A020233 -id:A020233 - cp's OEIS frontend

A020287 Strong pseudoprimes to base 61.

15, 217, 341, 1261, 2701, 3661, 6541, 6697, 7613, 13213, 16213, 22177, 23653, 23959, 31417, 50117, 61777, 63139, 67721, 76301, 77421, 79381, 80041, 102341, 113491, 115231, 145993, 160147, 163073, 164737, 170941, 178709, 197209, 210817, 249631

Offset: 1

David W. Wilson

nonn

The smallest number which is a strong pseudoprime to the bases 2 (A001262), 7 (A020233) and also 61 (here) is 4759123141 [Jaeschke]. - R. J. Mathar, Apr 05 2011

R. J. Mathar, Table of n, a(n) for n = 1..202
G. Jaeschke, On strong pseudoprimes to several bases, Mathematics of Computation, 61 (1993), 915-926.
Index entries for sequences related to pseudoprimes

A048684 Multiplicity of the maximum squarefree kernel function applied to the binomial coefficients C(n,k).

Original entry on oeis.org

2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer

nonn

			For n = 8, 9 or 10 the spectra of squarefree maximal divisors are {1,2,14,14,70,14,14,2,1}, {1,3,6,42,42,42,42,6,3,1} and {1,10,15,30,210,42,30,15,10,1}, respectively. The maxima (70,42 and 210) occur 1, 4 or 4 times. So a(8) = 1, a(9) = 4 and a(10) = 2.

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

Cf. A007947, A020233, A020738, A001221, A001222, A000005.

rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; a[n_] := Module[{r = rad /@ Table[Binomial[n, k], {k, 0, n}]}, Count[r, Max[r]]]; Array[a, 100] (* Amiram Eldar, Sep 17 2024 *)

rad(n) = vecprod(factor(n)[, 1]);
a(n) = {my(r = vector(n+1, k, rad(binomial(n,k-1))), rm = vecmax(r)); #select(x -> x==rm, r);} \\ Amiram Eldar, Sep 17 2024

A329759 Odd composite numbers k for which the number of witnesses for strong pseudoprimality of k equals phi(k)/4, where phi is the Euler totient function (A000010).

Original entry on oeis.org

15, 91, 703, 1891, 8911, 12403, 38503, 79003, 88831, 146611, 188191, 218791, 269011, 286903, 385003, 497503, 597871, 736291, 765703, 954271, 1024651, 1056331, 1152271, 1314631, 1869211, 2741311, 3270403, 3913003, 4255903, 4686391, 5292631, 5481451, 6186403, 6969511

Offset: 1

Amiram Eldar, Nov 20 2019

nonn

Odd numbers k such that A071294((k-1)/2) = A000010(k)/4.
For each odd composite number m > 9 the number of witnesses <= phi(m)/4. For numbers in this sequence the ratio reaches the maximal possible value 1/4.
The semiprime terms of this sequence are of the form (2*m+1)*(4*m+1) where 2*m+1 and 4*m+1 are primes and m is odd.

			15 is in the sequence since out of the phi(15) = 8 numbers 1 <= b < 15 that are coprime to 15, i.e., b = 1, 2, 4, 7, 8, 11, 13, and 14, 8/4 = 2 are witnesses for the strong pseudoprimality of 15: 1 and 14.

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, 2nd ed., Springer, 2005, Theorem 3.5.4., p. 136.

Amiram Eldar, Table of n, a(n) for n = 1..350
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.

Cf. A000010, A033181, A006945, A014233, A071294, A141768, A181782, A195328, A329468.

Cf. A001262, A020229, A020231, A020233.

o[n_] := (n - 1)/2^IntegerExponent[n - 1, 2];
a[n_?PrimeQ] := n - 1; a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, om = Length[p]; Product[GCD[o[n], o[p[[k]]]], {k, 1, om}] * (1 + (2^(om * Min[IntegerExponent[#, 2] & /@ (p - 1)]) - 1)/(2^om - 1))];
aQ[n_] := CompositeQ[n] && a[n] == EulerPhi[n]/4; s = Select[Range[3, 10^5, 2], aQ]

A298757 Numbers k with record value of the least strong pseudoprime to base k (A298756).

Original entry on oeis.org

2, 1320, 4712, 5628, 7252, 7852, 14787, 17340, 61380, 78750, 254923, 486605, 1804842, 4095086, 12772344, 42162995

Offset: 1

Amiram Eldar, Jan 26 2018

The record strong pseudoprimes are 2047, 4097, 4711, 5627, 7251, 7851, 9409, 10261, 11359, 13747, 18299, 25761, 32761, 38323, 40501, 97921, ...

Index entries for sequences related to pseudoprimes

Cf. A001262, A020229, A020230, A020231, A020232, A020233, A020234, A020235, A298756.

sppQ[n_?EvenQ, ] := False; sppQ[n?PrimeQ, ] := False; sppQ[n, b_] := Module[{ans=False},s = IntegerExponent[n-1, 2]; d = (n-1)/2^s; If[ PowerMod[b, d, n] == 1, ans=True, Do[If[PowerMod[b, d*2^r, n] == n-1, ans=True], {r, 0, s-1}]];ans]; smallestSPP[b_] := Module[ {k=3}, While[ !sppQ[k,b],k+=2];k ]; sm=0;a={};Do[s=smallestSPP[b];If[s>sm,sm=s;AppendTo[a,b]], {b,2,10^4}];a (* after Jean-François Alcover at A020229 *)

lista(nn) = {my(m=0); for (n=2, nn, my(r=a298756(n)); if (r>m, m =r; print1(n, ", ")););} \\ Michel Marcus, Jan 31 2022; using pari code in A298756

a(9)-a(16) from Jonathan Pappas, Jan 31 2022

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A020287 Strong pseudoprimes to base 61.

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A048684 Multiplicity of the maximum squarefree kernel function applied to the binomial coefficients C(n,k).

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A329759 Odd composite numbers k for which the number of witnesses for strong pseudoprimality of k equals phi(k)/4, where phi is the Euler totient function (A000010).

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A298757 Numbers k with record value of the least strong pseudoprime to base k (A298756).

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