A001567 -id:A001567 - cp's OEIS frontend

A208276 Number of Poulet numbers (or pseudoprimes to base 2, A001567) less than 2^n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 8, 13, 19, 32, 45, 64, 89, 124, 175, 251, 361, 502, 693, 944, 1264, 1713, 2361, 3169, 4232, 5749, 7750, 10403, 14011, 18667, 24958, 33389, 44540, 59565, 79343, 105659, 141147, 188231, 250568, 333737, 445316, 593366, 792172, 1059097, 1416055, 1893726, 2532703, 3390284, 4540673, 6086093, 8167163, 10964612, 14731767, 19806649, 26651383, 35893886, 48374139, 65247459, 88069251, 118968378

Offset: 1

Washington Bomfim, Feb 25 2012

nonn

Jan Feitsma, The pseudoprimes below 2^64 - statistics
Jan Feitsma and William Galway, Tables of pseudoprimes and related data

Cf. A001567, A055550, A108797, A225005

count=0;for(e=1,32,forcomposite(n=2^(e-1),2^e-1,if(n%2 && Mod(2,n)^(n-1)==1,count++)); print1(count", ")); \\ Hans Loeblich, May 15 2019

a(50)-a(64) from Feitsma's website, added by Max Alekseyev, Apr 23 2013

A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

31417, 130561, 41541241, 208969201, 224074369, 392099401, 851934601, 5186365801, 33847795741, 65096562721, 91625968981, 104070261901, 111794455501, 165814673473, 180007280041, 184020124201, 276032114281, 408164260141, 620052300421, 1240013784241, 1244667503017

Offset: 1

Max Alekseyev, Apr 25 2018

nonn

Numbers (k+1)/2 are listed in A298758.

Amiram Eldar, Table of n, a(n) for n = 1..3031 (terms below 2^64)

Subsequence of each of A001567, A216822, and A217465.

Cf. A005382, A298758, A303447, A303448.

Select[Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]], !PrimeQ[(#+1)/2] && PowerMod[2, (#-1)/2, (#+1)/2] == 1 &] (* Amiram Eldar, Nov 09 2023 *)

isF(n) = {Mod(2, n)^n==2 && !isprime(n) && n>1};
isok(n) = (n%2) && isF(n) && isF((n+1)/2); \\ Michel Marcus, Apr 26 2018

a(n) = 2*A298758(n) - 1.

A374027 Lexicographically earliest sequence of numbers whose partial products are all Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

341, 41, 61, 181, 721, 3061, 6121, 9181, 27541, 36721, 91801, 100981, 238681, 21242521, 67665781, 477361, 48690721, 7160401, 76377601, 35802001, 83394792001, 7500519001, 60004152001, 3420236664001, 1380095496001, 13110907212001, 56583915336001, 128003857254001

Offset: 1

Amiram Eldar, Jun 26 2024

nonn

			The partial products begin with 341 = A001567(1), 341 * 41 = 13981 = A001567(29), 341 * 41 * 61 = 852841 = A001567(234), 341 * 41 * 61 * 181 = 154364221 = A001567(2509), ... .

Daniel Suteu, Table of n, a(n) for n = 1..47
Index entries for sequences related to pseudoprimes.

Cf. A001567, A374028, A374029.

pspQ[n_] := PowerMod[2, n - 1, n] == 1; a[1] = 341; a[n_] := a[n] = Module[{k = 3, r = Product[a[i], {i, 1, n - 1}]}, While[!pspQ[k*r], k+=2]; k]; Array[a, 8]

ispsp(n) = Mod(2, n)^(n-1) == 1;
lista(len) = {my(prd = 1, c = 0, k = 341); while(c < len, while(!ispsp(prd * k), k += 2); prd *= k; print1(k,", "); c++; k = 3);}

my(S=List(341),base=2); my(m = vecprod(Vec(S))); my(L = znorder(Mod(base, m))); print1(S[1], ", "); while(1, forstep(k=lift(1/Mod(m, L)), oo, L, if(gcd(m,k) == 1 && k > 1 && base % k != 0, if((m*k-1) % znorder(Mod(base, k)) == 0, print1(k, ", "); listput(S, k); L = lcm(L, znorder(Mod(base, k))); m *= k; break)))); \\ Daniel Suteu, Jun 30 2024

a(21)-a(28) from Daniel Suteu, Jun 30 2024

A374028 Lexicographically earliest sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

