A001567 -id:A001567 - cp's OEIS frontend

A178997 Super-Poulet numbers having more than two different prime factors.

294409, 1398101, 1549411, 1840357, 12599233, 13421773, 15162941, 15732721, 28717483, 29593159, 61377109, 66384121, 67763803, 74658629, 78526729, 90341197, 96916279, 109322501

Offset: 1

T. D. Noe, Jan 11 2011

nonn

This is a subsequence of the super-Poulet numbers, A050217. Of the first 1000 super-Poulet numbers, only 18 have more than two prime factors.
a(10000) = A001567(5287334), so about 0.19% of the pseudoprimes in that range are in this sequence. - Charles R Greathouse IV, Sep 16 2016
The smallest Super-Poulet number with three prime factors not all distinct is 5654273717 = 4733*1093^2, which is not in this sequence. - Emmanuel Vantieghem, Sep 25 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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. García-Pulgarín, J. M. Velásquez-Soto and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.

okQ[n_] := CompositeQ[n] && PrimeNu[n] > 2 && AllTrue[Divisors[n], PowerMod[2, #, n] == 2&];
Reap[For[n = 1, n < 10^8, n = n+2, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Sep 11 2018 *) (* Corrected with PrimeNu instead of PrimeOmega by Emmanuel Vantieghem, Sep 24 2018 *)

is(n)=my(f=factor(n)); if(#f~ < 3, return(0)); fordiv(f, d, if(Mod(2, d)^d!=2, return(0))); 1 \\ Charles R Greathouse IV, Sep 01 2016

A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

Original entry on oeis.org

91625794219, 8796093022207, 1557609722332488343, 18216643597893471403

Offset: 1

V. Raman, Oct 04 2012

No other terms below 2^64.
The next term is <= 25790417485109157029391019 = A019320(258). - Emmanuel Vantieghem, Dec 01 2014
Terms A019320(k) belongs to this sequence for k in A297412. - Max Alekseyev, Dec 29 2017

			a(1) = 91625794219 = (2^38 - 2^19 + 1)/3 = A019320(114).
a(2) = 8796093022207 = 2^43 - 1 = A019320(43).

Mersenne Forum, Prime Conjecture

Cf. A069051, A216822, A219037, A217465.

Intersection of A001567 and { 2*A014945(k) + 1 }.

is_A217468(n)={Mod(2,(n-1)*n)^n==2 & !isprime(n)}  \\ - M. F. Hasler, Oct 07 2012

A085012 For p = prime(n), a(n) is the smallest prime q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq; a(n) is 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 31, 0, 257, 73, 89, 113, 11, 73, 61681, 127, 178481, 157, 233, 1321, 20857, 281, 19, 2731, 13367, 23, 193, 601, 307, 6361, 37, 29, 43, 2731, 953, 168749965921, 593, 31, 53, 2593, 499, 101653, 62020897, 54001, 2281, 97, 19707683773, 5347, 29191

Offset: 2

T. D. Noe, Jun 28 2003

nonn

Using a construction in Erdős's paper, it can be shown that every odd prime except 3, 5, 7 and 13 is a factor of some 2-factor pseudoprime. Note that the cofactor q can be very large; for p=317, the smallest is 381364611866507317969. Using a theorem of Lehmer, it can be shown that the possible values of q are among the prime factors of 2^(p-1)-1. The sequence A085014 gives the number of 2-factor pseudoprimes that have prime(n) as a factor.

Sequence A086019 gives the largest prime q such that q*prime(n) is a pseudoprime.

			a(11) = 11 because prime(11) = 31 and 11 is the smallest factor of 2^30-1 that yields a pseudoprime when multiplied by 31.

Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 105-112.

Amiram Eldar, Table of n, a(n) for n = 2..619
Amiram Eldar, Table of n, a(n) for n = 2..1000 with 2 missing terms (marked with value 0)
Paul Erdős, On the converse of Fermat's theorem, Amer. Math. Monthly 56 (1949), p. 623-624.
D. H. Lehmer, On the converse of Fermat's theorem, Amer. Math. Monthly 43 (1936), p. 347-354.
Index entries for sequences related to pseudoprimes

Cf. A001567 (base-2 pseudoprimes), A085014, A086019, A180471.

