A141232 -id:A141232 - cp's OEIS frontend

A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

178956971, 45812984491, 733007751851, 46912496118443, 3002399751580331, 192153584101141163, 49191317529892137643, 787061080478274202283, 3148244321913096809131, 3223802185639011132549803

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

If p-1 is squarefree, the terms are overpseudoprimes (see A141232). - Vladimir Shevelev, Jul 15 2008

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A000979, A000978, A124400, A126614, A127956, A127957.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 2, 30}]; a
Select[(2^Prime[Range[2,30]]+1)/3,CompositeQ] (* Harvey P. Dale, Feb 04 2015 *)

A141350 Overpseudoprimes to base 3.

Original entry on oeis.org

121, 703, 3281, 8401, 12403, 31621, 44287, 47197, 55969, 74593, 79003, 88573, 97567, 105163, 112141, 211411, 221761, 226801, 228073, 293401, 313447, 320167, 328021, 340033, 359341, 432821, 443713, 453259, 478297, 497503, 504913, 679057, 709873, 801139, 867043, 894781, 973241, 1042417

Offset: 1

Vladimir Shevelev, Jun 27 2008, corrected Sep 07 2008

nonn

If h_3(n) is the multiplicative order of 3 modulo n, r_3(n) is the number of cyclotomic cosets of 3 modulo n then, by the definition, n is an overpseudoprime to base 3 if h_3(n)*r_3(n)+1=n. These numbers are in A020229.

In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime of base 3 iff h_3(p_1)=...=h_3(p_k).

Amiram Eldar, Table of n, a(n) for n = 1..10798 (calculated using the b-file at A020229)
V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.
V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv preprint arXiv:1206:0606 [math.NT], 2012. - From _N. J. A. Sloane_, Oct 28 2012
V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.

Cf. A141232, A137576, A001262, A020229, A062117, A006694.

ops3Q[n_] := CompositeQ[n] && GCD[n, 3] == 1 && MultiplicativeOrder[3, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[3, #] &] - 1) + 1 == n; Select[Range[10^6], ops3Q] (* Amiram Eldar, Jun 24 2019 *)

isok(n) = (n!=1) && !isprime(n) && (gcd(n,3)==1) && (znorder(Mod(3,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(3, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018

a(10)-a(38) from Gilberto Garcia-Pulgarin added by Vladimir Shevelev, Feb 06 2012

A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 127, 257, 2047, 8191, 65537, 131071, 262657, 524287, 1082401, 8388607, 536870911, 2147483647, 4294967297, 137438953471, 2199023255551, 4432676798593, 8796093022207, 140737488355327, 9007199254740991, 18014398643699713, 576460752303423487

Offset: 1

Vladimir Shevelev, Jul 15 2008

nonn

Contains Fermat numbers A000215 (p=2) and Mersenne numbers A001348 (N=1). The terms of the sequence are either primes A000040 or overpseudoprimes A141232.

The values of A019320(n) for prime power n, sorted. This sequence is a subsequence of A064896, which means that all terms are sturdy numbers (A125121). It appears that the largest prime factor of each of these numbers is a sturdy prime (A143027). - T. D. Noe, Jul 21 2008

T. D. Noe, Table of n, a(n) for n=1..199
Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.

Cf. A000040, A000215, A001348, A141232.

nmax[p_] := Which[p == 2, 6, p == 3, 4, True, 2];
Reap[Do[If[IntegerQ[k = (2^p^n-1)/(2^p^(n-1)-1)] && k<10^18, Print[{p, n, k}]; Sow[k]], {p, Prime[Range[17]]}, {n, 1, nmax[p]}]][[2, 1]] // Union (* Jean-François Alcover, Dec 10 2018 *)

Definition corrected by and more terms from T. D. Noe, Jul 21 2008

A140803 Numbers of the form (2^(pq)-1) /((2^p-1)(2^q-1)), where p>q are primes.

Original entry on oeis.org

3, 11, 43, 151, 683, 2359, 2731, 43691, 174763, 599479, 2796203, 8727391, 9588151, 178956971, 715827883, 2454285751, 39268347319, 45812984491, 567767102431, 733007751851, 2932031007403, 10052678938039, 46912496118443, 145295143558111, 3002399751580331, 41175768098368951, 192153584101141163

Offset: 1

Vladimir Shevelev, Jul 15 2008, Jul 22 2008; corrected Sep 07 2008

nonn

The sequence contains, in particular, A126614 (q=2) and A143012 (q=3).

