A210461 -id:A210461 - 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

A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

Original entry on oeis.org

91, 703, 1891, 2701, 7381, 8911, 10585, 12403, 16471, 18721, 29161, 38503, 41041, 49141, 79003, 88831, 93961, 104653, 115921, 146611, 188191, 218791, 226801, 269011, 286903, 314821, 334153, 364231, 385003, 497503, 534061, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1237951

Offset: 1

Marius Coman, Oct 12 2012

nonn

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

Cf. A005935, A210461 (subsequence).

list(lim)=my(v=List(),n); lim\=1; forstep(k=27,sqrtint(8*lim+1),2, n=k^2>>3; if(Mod(3,n)^(n-1)==1, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

a(15)-a(18) and a(35) from Charles R Greathouse IV, Jun 30 2017

A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

Original entry on oeis.org

91, 7381, 597871, 48427561, 3922632451, 317733228541, 25736391511831, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset: 1

Marius Coman, Oct 12 2012

nonn

These numbers were obtained for values of k from 1 to 20, with the following exceptions: k = 10, 12, 13, 16, 17, 19, for which were obtained 3^n mod n = 3^7, 3^31, 3^37, 3^25, 3^31, 3^13.
Conjecture: There are infinitely many Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8, where k is a natural number.
It is true: for example, when 2k+1 is a prime number (see A210461). - Bruno Berselli, Jan 22 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..190
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A005935, A210461 (subsequence), A217841.

Select[Table[(3^(4k + 2) - 1)/8, {k, 80}], PowerMod[3, # - 1, #] == 1 &] (* Alonso del Arte, May 14 2019 *)

list(lim)=my(v=List(),t); lim\=1; for(k=1,(logint(8*lim+1,3)-2)\4, t=3^(4*k + 2)>>3; if(Mod(3,t)^t==3, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

A343009 a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.

Original entry on oeis.org

1, 5, 91, 4369, 406901, 62193781, 14129647351, 4467856773185, 1876182941212489, 1010101010101010101, 678356244890331342611, 555922008415320588345745, 546031727340884622966664381, 633213824057681722185793753109, 856031514432518244055765015738351

Offset: 1

Thomas Ordowski, Apr 02 2021

nonn

Conjecture: for n > 2, a(n) is a Fermat pseudoprime to base n.
If p is an odd prime, then a(p) is a Cipolla pseudoprime to base p.
Is a(m) a Fermat pseudoprime to base m for every composite m?
Amiram Eldar confirmed this up to m = 3800.
From Jianing Song, Aug 28 2022: (Start)
a(n) = Product_{d|(2n),d>2} Phi(d,n), where Phi(n,x) is the d-th cyclotomic polynomial. Note that Phi(n,x) > 1 for x >= 2 unless (n,x) = (1,2): suppose that n >= 3 and x >= 2, then Phi(n,x) = Product_{1<=j<=n,gcd(j,n)=1} (x - exp(2*j*Pi*i/n)) = Product_{1<=j<=n/2,gcd(j,n)=1} (x^2 - 2*cos(2*j*Pi/n)*x + 1) = Product_{1<=j<=n/2,gcd(j,n)=1} ((x - cos(2*j*Pi/n))^2 + (sin(2*j*Pi/n))^2) > 1 since x - cos(2*j*Pi/n) > 1. This shows that a(n) is composite for n > 2.
For n > 2, a(n) is a Fermat pseudoprime to base n, since n^(2*n) == 1 (mod a(n)) and 2*n divides a(n)-1 = n^2*(n^(2*n-2)-1)/(n^2-1): if n is even, then 2*n | n^2; if n is odd, then n | n^2 and 2 | n^2+1 = (n^4-1)/(n^2-1) | (n^(2*n-2)-1)/(n^2-1). (End)

			a(10) = (10^20-1)/99 = 1010101010101010101.

Cf. A005563, A023037, A117812, A124899, A132411, A210461.

a[n_] := (n^(2*n)-1)/(n^2-1); a[1] = 1; Array[a, 15] (* Amiram Eldar, Apr 02 2021 *)

a(n) = Sum_{k=0..n-1} n^(2*k). - Davide Rotondo, Aug 28 2022
From Alois P. Heinz, Aug 28 2022: (Start)
a(n) = A117812(n)/A005563(n-1) = A117812(n)/A132411(n-1) for n>=2.
Limit_{n -> 1} (n^(2*n)-1)/(n^2-1) = 1. (End).

More terms from Amiram Eldar, Apr 02 2021

a(1)=1 prepended and name adapted by Alois P. Heinz, Aug 28 2022

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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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A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

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A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

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A343009 a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.

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