11, 31, 41, 61, 181, 54001, 6841, 54721, 110017981, 13681, 20521, 61561, 123121, 225721, 246241, 205201, 410401, 1128601, 513001, 3078001, 4617001, 73442619001, 96993612810001, 55404001, 7188669001, 16773561001, 67094244001, 821904489001, 29370505311001

Offset: 1

Amiram Eldar, Jun 26 2024

nonn

			The partial products begin with 11, 11 * 31 = 341 = A001567(1), 11 * 31 * 41 = 13981 = A001567(29), 11 * 31 * 41 * 61 = 852841 = A001567(234), 11 * 31 * 41 * 61 * 181 = 154364221 = A001567(2509), ... .

Daniel Suteu, Table of n, a(n) for n = 1..43
Index entries for sequences related to pseudoprimes.

Cf. A001567, A374027, A374029.

pspQ[n_] := PowerMod[2, n - 1, n] == 1; a[1] = 11; a[n_] := a[n] = Module[{p = 2, r = Product[a[i], {i, 1, n - 1}]}, While[! pspQ[p*r], p = NextPrime[p]]; p]; Array[a, 10]

ispsp(n) = Mod(2, n)^(n-1) == 1;
lista(len) = {my(prd = 1, c = 0, k = 11); while(c < len, while(!ispsp(prd * k), k = nextprime(k+1)); prd *= k; print1(k,", "); c++; k = 3);}

my(P=List(11), base=2); my(m = vecprod(Vec(P))); my(L = znorder(Mod(base, m))); print1(P[1], ", "); while(1, forstep(p=lift(1/Mod(m, L)), oo, L, if(isprime(p) && m % p != 0 && base % p != 0, if((m*p-1) % znorder(Mod(base, p)) == 0, print1(p, ", "); listput(P, p); L = lcm(L, znorder(Mod(base, p))); m *= p; break)))); \\ Daniel Suteu, Jun 30 2024

a(23)-a(29) from Daniel Suteu, Jun 30 2024

A374029 Lexicographically earliest strictly increasing sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).

Original entry on oeis.org

11, 31, 41, 61, 181, 54001, 54721, 61561, 123121, 225721, 246241, 430921, 523261, 800281, 2400841, 9603361, 28810081, 76826881, 96033601, 15909022209601, 133133396385601, 5791302742773601, 15443473980729601, 61773895922918401, 386086849518240001, 13706083157897520001

Offset: 1

Amiram Eldar, Jun 26 2024

nonn

Are all the terms of the form 10*k+1?

			The partial products begin with 11, 11 * 31 = 341 = A001567(1), 11 * 31 * 41 = 13981 = A001567(29), 11 * 31 * 41 * 61 = 852841 = A001567(234), 11 * 31 * 41 * 61 * 181 = 154364221 = A001567(2509), ... .

Daniel Suteu, Table of n, a(n) for n = 1..28
Index entries for sequences related to pseudoprimes.

Cf. A001567, A374027, A374028.

pspQ[n_] := PowerMod[2, n - 1, n] == 1; a[1] = 11; a[n_] := a[n] = Module[{p = NextPrime[a[n-1]], r = Product[a[i], {i, 1, n - 1}]}, While[! pspQ[p*r], p = NextPrime[p]]; p]; Array[a, 10]

ispsp(n) = Mod(2, n)^(n-1) == 1;
lista(len) = {my(prd = 1, c = 0, k = 11); while(c < len, while(!ispsp(prd * k), k = nextprime(k+1)); prd *= k; print1(k,", "); c++);}

my(P=List(11), base=2); my(m = vecprod(Vec(P))); my(L = znorder(Mod(base, m))); print1(P[1], ", "); while(1, forstep(p=lift(1/Mod(m, L)), oo, L, if(p > P[#P] && isprime(p) && base % p != 0, if((m*p-1) % znorder(Mod(base, p)) == 0, print1(p, ", "); listput(P, p); L = lcm(L, znorder(Mod(base, p))); m *= p; break)))); \\ Daniel Suteu, Jun 30 2024

a(20)-a(26) from Daniel Suteu, Jun 30 2024

A074380 Sarrus numbers n (A001567) which satisfy mu(n) = -1 and which are not Super-Poulet numbers (A050217).

