A216170 -id:A216170 - cp's OEIS frontend

A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

341, 5461, 1398101, 22369621, 5726623061, 91625968981, 23456248059221, 96076792050570581, 1537228672809129301, 6296488643826193618261, 1611901092819505566274901, 25790417485112089060398421, 6602346876188694799461995861

Offset: 1

Bruno Berselli, Jan 21 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

This is the case a=2 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

Named after the Italian mathematician Michele Cipolla (1880-1947). - Amiram Eldar, Jun 15 2021

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013. The sequence is on page 3.
Michele Cipolla, Sui numeri composti P, che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica, Vol. 9 (1904), pp. 139-160.
Y. Hamahata and Y. Kokubun, Cipolla Pseudoprimes, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.6.

Cf. A001567, A002450, A210461, A215848, A216170.

a210454 = (`div` 3) . (subtract 1) . (4 ^) . a000040 . (+ 2)
-- Reinhard Zumkeller, Jan 22 2013

Magma
```
[(4^NthPrime(n)-1)/3: n in [3..15]];
```

P:=proc(q)local n;
for n from 3 to q do print((4^ithprime(n)-1)/3);
od; end: P(100); # Paolo P. Lava, Oct 11 2013

Mathematica
```
(4^# - 1)/3 & /@ Prime[Range[3, 15]]
```

Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
makelist((4^Prime(n)-1)/3, n, 3, 15);

a(n)=4^prime(n+2)\3 \\ Charles R Greathouse IV, Jul 09 2015

A321868 Fermat pseudoprimes to base 2 that are octagonal.

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

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A000567.
The corresponding indices of the octagonal numbers are 11, 15, 29, 31, 37, 43, 53, 73, 93, 103, 151, 167, 233, 271, 307, 341, 421, 443, 467, 487, 491, 577, 593, 601, 631, 645, 667, 673, 683, 705, 727, 751, 761, 901, 911, 919, 991, ...
First differs from A216170 at n = 505.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A000567, A001567, A216170, A293623, A293624, A321869.

oct[n_]:=n(3n-2); Select[oct[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = (n>1) && ispolygonal(n, 8) && !isprime(n) && (Mod(2, n)^n==2); \\ Daniel Suteu, Nov 29 2018

A216276 Fermat pseudoprimes to base 2 of the form (p^2 + 2*p)/3, where p is also a Fermat pseudoprime to base 2.

Original entry on oeis.org

997633, 1398101, 2433601, 3581761, 26474581, 37354465, 63002501, 70006021, 82268033, 93030145, 561481921, 804978721, 1231726981, 2602378721, 2942952481, 12817618945, 15516020833, 16627811905, 22016333333, 25862624705, 53707855201, 67220090785, 95074073281, 144278347201

Offset: 1

Marius Coman, Sep 03 2012

nonn

The corresponding values of the Fermat pseudoprime p: 1729, 2047, 3277, 8911, 10585, 13747, 14491, 15709, 16705, 41041, 49141, 60787, 88357, 196093, 215749, 223345, 256999, 278545, 401401, 449065, 657901.
Conjecture: For any Fermat pseudoprime to base 2, p1, there exist infinitely many Fermat pseudoprimes to base 2, of the form p2 = (p1^n + n*p1)/(n+1), where n > 1.
Conjecture: For any Carmichael number c there exist infinitely many Carmichael numbers of the form (c^n + n*c)/(n + 1) with n > 1.

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

Cf. A001567, A216170.

is(n)=my(s); issquare(3*n+1,&s) && Mod(2,s-1)^(s-2)==1 && !isprime(s-1) && Mod(2,n)^n==2 && n>1 \\ Charles R Greathouse IV, Jul 07 2017

forcomposite(p=1729,1e6, n=p*(p+2)/3; if(Mod(2,p)^p==2 && Mod(2,n)^n==2, print1(n", "))) \\ Charles R Greathouse IV, Jul 07 2017

a(3) and a(15) inserted by Charles R Greathouse IV, Jul 07 2017

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A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

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A321868 Fermat pseudoprimes to base 2 that are octagonal.

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A216276 Fermat pseudoprimes to base 2 of the form (p^2 + 2*p)/3, where p is also a Fermat pseudoprime to base 2.

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