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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. 1
997633, 1398101, 2433601, 3581761, 26474581, 37354465, 63002501, 70006021, 82268033, 93030145, 561481921, 804978721, 1231726981, 2602378721, 2942952481, 12817618945, 15516020833, 16627811905, 22016333333, 25862624705, 53707855201, 67220090785, 95074073281, 144278347201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

Note that for 22 from the first 200 Fermat pseudoprimes to base 2 were obtained also Fermat pseudoprimes to base 2 through this formula!

Conjecture: For any Fermat pseudoprime to base 2, p1, there exist infinite many Fermat pseudoprimes to base 2, p2, formed this way: p2 = (p1^n + n*p1)/(n+1), where n natural, n > 1.

Conjecture: For any Carmichael number c there exist infinite many Carmichael numbers of the form (c^n + n*c)/(n + 1) with n > 1. Note that, in the sequence above, from Fermat pseudoprimes to base 2 that are also Carmichael numbers (1729, 8911, 10585, 41041, 278545, 449065) were obtained too Carmichael numbers.

LINKS

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

PROG

(PARI) 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

(PARI) 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

CROSSREFS

Cf. A001567, A216170.

Sequence in context: A055617 A055618 A341551 * A252903 A289140 A033426

Adjacent sequences:  A216273 A216274 A216275 * A216277 A216278 A216279

KEYWORD

nonn

AUTHOR

Marius Coman, Sep 03 2012

EXTENSIONS

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

STATUS

approved

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Last modified November 29 12:24 EST 2021. Contains 349416 sequences. (Running on oeis4.)