A001262 -id:A001262 - cp's OEIS frontend

A112453 Strong pseudoprimes (base-2) equal to product of 6 primes not necessarily distinct.

10761055201, 26244332101, 49430153305, 53125756201, 247173316831, 276228879031, 360187750105, 516311394481, 558417883465, 605526139765, 765771364801, 823307740165, 877094650081, 904455227845, 1113985668601, 1237898798461, 1324214598565, 1353668162185, 1370368908805

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

			a(1) = 10761055201 = 13*29*41*61*101*113.

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

Intersection of A046306 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A113639 Pandigital strong pseudoprimes (base-2).

Original entry on oeis.org

13069482857, 16974853201, 18260734519, 23759084161, 24785139601, 25089467413, 29384567041, 38706945421, 41390257861, 49065821737, 65829130417, 67903858241, 76953210841, 98642074513, 100695314287, 106237698451, 106751489293, 110586343297, 113646829507, 116548907329

Offset: 1

Shyam Sunder Gupta, Jan 15 2006

			a(1) = 13069482857 is a term because 13069482857 is a strong pseudoprime (base-2) and also contains all digits from 0 to 9 so pandigital also.

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

Intersection of A171102 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A113640 Zeroless pandigital strong pseudoprimes (base-2).

Original entry on oeis.org

6913548721, 11396574289, 13842496537, 14872531669, 16952474813, 16963825147, 16974152381, 18632427859, 27569184173, 28239546721, 47569352881, 54689213377, 54987116329, 68719214593, 79633524181, 93541462873, 95126134837, 118214685793, 121899457633, 123572846953

Offset: 1

Shyam Sunder Gupta, Jan 15 2006

			a(1) = 6913548721 is a term because 6913548721 is a strong pseudoprime (base-2) and also contains all digits from 1 to 9 so it is also zeroless pandigital.

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

Intersection of A050289 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A140658 Overpseudoprimes to bases 2 and 3.

Original entry on oeis.org

5173601, 13694761, 16070429, 27509653, 54029741, 66096253, 102690677, 117987841, 193949641, 206304961, 314184487, 390612221, 393611653, 717653129, 960946321, 1157839381, 1236313501, 1481626513, 1860373241, 1921309633, 2217879901, 2412172153, 2626783921

Offset: 1

Vladimir Shevelev, Jul 10 2008

nonn

From the first 19 strong pseudoprimes to bases 2 and 3 (A072276) only 6 are overpseudoprimes to the same bases.

Daniel Suteu, Table of n, a(n) for n = 1..10000 (terms 1..105 from Amiram Eldar)
Index entries for sequences related to pseudoprimes

Intersection of A141232 and A141350; subsequence of A072276.

Cf. A001262, A020229.

More terms from Amiram Eldar, Jun 24 2019

A143012 Numbers of the form (4^p + 2^p + 1)/7, where p > 3 is prime.

Original entry on oeis.org

151, 2359, 599479, 9588151, 2454285751, 39268347319, 10052678938039, 41175768098368951, 658812288653553079, 2698495133088002829751, 690814754065816531725751, 11053036065049294753459639, 2829577232652317876553477559, 11589948344943812957569751412151

Offset: 1

Vladimir Shevelev, Jul 15 2008, Jul 21 2008

nonn

If 8^p-1 is squarefree then the terms of the sequence are either primes (A000040) or overpseudoprimes to base 2 (A141232). In particular, composite numbers of the sequence are strong pseudoprimes to base 2 (A001262). E.g., a(5)=2454285751 is A001262(1828).

V. Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.

Cf. A001262, A141232, A000040, A126614.

p:=ithprime: seq((4^p(n)+2^p(n)+1)*1/7, n=3..14); # Emeric Deutsch, Aug 16 2008

(4^#+2^#+1)/7&/@Prime[Range[3,30]] (* Harvey P. Dale, Feb 19 2013 *)

Extended by Emeric Deutsch, Aug 16 2008

More terms from Harvey P. Dale, Feb 19 2013

A180066 Smallest prime factor of 2-strong pseudoprimes.

Original entry on oeis.org

23, 29, 37, 31, 53, 7, 13, 127, 157, 7, 97, 5, 61, 53, 149, 151, 229, 137, 157, 103, 43, 41, 157, 233, 151, 277, 13, 131, 313, 277, 11, 233, 157, 359, 499, 13, 79, 431, 5, 101, 397, 661, 307, 479, 313, 331, 13, 251, 11, 601, 103, 1093, 449, 409, 733, 571, 499, 673, 829

Offset: 1

Kevin Batista (kevin762401(AT)yahoo.com), Aug 09 2010

nonn

			The 8th pseudoprime, 42799, is divisible by 127.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to pseudoprimes

a(n) = A020639(A001262(n)).

b-file and edits from Charles R Greathouse IV, Aug 28 2010

A273618 Numbers m = 2*k+1 where k is odd with the property that 3^k mod m = 1 and k^k mod m = 1.

