A178997 -id:A178997 - cp's OEIS frontend

A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751

Offset: 1

Eric W. Weisstein

nonn

Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - T. D. Noe, Jan 11 2011
Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016
Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016
Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - Thomas Ordowski, Sep 12 2016
Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - Robert Israel, Sep 13 2016
Composite numbers n such that 2^A258409(n) == 1 (mod n). - Thomas Ordowski, Sep 15 2016

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

A214305 is a subsequence.
A065341 is a subsequence. - Thomas Ordowski, Nov 20 2016
Cf. A001220, A001567, A178997.

filter:= = proc(n)
    not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Sep 13 2016

Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]

is(n)=if(isprime(n), return(0)); fordiv(n,d, if(Mod(2,d)^d!=2, return(0))); n>1 \\ Charles R Greathouse IV, Aug 27 2016

A328663 Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

7381, 512461, 532171, 1018601, 2044657, 3882139, 5934391, 8624851, 10802017, 14396449, 19383673, 25708453, 32285041, 35728129, 35807461, 38316961, 43040161, 53369149, 58546753, 59162891, 64464919, 71386849, 75397891, 79511671, 81276859, 83083001, 84890737, 85636609

Offset: 1

Amiram Eldar, Oct 24 2019

nonn

Super pseudoprimes to base 3 are Fermat pseudoprimes to base 3 all of whose composite divisors are also Fermat pseudoprimes to base 3. Therefore all the Fermat pseudoprimes to base 3 that are semiprimes are super pseudoprimes. This sequence contains the nontrivial terms of A328662, i.e. terms with at least one composite proper divisor.
Fehér and Kiss proved that there are infinitely many terms with 3 distinct prime factors (their proof was for all bases a > 1 that are not divisible by 4. Phong proved it for all bases a > 1).
The first term, 7381, is not squarefree. What is the next such term?

			512461 is in the sequence since it is a Fermat pseudoprime to base 3, 3^512460 == 1 (mod 512461), and all of its divisors that are larger than 1 are either primes (31, 61, and 271), or Fermat pseudoprimes to base 3 (1891, 8401, 16531, 512461).

Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.

Amiram Eldar, Table of n, a(n) for n = 1..400
J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.

Cf. A178997

Subsequence of A005935, A328662.

aQ[n_]:=  PrimeOmega[n] > 2 && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]

A291637 Carmichael numbers (A002997) that are super-Poulet numbers (A050217).

Original entry on oeis.org

294409, 1299963601, 4215885697, 4562359201, 7629221377, 13079177569, 19742849041, 45983665729, 65700513721, 147523256371, 168003672409, 227959335001, 459814831561, 582561482161, 1042789205881, 1297472175451, 1544001719761, 2718557844481, 3253891093249, 4116931056001, 4226818060921, 4406163138721, 4764162536641, 4790779641001, 5419967134849, 7298963852041, 8470346587201

Offset: 1

Max Alekseyev and Thomas Ordowski, Aug 28 2017

nonn

Problem: are there infinitely many such numbers?
From Daniel Suteu, Sep 17 2020: (Start)
If we consider f(n) to be the smallest number in the sequence with n prime factors, then we have:
f(3) = 294409,
f(4) = 3018694485093841,
f(5) <= 521635331852681575100906881,
f(6) <= 2835402730651853232634509813787097410561,
f(7) <= 165784025660216242122027716057592895796242004385542265601. (End)

Amiram Eldar, Table of n, a(n) for n = 1..5328 (calculated using data from Claude Goutier; terms 1..983 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Intersection of A178997 and A002997.

A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

Original entry on oeis.org

11500521553, 13079177569, 52474339009, 168003672409, 229352039821, 280792563977, 318289021201, 428178002569, 918660756421, 2015841188197, 2367478228501, 2544457029601, 2639665216117, 3023595814801, 3457449931321, 3712164285421, 4348114583017, 6046196043229

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

Up to 2^64 all the 1085 terms are nonsquarefree, 2 terms have 4 prime factors: a(163) = 18362297383286473 = 3037 * 6073 * 9109 * 109297 and a(651) = 2587580959818925201 = 18121 * 36241 * 54361 * 72481, and no term have more than 4 prime factors.