Original entry on oeis.org

561, 645, 1105, 1729, 1905, 2465, 2821, 4371, 6601, 8481, 8911, 10585, 12801, 13741, 13981, 15841, 16705, 25761, 29341, 30121, 30889, 33153, 34945, 41665, 46657, 52633, 57421, 68101, 74665, 83665, 87249, 88561, 91001, 93961, 113201

Offset: 1

Jani Melik, Sep 24 2002

nonn

Some of these are Carmichael numbers, A002997: 561, 1105, 1729, ....

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

Cf. A001567, A050217, A002997.

Select[ Range[115000], !PrimeQ[ # ] && OddQ[ # ] && PowerMod[2, # - 1, # ] == 1 && Union[ PowerMod[2, Drop[Divisors[ # ], 1], # ]] != {2} && MoebiusMu[ # ] != 1 &]

is(n)=if(isprime(n) || Mod(2,n)^(n-1)!=1 || moebius(n)>=0, return(0)); fordiv(n, d, if(Mod(2, d)^d!=2, return(1))); 0 \\ Charles R Greathouse IV, Sep 01 2016

Edited and extended by Robert G. Wilson v, Oct 03 2002

A212601 Intersection of A001567 and A212502.

Original entry on oeis.org

4033, 6601, 8321, 15841, 25761, 29341, 41041, 46657, 75361, 115921, 162401, 172081, 252601, 294409, 314821, 332949, 401401, 410041, 488881, 530881, 552721, 642001, 721801, 873181, 934021, 1004653, 1207361, 1461241, 1876393, 1909001, 2081713, 2085301, 2113921

Offset: 1

José María Grau Ribas, May 22 2012

nonn

Only 1 (mod 4) numbers have been found.

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

Cf. A201629.
Cf. A001567 (pseudoprimes to base 2).
Cf. A212502 (pseudoprimes to base 1+2i).

t[n_] := Which[Mod[n, 4] == 3, n + 1, Mod[n, 4] == 1, n - 1,  True,  n]; Select[1 + Range[99000], PowerMod[2, # - 1, #] == 1 && !PrimeQ[#] && Im[PowerMod[1 + 2I, t[#], #]] == 0 &]

A227905 Numbers of the form 4k+3 (A004767) that are Lucas pseudoprimes and Fermat pseudoprimes to base 2 (intersection of A005845 and A001567).

Original entry on oeis.org

741751, 1024651, 5481451, 31150351, 109437751, 139952671, 178482151, 284301751, 383425351, 395044651, 407282851, 417027451, 498706651, 582799951, 612816751, 620072251, 652969351, 738820351, 977755351, 1126587151, 1204176751, 1397357851, 1588247851, 1789167931

Offset: 1

José María Grau Ribas, Oct 12 2013

nonn

This sequence uses the Bruckman definition of "Lucas pseudoprime". There are 400,114 examples less than 2^64. - Dana Jacobsen, Jan 07 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
W. Galway, Tables of pseudoprimes and related data [Includes a file with base-2 Fermat pseudoprimes up to 2^64.]

Cf. A004767 (4n+3).

Cf. A001567 (Fermat pseudoprimes to base 2), A005845 (Lucas pseudoprimes).

