A328662 -id:A328662 - cp's OEIS frontend

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

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]

A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 90751, 104653, 226801, 282133, 653333, 665281, 721801, 873181, 1373653, 1530787, 1537381, 1584133, 1690501, 1755001, 1987021, 2008597, 2035153, 2284453, 2746589, 2944261, 3059101, 3116107, 3363121, 3375041, 3375487, 4082653, 4314967

Offset: 1

Amiram Eldar, Mar 08 2020

nonn

The first term that has more than 2 prime factors is a(1067) = A333131(1) = 11500521553.

The first term that is also a Carmichael number is a(1131) = 13079177569.

			2701 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 all primes: 37 and 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A050217 and A328662.

Subsequence of A001567, A005935, A052155 and A153513.

pspQ[n_] := CompositeQ[n] && AllTrue[Rest @ Divisors[n], PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 &]; Select[Range[10^6], pspQ]

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]

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

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A333130 Numbers that are super pseudoprimes to both bases 2 and 3.

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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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