			11500521553 is a term since it is a Fermat pseudoprime to both bases 2 and 3, and its proper divisors that are larger than 1 are either primes (937, 1873, 6553) or Fermat pseudoprimes to both bases 2 and 3 (1755001, 6140161, 12273769, 11500521553).

Amiram Eldar, Table of n, a(n) for n = 1..1085 (terms below 2^64)

Subsequence of A001567, A005935, A050217, A052155, A153513, A178997, A328662, A328663, A333130.

pspQ[n_] := PrimeOmega[n] > 2 && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; seq = {}; Do[If[pspQ[n], AppendTo[seq, n]], {n, 1, 6*10^10}]; seq

A328664 Least super pseudoprime to base n that is not a semiprime.

Original entry on oeis.org

294409, 7381, 13981, 342271, 9331, 747289, 63, 8, 99, 4921, 1729, 12, 195, 355957, 255, 8, 325, 18, 399, 20, 483, 1183, 575, 8, 27, 1729, 27, 28, 637, 30, 1023, 8, 105, 153, 1295, 12, 1105, 29659, 1599, 8, 12167, 42, 45, 44, 45, 1105, 637, 8, 147, 50, 2703, 27

Offset: 2

Amiram Eldar, Oct 24 2019

nonn

A number is super pseudoprime to base n > 1 if it is a Fermat pseudoprime to base n and of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base n.

The semiprime Fermat pseudoprimes are trivial terms since they do not have composite proper divisors.

			a(2) = 294409 = 37 * 73 * 109 is the first term of A178997.
a(3) = 7381 = 11^2 * 61 is the first term of A328663.

Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.

Amiram Eldar, Table of n, a(n) for n = 2..10000
J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.

Cf. A178997, A328662, A328663.

a[n_] := Module[{k=1}, While[PrimeOmega[k] < 3 || !AllTrue[Rest[Divisors[k]], PowerMod[n, #-1, #] == 1 &], k++]; k]; Array[a, 10, 2]

A328665 Least super-Poulet number (A050217) with n distinct prime factors.

Original entry on oeis.org

341, 294409, 9972894583, 1264022137981459, 14054662152215842621

Offset: 2

Amiram Eldar, Oct 24 2019

a(7) <= 1842158622953082708177091, and a(8) <= 317565023788749598474704753433331761 (from Michon's site).
From Daniel Suteu, Oct 28 2019: (Start)
a(8)  <= 192463418472849397730107809253922101,
a(9)  <= 1347320741392600160214289343906212762456021,
a(10) <= 70865138168006643427403953978871929070133095474701,
a(11) <= 3363391752747838578311772729701478698952546288306688208857,
a(12) <= 132153369641266990823936945628293401491197666138621036175881960329,
a(13) <= 9105096650335639994239038954861714246150666715328403635257215036295306537. (End)

Gérard P. Michon, Super-pseudoprimes to Base a, Numericana, 2005.
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

Cf. A001567, A007011, A050217, A178997, A300327.

a[n_] := Module[{k=1}, While[PrimeNu[k] < n || PowerMod[2, k - 1, k] != 1 || Union @ PowerMod[2, Rest[Divisors[k]], k] != {2}, k++]; k]; Array[a, 3, 2]

isok(k, n) = if (omega(k) == n, fordiv(k, d, if(Mod(2, d)^d!=2, return(0))); return(1));
a(n) = my(k=4); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2019

isupperbound(n,k) = my(f=factor(k)); omega(f) == n && Mod(2, k)^gcd(vector(#f~, i, f[i,1]-1)) == 1; \\ Daniel Suteu, Oct 28 2019

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A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

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A328663 Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).

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A291637 Carmichael numbers (A002997) that are super-Poulet numbers (A050217).

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A333131 Super pseudoprimes to both bases 2 and 3 (A333130) with more than two prime factors (counted with multiplicity).

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A328664 Least super pseudoprime to base n that is not a semiprime.

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A328665 Least super-Poulet number (A050217) with n distinct prime factors.

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