Table[p=Prime[n]; q=Transpose[FactorInteger[2^(p-1)-1]][[1]]; i=1; While[i<=Length[q] && (PowerMod[2, p*q[[i]]-1, p*q[[i]]]>1), i++ ]; If[i>Length[q], 0, q[[i]]], {n, 2, 56}]

A129492 Composite numbers k such that 2^k mod k is a power of 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 38, 39, 40, 44, 46, 48, 51, 52, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 72, 74, 76, 78, 80, 82, 84, 85, 86, 87, 90, 92, 93, 94, 96, 102, 106, 111, 112, 114, 116, 118, 120, 122, 123, 124, 126, 129, 132, 133, 134, 138

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 4, 8, 16, 18, 25, 27, 32, 35, 36, 42, 45, 49, 50, 54, 55, 64, 70, 75, 77, 81, 88, 91, 95, 98, 99, ....

			15 is a term since 2^15 mod 15 = 8.

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

Cf. A036236, A129493, A129494, A129495, A129496, A129497.

Contains A001567.

[k:k in [2..150]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [2]) where a is 2^k mod k ]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k;
  if isprime(n) then return false fi;
  k:= 2 &^ n mod n;
  k > 1 and k = 2^padic:-ordp(k,2)
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[Range@ 141, IntegerQ@ Log[2, PowerMod[2, #, # ]] &]

A163209 Catalan pseudoprimes: odd composite integers n=2m+1 satisfying A000108(m) == (-1)^m 2 (mod n).

Original entry on oeis.org

5907, 1194649, 12327121

Offset: 1

Peter Luschny, Jul 24 2009

Also, Wilson spoilers: composite n which divide A056040(n-1) - (-1)^floor(n/2). For the factorial function, a Wilson spoiler is a composite n that divides (n-1)! + (-1). Lagrange proved that no such n exists. For the swinging factorial (A056040), the situation is different.
Also, composite odd integers n=2*m+1 such that A000984(m) == (-1)^m (mod n).
Contains squares of A001220. In particular, a(2) = A001220(1)^2 = 1093^2 = 1194649 = A001567(274) and a(3) = A001220(2)^2 = 3511^2 = 12327121 = A001567(824).
See the Vardi reference for a binomial setup.
Aebi and Cairns 2008, page 9: a(4) either has more than 2 factors or is > 10^10. - Dana Jacobsen, May 27 2015
a(4) > 10^10. - Dana Jacobsen, Mar 03 2018

I. Vardi, Computational Recreations in Mathematica, 1991, p. 66.

C. Aebi, G. Cairns (2008). "Catalan numbers, primes and twin primes". Elemente der Mathematik 63 (4): 153-164. doi:10.4171/EM/103
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Index entries for sequences related to pseudoprimes

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WS := proc(f,r,n) select(p->(f(p-1)+r(p)) mod p = 0,[$2..n]);
select(q -> not isprime(q),%) end:
A163209 := n -> WS(swing,p->(-1)^iquo(p+2,2),n);

v(n,p)=my(s); n*=2; while(n\=p, s+=n%2); s
is(n)=if(n%2==0,return(0)); my(m=Mod(1,n),a=n\2); fordiv(n,d,if(isprime(d) && v(a,d), return(0))); forprime(p=2,a, m*=p^v(a,p)); forprime(p=a+1,n,m*=p); m==(-1)^a
forcomposite(n=4,2e7, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, Mar 06 2015

# Reasonable for isolated values, slow for the sequence:
use ntheory ":all";
sub is { my $m = ($[0]-1)>>1; (binomial($m<<1,$m) % $[0]) == (($m&1) ? $_[0]-1 : 1); }
foroddcomposites { say if is($) } 2e7;  # _Dana Jacobsen, May 03 2015

# Much faster for sequential testing:
use Math::GMPz; use ntheory ":all"; { my($c,$l)=(Math::GMPz->new(1),1); sub catalan { while ($[0] > $l) { $l++; $c *= 4*$l-2; Math::GMPz::Rmpz_divexact_ui($c,$c,$l+1); } $c; } } my $m; foroddcomposites { $m = ($-1)>>1; say if (catalan($m) % $) == (($m&1) ? $-2 : 2); } 2e7;  # Dana Jacobsen, May 03 2015

a(1) = 5907 = 3*11*179 was found by S. Skiena
Typo corrected Peter Luschny, Jul 25 2009
Edited by Max Alekseyev, Jun 22 2012

A175865 Numbers k with property that 2^(k-1) == 1 (mod k) and 2^((3*k-1)/2) - 2^((k-1)/2) + 1 == 0 (mod k).