Original entry on oeis.org

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

Alzhekeyev Ascar M, May 26 2016

nonn

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.
The subsequence of these composites begins: 143193768587, 440097066011, 1188059560451, 1392770336147, 1640446291859, 2526966350771, 3639120872171, 3989703695867, 4202422108523, ....
Perhaps this sequence contains all the terms of the sequence A107007 (except 3) or A168539.

			m=131; 131=2*65+1; 3^65 mod 131 = 1 and 65^65 mod 131 = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A176997.

Cf. A001262, A001567, A107007, A168539, A214151.

filter:= proc(n) local k;
  k:= (n-1)/2;
  3 &^ k mod n = 1 and k &^ k mod n = 1
end proc:
select(filter, [seq(i,i=3..3000, 4)]); # Robert Israel, Nov 28 2019

2#+1&/@Select[Range[1,1200,2],PowerMod[3,#,2#+1]==PowerMod[ #,#,2#+1] == 1&] (* Harvey P. Dale, May 05 2022 *)

A288153 Plumb pseudoprimes: odd composites that pass Colin Plumb's extended Euler criterion test.

Original entry on oeis.org

1729, 1905, 2047, 2465, 3277, 4033, 4681, 8321, 12801, 15841, 16705, 18705, 25761, 29341, 33153, 34945, 41041, 42799, 46657, 49141, 52633, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 90751, 104653, 113201

Offset: 1

Charles R Greathouse IV, Jun 05 2017

nonn

Suppose n is composite. Then if n = 1 mod 8, it is in the sequence if 2^((n-1)/4) = 1 or -1 mod n; if n = 3 or 5 mod 8, it is in the sequence if 2^((n-1)/2) = -1 mod n; and if n = 3 mod 8, it is in the sequence if 2^((n-1)/2) = 1 mod n.

a(1) = 1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jun 05 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A001567; A001262 is a subsequence.

is(n)=if(n<2 || isprime(n) || n%2==0, return(0)); my(n8=n%8, e=n>>((n8==1)+1), t=Mod(2,n)^e); if(t==1, n8==1 || n8==7, if(t==-1, n8==1 || n8==3 || n8==5, 0))

use Math::Prime::Util ':all'
foroddcomposites { print "$, " if is_euler_plumb_pseudoprime($); } 9,999999;

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]

A376253 Composite numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2).

Original entry on oeis.org

4681, 15841, 42799, 52633, 220729, 647089, 951481, 1082401, 1145257, 1969417, 2215441, 3567481, 4835209, 5049001, 5681809, 6140161, 6334351, 8725753, 10712857, 11777599, 12327121, 13500313, 14709241, 22564081, 22849481, 22953673, 23828017, 27271151, 28758601, 30576151

Offset: 1

Thomas Ordowski, Sep 17 2024

nonn

If 2^(k-1) == 1 (mod k) and 2^(2^(k-1)-1) == 1 (mod k), then 2^(2^(k-1)-1) == 1 (mod k^2). In fact, all such pseudoprimes are strong pseudoprimes to base 2.

Other terms; 951481 = 271*3511, 2215441 = 631*3511, 28758601 = 8191*3511, ... are not Fermat pseudoprimes to base 2, where 3511 is a Wieferich prime. The Wieferich prime 1093 cannot be a factor of these numbers (see A374953).

Cf. A001220, A001262, A001567, A188465, A374841, A374953.

q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k-1, m] == 1]; Select[Range[1, 10^6, 2], CompositeQ[#] && q[#] &] (* Amiram Eldar, Sep 17 2024 *)

is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 17 2024

More terms from Amiram Eldar, Sep 17 2024

cp's OEIS Frontend

A112453 Strong pseudoprimes (base-2) equal to product of 6 primes not necessarily distinct.

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A113639 Pandigital strong pseudoprimes (base-2).

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A113640 Zeroless pandigital strong pseudoprimes (base-2).

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A140658 Overpseudoprimes to bases 2 and 3.

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A143012 Numbers of the form (4^p + 2^p + 1)/7, where p > 3 is prime.

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A180066 Smallest prime factor of 2-strong pseudoprimes.

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A273618 Numbers m = 2*k+1 where k is odd with the property that 3^k mod m = 1 and k^k mod m = 1.

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Maple

Mathematica

A288153 Plumb pseudoprimes: odd composites that pass Colin Plumb's extended Euler criterion test.

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PARI

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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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