A055552 -id:A055552

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A055550

Number of Poulet numbers (or pseudoprimes to base 2, A001567) less than 10^n.

+10
4

0, 0, 3, 22, 78, 245, 750, 2057, 5597, 14884, 38975, 101629, 264239, 687007, 1801533, 4744920, 12604009, 33763684, 91210364

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 245, pp 68, Ellipses, Paris 2008. - Lekraj Beedassy, Jul 20 2008

LINKS

Table of n, a(n) for n=1..19.

Jan Feitsma and William Galway, Tables of pseudoprimes and related data

Eric Weisstein's World of Mathematics, Poulet Number

Eric Weisstein's World of Mathematics, Pseudoprime

MATHEMATICA

Table[Count[Select[Range[2, 10^6], ! PrimeQ[#] && PowerMod[2, # - 1, #] == 1 &], x_ /; x < 10^n], {n, 6}] (* Robert Price, Jun 09 2019 *)

CROSSREFS

Cf. A001567, A055552.

KEYWORD

nonn,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

a(14) and a(15) computed by William F. Galway and communicated by Arthur Livingstone (arthur.livingstone(AT)gmail.com), Apr 10 2007

Edited by N. J. A. Sloane, Feb 01 2009 at the suggestion of Charles R Greathouse IV

a(14)-a(15) recomputed (were missing) and a(16)-a(19) added by Charles R Greathouse IV, Jan 28 2011, based on the calculations of Jan Feitsma

STATUS

approved

A108797

Number of base-2 strong pseudoprimes (A001262) less than 2^n.

+10
3

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 4, 6, 7, 11, 18, 24, 34, 49, 75, 104, 147, 210, 296, 409, 552, 734, 981, 1311, 1736, 2314, 3093, 4139, 5511, 7396, 9835, 13106, 17493, 23270, 31115, 41664, 55763, 74739, 100342, 134559, 180725, 243566, 327731, 441270, 594585, 803252, 1085426, 1468777, 1988905, 2697846, 3662239, 4976375, 6767707, 9212942, 12552513, 17114780, 23355139, 31894014

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

LINKS

Table of n, a(n) for n=1..64.

Eric Weisstein's World of Mathematics, Strong Pseudoprime

Jan Feitsma and William Galway, Tables of pseudoprimes and related data

EXAMPLE

a(12)=3 since 2047, 3277 and 4033 are the 2-SPSPs less than 4096.

CROSSREFS

Cf. A001262, A055552, A208276

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Dec 03 2005

EXTENSIONS

a(24)-a(64) from Charles R Greathouse IV, Jan 28 2011, based on the calculations of Jan Feitsma.

a(1)=...=a(10)=0 prepended by Max Alekseyev, Apr 23 2013

STATUS

approved

A055551

Number of base-2 Euler-Jacobi pseudoprimes (A047713) less than 10^n.

+10
1

0, 0, 1, 12, 36, 114, 375, 1071, 2939, 7706, 20417, 53332, 139597, 364217, 957111, 2526795, 6725234, 18069359, 48961462

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Pomerance et al. gave the terms a(3)-a(10). Pinch gave the terms a(4)-a(13), but a(13)=124882 was wrong. He later calculated the correct value, which appears in Guy's book. - Amiram Eldar, Nov 08 2019

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, section A12, p. 44.

LINKS

Table of n, a(n) for n=1..19.

Jan Feitsma and William F. Galway, Tables of pseudoprimes and related data.

Richard G.E. Pinch, The pseudoprimes up to 10^13, Algorithmic Number Theory, 4th International Symposium, ANTS-IV, Leiden, The Netherlands, July 2-7, 2000, Proceedings, Springer, Berlin, Heidelberg, 2000, pp. 459-473, alternative link.

Carl Pomerance, John L. Selfridge, and Samuel S. Wagstaff, The pseudoprimes to 25*10^9, Mathematics of Computation, Vol. 35, No. 151 (1980), pp. 1003-1026.

Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.

Eric Weisstein's World of Mathematics, Pseudoprime.

EXAMPLE

Below 10^3 there is only one Euler-Jacobi pseudoprime, 561. Therefore a(3) = 1.

MATHEMATICA

ejpspQ[n_] := CompositeQ[n] && PowerMod[2, (n - 1)/2, n] == Mod[JacobiSymbol[2, n], n]; s = {}; c = 0; p = 10; n = 1; Do[If[ejpspQ[n], c++]; If[n > p, AppendTo[s, c]; p *= 10], {n, 1, 1000001, 2}]; s (* Amiram Eldar, Nov 08 2019 *)

CROSSREFS

Cf. A047713, A055550, A055552.

KEYWORD

nonn,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

a(13) corrected and a(14)-a(19) added by Amiram Eldar, Nov 08 2019 (calculated from Feitsma & Galway's tables)

STATUS

approved

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