If pq-1 is squarefree then the terms of the sequence are either primes or overpseudoprimes to base 2 (see A141232). In particular, they are strong pseudoprimes to base 2 (A001262).

			Entry 3 from (q=2,p=3), entry 11 from (q=2,p=5), entry 43 from (q=2,p=7), entry 151 from (q=3,p=5), entry 683 from (q=2,p=11).

Robert Israel, Table of n, a(n) for n = 1..825
V. Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.
S. Wagstaff, Factorizations of 2^n-1

Cf. A001262, A141232, A126614, A143012.

N:= 100: # to use all (p,q) with p*q < N
Primes:= select(isprime,[$2..floor(N/2)]):
A:= {}:
for i from 1 to nops(Primes) do
  p:= Primes[i];
  Qs:= select(q -> q < N/p, [seq(Primes[j],j=1..i-1)]);
  A:= A union {seq((2^(p*q)-1)/(2^p-1)/(2^q-1),q=Qs)};
od:
A; # Robert Israel, Sep 02 2014

terms = 27; Clear[seq]; seq[m_] := seq[m] = Table[(2^(p q)-1)/((2^p-1) (2^q-1)), {q, Prime[Range[m]]}, {p, Prime[Range[PrimePi[q]+1, terms]]}] // Flatten // Union // PadRight[#, terms]&;
seq[1]; seq[m=2]; While[seq[m] != seq[m-1], m++]; seq[m] (* Jean-François Alcover, Sep 17 2018 *)

a(17) to a(27) from Robert Israel, Sep 03 2014

A141390 Overpseudoprimes to base 5.

Original entry on oeis.org

781, 1541, 5461, 13021, 15751, 25351, 29539, 38081, 40501, 79381, 100651, 121463, 133141, 195313, 216457, 315121, 318551, 319507, 326929, 341531, 353827, 375601, 416641, 432821, 453331, 464881, 498451, 555397, 556421, 753667, 764941, 863329, 872101, 886411

Offset: 1

Vladimir Shevelev, Jun 29 2008

nonn

If h_5(n) is the multiplicative order of 5 modulo n, r_5(n) is the number of cyclotomic cosets of 5 modulo n then, by the definition, n is an overpseudoprime of base 5 if h_5(n)*r_5(n)+1=n. These numbers are in A020231. In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime to base 5 iff h_5(p_1)=...=h_5(p_k). E.g., since h_5(101)=h_5(251)=h_5(401)=25, the number 101*251*401=10165751 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..327 (calculated from the b-file at A020231)
V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.
V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv preprint arXiv:1206:0606, 2012. - From _N. J. A. Sloane_, Oct 28 2012
V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.

Cf. A141232, A141350, A020231, A020229.

ops5Q[n_] := CompositeQ[n] && GCD[n, 5] == 1 && MultiplicativeOrder[5, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[5, #] &] - 1) + 1 == n; Select[Range[6, 10^6], ops5Q] (* Amiram Eldar, Jun 24 2019 *)

isok(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018

Inserted a(2) and a(8) and extended at the suggestion of Gilberto Garcia-Pulgarin by Vladimir Shevelev, Feb 06 2012

A131952 a(n) is the maximal overpseudoprime q to base 2 such that the multiplicative order of 2 mod q equals A143584(n).

Original entry on oeis.org

2047, 8388607, 1082401, 3277, 536870911, 8727391, 4033, 137438953471, 9588151, 2199023255551, 8796093022207, 838861, 14709241, 140737488355327, 65281, 1016801, 2454285751, 13421773, 9007199254740991, 567767102431, 39268347319, 178956971, 576460752303423487, 80581

Offset: 1

Vladimir Shevelev, Aug 26 2008

nonn

Or composite terms of A064078.

