A001262 -id:A001262 - cp's OEIS frontend

A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

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, Jul 05 2012

nonn

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.
The subsequence of these composites begins: 1441091, 3587553971, 4528686251, 23260036451, 47535120323, 61070250323, 90474845819, 143193768587, 162016315907, 173868807611, 180998962187, 238364070323, 285370693931, 298577370323, ...
Perhaps this sequence contains all the terms of the sequence A107007 or A168539.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A176997.

Cf. A003629, A006970, A175865.

isA214151 := proc(n)
    if (n mod 6 = 5) and modp(2 &^ ((n-1)/2),n)  = n-1 and modp(3 &^ ((3*n-1)/2),n)  = 3 then
        true;
    else
        false;
    end if;
end proc:
for n from 5 by 6 do
    if isA214151(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 20 2012

Select[Range[5,2500,6],PowerMod[3,(3#-1)/2,#]==3&&PowerMod[2,(#-1)/2,#] == #-1&] (* Harvey P. Dale, Mar 14 2022 *)

for(n=0, 200, b=6*n+5; if(Mod(3, b)^((3*b-1)/2)==3, if(Mod(2, b)^((b-1)/2)==b-1 , print1(b, ", "))));

A230810 Smallest of 3 consecutive odd numbers that are primes or strong pseudoprime to base 2.

Original entry on oeis.org

3, 3465253618399, 374166120095639, 12959269432331237, 44202753561285409, 1640293473202755797, 1640293473202755799, 10623546148468360249

Offset: 1

Shyam Sunder Gupta, Oct 30 2013

Since a(7) = a(6) + 2, the number a(6) = 1640293473202755797 actually starts a sequence of four consecutive odd numbers which are prime or strong pseudoprimes to base 2. - M. F. Hasler, Dec 08 2016

			3465253618399 is in sequence because 3465253618399, 3465253618401, 3465253618403 is a set of 3 consecutive odd number where 3465253618401 is strong pseudoprime base 2 and others are prime.

S. S. Gupta, Can You Find no. 48, Oct. 2016.

Cf. A001262, A001567, A230667, A230808.

Edited by M. F. Hasler, Dec 08 2016

A360184 Square array A(n, k) read by antidiagonals downwards: smallest base-n strong Fermat pseudoprime with k distinct prime factors for k, n >= 2.

Original entry on oeis.org

2047, 15841, 703, 800605, 8911, 341, 293609485, 152551, 4371, 781, 10761055201, 41341321, 129921, 24211, 217, 5478598723585, 12283706701, 9224391, 4382191, 29341, 325, 713808066913201, 1064404682551, 2592053871, 381347461, 3405961, 58825, 65, 90614118359482705

Offset: 2

Daniel Suteu, Mar 04 2023

The array A(n, k) starts as follows:
   k  =   2     3       4         5            6
n = 2: 2047 15841  800605 293609485  10761055201
n = 3:  703  8911  152551  41341321  12283706701
n = 4:  341  4371  129921   9224391   2592053871
n = 5:  781 24211 4382191 381347461   9075517561
n = 6:  217 29341 3405961 557795161 333515107081

Daniel Suteu, Table of n, a(n) for n = 2..137

Cf. A001262, A180065 (row n=2), A271873.

strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;
strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k, e, r) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1 && strong_check(p, base, e, r), my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; strong_check(p, base, e, r) || next; my(z=znorder(Mod(base, p))); gcd(m,z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1, e, r)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); my(res=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 2, k, v, -1))); vecsort(Set(res));
T(n, k) = if(n < 2, return()); my(x=vecprod(primes(k)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, k, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);
print_table(n, k) = for(x=2, n, for(y=2, k, print1(T(x, y), ", ")); print(""));
for(k=2, 9, for(n=2, k, print1(T(n, k-n+2)", ")));

A020240 Strong pseudoprimes to base 14.

Original entry on oeis.org

15, 841, 2743, 3277, 5713, 6541, 7171, 9073, 18721, 21667, 22261, 23521, 38221, 38417, 40501, 41371, 49471, 58255, 68401, 71969, 79003, 88381, 91681, 95033, 96049, 97469, 110309, 115417, 119341, 124609, 134413, 141373, 165677, 211951, 226801

Offset: 1

David W. Wilson

nonn

R. J. Mathar, Table of n, a(n) for n = 1..218
Index entries for sequences related to pseudoprimes

Cf. A001262 (base 2), A020233 (base 7)

A020250 Strong pseudoprimes to base 24.

