A001262 -id:A001262 - cp's OEIS frontend

A376255 Strong pseudoprimes to base 2 which are multiples of 3.

5455590801, 186082586691, 719565156651, 2266886795811, 2307208917651, 2641415618451, 3465253618401, 5115406263771, 6300800933001, 26054428010601, 44612081317851, 73150641067251, 85382703388851, 92522271877491, 97971365867331, 133539279298491, 167608677353601

Offset: 1

Shyam Sunder Gupta, Sep 17 2024

nonn

			a(1) = 5455590801 since it is the first Strong pseudoprimes to base 2 which is divisible by 3.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..299

Intersection of A001262 and A008585.

A376304 Strong pseudoprimes to base 2 that are not squarefree.

Original entry on oeis.org

1194649, 12327121, 3914864773, 5654273717, 26092328809, 58706246509, 74795779241, 237865367741, 467032496113, 601401837037, 1101047056201, 1629827375177, 2327330361721, 3427506518801, 3950198906473, 6151420925105, 7816904988985, 16034307692677

Offset: 1

Shyam Sunder Gupta, Sep 20 2024

nonn

The intersection of A013929 and A001262.

All members of this sequence are divisible by a square of a Wieferich prime.

			a(5) = 26092328809 = 21841 * 1093^2 is a strong pseudoprime that is not squarefree.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..626

Cf. A001262, A013929, A158358, A001220.

A209395 Strong pseudoprimes to bases 19, 23 and 29.

Original entry on oeis.org

4224533, 5903497, 16462297, 22028203, 44068001, 336273211, 1067437801, 1813073653, 1876485691, 1894909141, 2072488771, 2458231903, 2791053541, 2827961221, 3733646491, 4333572253

Offset: 1

R. J. Mathar, Mar 07 2012

nonn

Intersection of A020245, A020249 and A020255.

Cf. A074773, A188755.

PARI
```
/* see A001262, A020237, A020239 */
```

A230811 Smallest starting number for n consecutive odd numbers that are primes or strong pseudoprime (base 2).

Original entry on oeis.org

3, 3, 3, 1640293473202755797

Offset: 1

Shyam Sunder Gupta, Oct 30 2013

It is conjectured that there can be at most one more term, a(5).

			First and smallest occurrence of n, n >= 1, consecutive odd numbers that are primes or strong pseudoprime (base 2):
a(1) = 3: (3) ;
a(2) = 3: (3, 5) ;
a(3) = 3: (3, 5, 7);
a(4) = 1640293473202755797: (1640293473202755797, 1640293473202755799, 1640293473202755801, 1640293473202755803); (1640293473202755801 is strong pseudoprime base 2 and others are prime);

Cf. A001567, A001262, A230667, A230810.

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

A329223 Poulet numbers (Fermat pseudoprimes to base 2) that are congruent to either 3 or 27 (mod 80) and each prime factor is congruent to 3 mod 80.

Original entry on oeis.org

51962615262396907, 330468624532072027, 2255490055253468347, 18436227497407654507

Offset: 1

Daniel Suteu, Nov 08 2019

If a term of this sequence is also a Carmichael number (A002997) and a Lucas-Carmichael number (A006972), then it would be a counterexample to Agrawal's conjecture, as Hendrick Lenstra and Carl Pomerance showed.
330468624532072027 is the only Carmichael number below 2^64 that is a term of this sequence. However, it is not a Lucas-Carmichael number.
The sequence also includes: 68435117188079800987, 164853581396047908970027, 522925572082528736632187, 1820034970687975620484907, 4263739170243679206753787, 4360728281510798266333387, 28541906071781213329174507, 33833150661360980271172507, 84444874320158644422192427, 175352076630428496579381067, 270136290676063386556053067, 615437738523352001584590187, 3408560627000081376639770587, 11260257876970792445537580187.
No term with 5 prime factors (which would be congruent to 3 mod 80) is known to the author.
Are all terms also strong pseudoprimes to base 2 (A001262)?

			51962615262396907 is a term because it is a Fermat pseudoprime to base 2 and it is congruent to 27 (mod 80) and all of its prime factors (643, 154723, 522306163) are congruent to 3 mod 80.

H. W. Lenstra, and Carl Pomerance, Remarks on Agrawal's conjecture, American Institute of Mathematics (2003), pp. 30-32.
Tomáš Váňa, Agrawal's Conjecture and Carmichael Numbers, student scientific conference, pp. 13-22.
Wikipedia, Agrawal's conjecture

Cf. A001567.

isok(n) = ((n%80==3) || (n%80==27)) && (Mod(2, n)^(n-1) == 1) || return(0); my(f=factor(n)[,1]); (#f > 1) && (#select(p->p%80==3, f) == #f);

A361256 Smallest base-n strong Fermat pseudoprime with n distinct prime factors.

Original entry on oeis.org

2047, 8911, 129921, 381347461, 333515107081, 37388680793101, 713808066913201, 665242007427361, 179042026797485691841, 8915864307267517099501, 331537694571170093744101, 2359851544225139066759651401, 17890806687914532842449765082011

Offset: 2

Daniel Suteu, Mar 06 2023

nonn

Main diagonal of A360184.

Cf. A001262, A180065, A271874, A360184.

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));
a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);

A376473 Numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2) and 2^(k-1) =/= 1 (mod k).

Original entry on oeis.org

951481, 2215441, 28758601, 81844921, 1221936841, 10370479321, 16287076081, 26946809137, 33663998161, 35094800881, 134619011281, 188455112353, 299226038833, 314240366881, 383116075201, 594981050401, 1230227375833, 1572186445201, 2096189123113, 2377714473001

Offset: 1

Thomas Ordowski, Sep 24 2024

nonn

The terms k of A374953 for which A002326((k-1)/2) is odd.
Numbers k in A376253 that are not strong pseudoprimes to base 2.
Every term of this sequence must have a Wieferich prime factor (for example, 951481 = 271 * 3511). The Wieferich prime 1093 cannot divide such a number (see A374953).

Subsequence of A374953.

Cf. A001220, A002326, A001262, A001567, A376253.

q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k - 1, m] == 1]; Select[Range[1, 2300000, 2], PowerMod[2, # - 1, #] != 1 && q[#] &] (* Amiram Eldar, Sep 24 2024 *)

is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, k)^(k-1) != 1 && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 24 2024

list(lim)=my(v=List()); if(lim>3<<64, warning("May miss multiples of Wieferich primes > 2^64.")); forstep(n=10533,lim,7022, if(Mod(2, znorder(Mod(2, n^2)))^(n-1) == 1 && Mod(2,n)^n != 2, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Sep 24 2024

More terms from Amiram Eldar, Sep 24 2024

cp's OEIS Frontend

A376255 Strong pseudoprimes to base 2 which are multiples of 3.

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A376304 Strong pseudoprimes to base 2 that are not squarefree.

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A209395 Strong pseudoprimes to bases 19, 23 and 29.

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A230811 Smallest starting number for n consecutive odd numbers that are primes or strong pseudoprime (base 2).

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

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A329223 Poulet numbers (Fermat pseudoprimes to base 2) that are congruent to either 3 or 27 (mod 80) and each prime factor is congruent to 3 mod 80.

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A361256 Smallest base-n strong Fermat pseudoprime with n distinct prime factors.

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A376473 Numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2) and 2^(k-1) =/= 1 (mod k).

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