			For q=256999, 486737, 2304167 and 536870911, the multiplicative order of 2 mod q is A143584(5) = 29, so a(5) = 536870911.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000

Cf. A143584, A064078, A141232, A122929, A141629, A002326.

for(k=1,200,m=polcyclo(k,2);m/=gcd(m,k);m!=1&&!isprime(m)&&print1(m,", ")) \\ Jeppe Stig Nielsen, Aug 31 2020

More terms from Hugo Pfoertner, Aug 31 2020

A141629 a(n) is the least base-2 overpseudoprime k such that the multiplicative order of 2 mod k equals 8*n+20.

Original entry on oeis.org

3277, 4033, 838861, 8321, 80581, 130561, 104653, 20647621, 280601, 818201, 68719214593, 57646075230342349, 48448661, 1353244757701, 351479006145541, 88357, 390937, 1846171781, 17585969, 9774181, 28147501026509, 3882413703281, 1251949, 9007199388958721

Offset: 1

Vladimir Shevelev, Aug 24 2008

C. Pomerance proved (private correspondence) that for every n>=1 there exists at least one overpseudoprime (a(n)) for which the multiplicative order of 2 mod a(n) equals 8n+20.

a(25) > 2^64. - Amiram Eldar, Nov 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..1000, with zeros for 277 unknown values that are > 2^64.
Vladimir Shevelev, An upper estimate for the overpseudoprime counting function, arXiv:0807.1975 [math.NT], 2008.

Cf. A141232, A122929, A140452, A140797, A140803.

a(4) corrected and a(12)-a(24) added by Amiram Eldar, Nov 09 2023

A143584 Integers that are equal to the multiplicative order of 2 modulo some overpseudoprime to base 2.

Original entry on oeis.org

11, 23, 25, 28, 29, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 79, 81, 82, 83, 84, 87, 88, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112

Offset: 1

Vladimir Shevelev, Aug 25 2008

nonn

A064078(a(n)) is a composite number. The sequence has a positive density since it contains, in particular, numbers of the form 8n+20 for n >= 1 (C. Pomerance, private correspondence). Since, e.g., 38 is not in the sequence, there is not an overpseudoprime m such that ord_m(2)=38.
Phi_{a(n)}(2), the a(n)-th cyclotomic polynomial of x evaluated at x=2 has at least 2 distinct prime factors that are not prime factors of the Phi_k(2) for any positive integer k < a(n). For example, Phi_11(2) = 2^11 - 1 = 2047 = 23 * 89 and Phi_25(2) = 2^20 + 2^15 + 2^10 + 2^5 + 1 = 1082401 = 601 * 1801. Note that p = a(n) is prime if and only if Phi_p(2) = 2^p - 1 is composite. - David Terr, Sep 09 2018
It is easy to prove the statement above. We use the fact that Phi_j(n) and Phi_k(n) are coprime whenever j and k are coprime as well as the fact that an overpseudoprime has at least 2 distinct prime factors. - David Terr, Oct 10 2018
A number k is included iff either 2^k-1 has more than one primitive prime factor (cf. A086251, A161508) or the only primitive prime factor of 2^k-1 is a Wieferich prime (no examples known). - Jeppe Stig Nielsen, Sep 01 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000

Cf. A002326, A014664, A064078, A086251, A122929, A141232, A161508.

Cf. A131952 (for the corresponding maximal overpseudoprimes).

isok(k) = my(m=polcyclo(k,2)); m/=gcd(m,k); m!=1&&!isprime(m) \\ Jeppe Stig Nielsen, Sep 01 2020

Name edited by Michel Marcus, Oct 06 2018
More terms from Michel Marcus, Oct 11 2018
Data for terms >= 100 corrected by Jeppe Stig Nielsen, Sep 01 2020

A192297 Lesser of pseudo twin primes to base 2.

Original entry on oeis.org

561, 643, 645, 1103, 1905, 2465, 2699, 2819, 4369, 4371, 4679, 6599, 10259, 12799, 14489, 16703, 18719, 19949, 23001, 25759, 25761, 29339, 30119, 31607, 33151, 39863, 41039, 42797, 49139, 52631, 55243, 60701, 62743, 68099, 72883, 83663, 85487, 87249, 90749

Offset: 1

Vladimir Shevelev, Oct 11 2011

nonn

We call numbers {k,k+2} pseudo twin primes to base 2 if at least one of them is composite, while 2^(k-1) == 1 (mod k) and 2^(k+1) == 1 mod (k+2).