Original entry on oeis.org

25, 175, 553, 949, 1541, 1975, 2701, 4537, 6931, 7501, 13825, 21349, 25273, 25477, 29341, 38323, 42121, 42127, 47617, 49141, 52417, 63701, 80137, 83333, 85609, 94753, 105121, 113527, 128143, 144841, 167869, 169027, 179521, 181351, 197209, 201373

Offset: 1

David W. Wilson

nonn

R. J. Mathar, Table of n, a(n) for n = 1..203
Index entries for sequences related to pseudoprimes

strongPseudoprimeQ[b_, n_] := Module[{rems = Table[PowerMod[b, (n - 1)/2^expo, n], {expo, 0, IntegerExponent[n - 1, 2]}]}, (rems[[-1]] == 1 || MemberQ[rems, n - 1]) && PowerMod[b, n - 1, n] == 1]; max = 5000; Select[Complement[Range[2, max], Prime[Range[PrimePi[max]]]], strongPseudoprimeQ[24, #] &] (* Alonso del Arte, Aug 03 2018 *)

is(n, a=24)= (bittest(n, 0) && !isprime(n) && n>8) || return; my(s=valuation(n-1, 2)); if(1==a=Mod(a, n)^(n>>s), return(1)); while(a!=-1 && s--, a=a^2); a==-1 \\ Felix Fröhlich, Aug 03 2018, adapted from code by M. F. Hasler in A001262

A020261 Strong pseudoprimes to base 35.

Original entry on oeis.org

9, 1261, 2701, 2871, 5083, 11041, 13051, 15051, 16441, 16589, 22681, 23959, 31201, 31621, 38081, 39091, 44749, 49601, 49771, 50737, 54223, 74023, 79381, 100081, 113527, 117157, 151061, 151313, 154201, 160147, 169801, 203841, 282133, 304057

Offset: 1

David W. Wilson

nonn

R. J. Mathar, Table of n, a(n) for n = 1..300
Index entries for sequences related to pseudoprimes

for n from 2 do
    if isStrongPsp(n,35) then # calls code in A001262 and A007814
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 30 2024

A020287 Strong pseudoprimes to base 61.

Original entry on oeis.org

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

A112450 Strong pseudoprimes (base-2) equal to product of 3 primes not necessarily distinct.

Original entry on oeis.org

15841, 29341, 52633, 74665, 252601, 314821, 476971, 635401, 1004653, 1023121, 1907851, 1909001, 2419385, 2953711, 3581761, 4335241, 4682833, 5049001, 5444489, 5599765, 5681809, 9069229, 13421773, 15247621, 15510041, 15603391, 17509501, 26254801, 26758057, 27966709

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

The term a(11)=1907851 is also a strong pseudoprime to base 5, cf. A020231. M. F. Hasler, Aug 16 2012

			a(1) = 15841 = 7*31*73.

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

Intersection of A014612 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A112451 Strong pseudoprimes (base-2) equal to product of 4 primes not necessarily distinct.

Original entry on oeis.org

800605, 4502485, 5310721, 7177105, 9995671, 10655905, 11473885, 13216141, 15698431, 21417991, 22564081, 29878381, 36338653, 38624041, 43397551, 45485881, 45769645, 48369727, 51340807, 57561085, 77576401, 81445585, 86067241, 90626185, 104852881, 112792519, 115007581

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

			a(1) = 800605 = 5*13*109*113.

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

Intersection of A014613 and A001262.

More terms from Amiram Eldar, Nov 10 2019

A112452 Strong pseudoprimes (base-2) equal to product of 5 primes not necessarily distinct.

Original entry on oeis.org

293609485, 440707345, 606057985, 831807145, 958970545, 1816572745, 2395916965, 2708826841, 2907393385, 3246238801, 4340265931, 4953963781, 5949820045, 6845182669, 9580649065, 10121349421, 11360308765, 11892462985, 13560708421, 16034618701, 16720656121, 16765381165

Offset: 1

Shyam Sunder Gupta, Dec 12 2005

nonn

			a(1) = 293609485 = 5*13*29*109*1429.

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

Intersection of A014614 and A001262.

More terms from Amiram Eldar, Nov 10 2019

cp's OEIS Frontend

A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A230810 Smallest of 3 consecutive odd numbers that are primes or strong pseudoprime to base 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A360184 Square array A(n, k) read by antidiagonals downwards: smallest base-n strong Fermat pseudoprime with k distinct prime factors for k, n >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A020240 Strong pseudoprimes to base 14.

Views

Author

Keywords

Links

Crossrefs

A020250 Strong pseudoprimes to base 24.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

A020261 Strong pseudoprimes to base 35.

Views

Author

Keywords

Links

Programs

Maple

A020287 Strong pseudoprimes to base 61.

Views

Author

Keywords

Comments

Links

A112450 Strong pseudoprimes (base-2) equal to product of 3 primes not necessarily distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A112451 Strong pseudoprimes (base-2) equal to product of 4 primes not necessarily distinct.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A112452 Strong pseudoprimes (base-2) equal to product of 5 primes not necessarily distinct.

Views

Author

Keywords

Examples

Links