [1..271 by 2] | not n mod 3 eq 0 and IsOne(4^(n-1) mod n)];

cp's OEIS Frontend

A210461 Cipolla pseudoprimes to base 3: (9^p-1)/8 for any odd prime p.

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Maxima

A216170 Fermat pseudoprimes to base 2 of the form (n^2 + 2*n)/3.

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PARI

A293626 Numbers of the form (2^(2p) + 1)/5, where p is a prime > 5.

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Mathematica

A371757 Numbers of the form (4^(p^(k+1)) + 1)/(4^(p^k) + 1), where k >= 1 and p is an odd prime other than 5.

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Mathematica

A243292 Odd numbers n such that (4^n - 1)/3 is a base-2 pseudoprime.

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Magma