4369 is the only known term such that both k and k+2 are composite (cf. A173619). - Jianing Song, Nov 20 2021

Amiram Eldar, Table of n, a(n) for n = 1..15587 (terms below 10^12; terms 1..1000 from Alois P. Heinz)

Cf. A001567, A002997, A141232.

Cf. A173619.

a:= proc(n) option remember; local k;
      for k from 2+`if` (n=1, 1, a(n-1)) by 2 while
        isprime(k) and isprime(k+2) or
          (2&^(k-1) mod k)<>1 or (2&^(k+1) mod (k+2))<>1
      do od; k
    end:
seq (a(n), n=1..40);  # Alois P. Heinz, Oct 13 2011

fQ[n_] := (! PrimeQ[n] || ! PrimeQ[n + 2]) && PowerMod[2, n - 1, n] == 1 && PowerMod[2, n + 1, n + 2] == 1; Select[2 Range@ 32000 + 1, fQ] (* Robert G. Wilson v, Oct 11 2011 *)

is(n)=Mod(2,n^2+2*n)^(n+2)==3*n+8 && (!isprime(n) || !isprime(n+2)) && n>1 \\ Charles R Greathouse IV, Dec 02 2014

2^(a(n) + 2) == 3*a(n) + 8 (mod a(n)*(a(n)+2)).

4*(2^(a(n)-1)-1) == -a(n)*((a(n)-1)/2) (mod a(n)*(a(n)+2)). - Davide Rotondo, Nov 07 2021

A194231 Numbers k such that at least one of k and k+2 is composite, while for every b coprime to k*(k+2), b^(k-1) == 1 (mod k) and b^(k+1) == 1 (mod k+2).

Original entry on oeis.org

561, 1103, 2465, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 656601, 670031, 1033667, 1909001, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619

Offset: 1

Vladimir Shevelev, Oct 12 2011

nonn

These might be called "Carmichael pseudo-twin-primes".

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

Cf. A002997, A001567, A141232.

Subsequences: A272754, A290692.

with(numtheory):
ic:= proc(n) local p;
       if not issqrfree(n) then false
     else for p in factorset(n) do
            if irem (n-1, p-1)<>0 then return false fi
          od; true
       fi
     end:
a:= proc(n) option remember; local k;
      for k from 2 +`if`(n=1, 1, a(n-1)) by 2 while
        isprime(k) and isprime(k+2) or not (ic(k) and ic(k+2))
      do od; k
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Oct 12 2011

terms = 31; bMax = 20(* sufficient for 31 terms *); coprimes[n_] := Select[ Range[bMax], CoprimeQ[#, n]&]; Reap[For[n = m = 1, m <= terms, n += 2, If[CompositeQ[n] || CompositeQ[n+2], If[AllTrue[coprimes[n(n+2)], PowerMod[#, n-1, n] == 1 && PowerMod[#, n+1, n+2] == 1&], Print["a(", m, ") = ", n]; Sow[n]; m++]]]][[2, 1]] (* Jean-François Alcover, Mar 28 2017 *)

For every b coprime to a(n)*(a(n)+2), 2*b^(a(n)+1) == (b^2-1)*(a(n)+2) (mod a(n)*(a(n)+2)). Conversely (Max Alekseyev), if for every b coprime to N*(N+2), 2*b^(N+1) == (b^2-1)*(N+2) (mod N*(N+2)), then N is in the sequence. - Vladimir Shevelev, Oct 14 2011

cp's OEIS Frontend

A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

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A141350 Overpseudoprimes to base 3.

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A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

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A140803 Numbers of the form (2^(p*q)-1) /((2^p-1)*(2^q-1)), where p>q are primes.

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A141390 Overpseudoprimes to base 5.

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A131952 a(n) is the maximal overpseudoprime q to base 2 such that the multiplicative order of 2 mod q equals A143584(n).

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A141629 a(n) is the least base-2 overpseudoprime k such that the multiplicative order of 2 mod k equals 8*n+20.

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A143584 Integers that are equal to the multiplicative order of 2 modulo some overpseudoprime to base 2.

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A140803 Numbers of the form (2^(pq)-1) /((2^p-1)(2^q-1)), where p>q are primes.