Select[4*Range[8000000]+3, CompositeQ[#] && PowerMod[2, (# - 1), # ] == 1 && Divisible[LucasL[#]-1, #] &] (* Amiram Eldar, Jun 27 2019 *)

perl -Mntheory=:all -nE 'chomp; say if ($%4)==3 && (lucas_sequence($,1,-1,$))[1] == 1' psps-below-2-to-64.txt  # _Dana Jacobsen, Jan 07 2015

perl -Mntheory=:all -E 'foroddcomposites { say if $%4 == 3 && ispseudoprime($,2) && (lucas_sequence($,1,-1,$))[1] == 1 } 1e14'  # Dana Jacobsen, Jan 10 2015

More terms from Dana Jacobsen, Jan 07 2015

a(16)-a(24) from Amiram Eldar, Jun 27 2019

A346568 Fermat pseudoprimes to base 2 (A001567) k such that A003961(k) is also a Fermat pseudoprime to base 2.

Original entry on oeis.org

710533, 915981, 1293337, 2134277, 3542533, 13747361, 161216021, 206304961, 284166877, 748419127, 968283247, 1265740717, 2582246701, 4297753027, 10891270501, 11176136947, 11273608417, 11606768801, 12169503061, 13321141597, 14241379237, 17005529227, 19600350001

Offset: 1

Amiram Eldar, Jul 23 2021

nonn

a(1) = 710533 = 487 * 1459 has 2 distinct prime divisors.
a(2) = 915981 = 3 * 11 * 41 * 677 has 4 distinct prime divisors.
a(58) = 176529862601 = 2141 * 6421 * 12841 is the least term with 3 distinct prime divisors.
a(6884) = 15314196673937701 = 19 * 31 * 41 * 71 * 109 * 281 * 331 * 881 is the least term with 8 distinct prime divisors.
a(111) = 619303584901 is the least term k such that A003961(k) is also a term.
a(30430) = 507728732614597601 is the least term k such that both A003961(k) and A003961(A003961(k)) are also terms.

			710533 = 487 * 1459 is a term since it is a Fermat pseudoprime to base 2, and A003961(710533) = 491 * 1471 = 722261 is also a Fermat pseudoprime to base 2.

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

Cf. A001567, A003961.

A346569 is a subsequence.

psp = Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; pspQ[n_] := PowerMod[2, n - 1, n] == 1; Select[psp, pspQ[s[#]] &]

A371759 a(n) is the smallest n-gonal number that is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Original entry on oeis.org

561, 1194649, 7957, 561, 23377, 341, 129889, 1105, 35333, 561, 204001, 31609, 2940337, 1105, 493697, 8481, 13981, 1905, 88561, 41665, 10680265, 1729, 107185, 264773, 449065, 6601, 2165801, 23001, 1141141, 13981, 272251, 4369, 17590957, 15841, 137149, 2821, 561

Offset: 3

Amiram Eldar, Apr 05 2024

nonn

The corresponding indices of the n-gonal numbers are 33, 1093, 73, 17, 97, ... (A371760).

			a(4) = A001220(1)^2 = 1093^2 = 1194649. The only known square base-2 pseudoprimes are the squares of the Wieferich primes (A001220).

Amiram Eldar, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Polygonal Number.
Eric Weisstein's World of Mathematics, Poulet Number.
Wikipedia, Polygonal number.
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A001220, A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371760.

p[k_, n_] := ((n-2)*k^2 - (n-4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; p[k, n]]; Array[a, 50, 3]

p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); p(k, n);}

a(n) = ((n-2)*k^2 - (n-4)*k)/2, where k = A371760(n).

cp's OEIS Frontend

A208276 Number of Poulet numbers (or pseudoprimes to base 2, A001567) less than 2^n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A303531 Numbers k such that both k and (k+1)/2 are Fermat pseudoprimes to base 2 (A001567).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374027 Lexicographically earliest sequence of numbers whose partial products are all Fermat pseudoprimes to base 2 (A001567).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A374028 Lexicographically earliest sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A374029 Lexicographically earliest strictly increasing sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A074380 Sarrus numbers n (A001567) which satisfy mu(n) = -1 and which are not Super-Poulet numbers (A050217).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A212601 Intersection of A001567 and A212502.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A227905 Numbers of the form 4k+3 (A004767) that are Lucas pseudoprimes and Fermat pseudoprimes to base 2 (intersection of A005845 and A001567).

Views