Original entry on oeis.org

3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557, 563

Offset: 1

Alzhekeyev Ascar M, Sep 30 2010

nonn

All composites in this sequence are 2-pseudoprimes, see A001567.
The subsequence of composites begins: 3277, 29341, 49141, 80581, 88357, 104653, 196093, 314821, 458989, 476971, 489997, ..., . - Robert G. Wilson v, Oct 02 2010
The sequence includes all the primes of A003629. - Alzhekeyev Ascar M, Mar 09 2011
If we consider the composites in this sequence which are in the modulo classes == 3 (mod 8) or == 5 (mod 8), they are moreover strong pseudoprimes to base 2 (see A001262). - Alzhekeyev Ascar M, Mar 09 2011
Are there any composites in this sequence which are *not* in the two modulo classes == {3,5} (mod 8)? - R. J. Mathar, Mar 29 2011

			3 is a term since 2^(3-1)-1 = 3 is divisible by 3, and 2^((3*3-1)/2) - 2^((3-1)/2) + 1 = 15 is divisible by 3.

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

Cf. A001262, A001567, A003629.

fQ[n_] := PowerMod[2, n - 1, n] == 1 && Mod[ PowerMod[2, (3 n - 1)/2, n] - PowerMod[2, (n - 1)/2, n], n] == n - 1; Select[ Range@ 570, fQ] (* Robert G. Wilson v, Oct 02 2010 *)

More terms from Robert G. Wilson v, Oct 02 2010

A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

Original entry on oeis.org

341, 5461, 1398101, 22369621, 5726623061, 91625968981, 23456248059221, 96076792050570581, 1537228672809129301, 6296488643826193618261, 1611901092819505566274901, 25790417485112089060398421, 6602346876188694799461995861

Offset: 1

Bruno Berselli, Jan 21 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

This is the case a=2 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

Named after the Italian mathematician Michele Cipolla (1880-1947). - Amiram Eldar, Jun 15 2021

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013. The sequence is on page 3.
Michele Cipolla, Sui numeri composti P, che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica, Vol. 9 (1904), pp. 139-160.
Y. Hamahata and Y. Kokubun, Cipolla Pseudoprimes, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.6.

Cf. A001567, A002450, A210461, A215848, A216170.

a210454 = (`div` 3) . (subtract 1) . (4 ^) . a000040 . (+ 2)
-- Reinhard Zumkeller, Jan 22 2013

Magma
```
[(4^NthPrime(n)-1)/3: n in [3..15]];
```

P:=proc(q)local n;
for n from 3 to q do print((4^ithprime(n)-1)/3);
od; end: P(100); # Paolo P. Lava, Oct 11 2013

Mathematica
```
(4^# - 1)/3 & /@ Prime[Range[3, 15]]
```

Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
makelist((4^Prime(n)-1)/3, n, 3, 15);

a(n)=4^prime(n+2)\3 \\ Charles R Greathouse IV, Jul 09 2015

A247831 Pseudoprimes to base 2 divisible by 3511^2, including the even pseudoprimes.

Original entry on oeis.org

12327121, 129816911251, 259621495381, 346157884801, 605767053061, 6317168754781, 6922923480721, 12634325182441, 18518799663001, 21634109682121, 24273469559431, 57114029344321, 65681131896901, 102718706568661, 135083316211741, 135818875521811, 153342494379361

Offset: 1

Felix Fröhlich, Sep 24 2014

nonn

Numbers k such that 2^k == 2 (mod k) and k is divisible by 3511^2.

Unless there are other Wieferich primes (A001220) besides 1093 and 3511, the intersection and the union of this sequence with A247830 are given by A219346 and A158358, respectively, and the even terms are given by A295740. - Max Alekseyev, Nov 26 2017 [The indices of the even terms in this sequence are 430, 525, 543, 701, 811, 826, 937, 1235, 1277, 1388, ... - Jianing Song, Feb 08 2019]

Jianing Song, Table of n, a(n) for n = 1..1470
R. G. E. Pinch, The pseudoprimes up to 10^13, Lecture Notes in Computer Science, 1838 (2000), 459-473.
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Mathematics of Computation 35 (1980), pp. 1003-1026.

Subsequence of each of (A001567 U A006935), A015919, A158358 composed of the terms divisible by 3511^2.

Cf. A001220, A013929, A219346, A247830, A295740.

vi=readvec("b158358.txt")
for(n=1, #vi, if(Mod(vi[n], 3511^2)==0, print1(vi[n], ", ")))

list(N)=select(k->Mod(2, k)^k==2, 3511^2*vector(N\3511^2\2, i, i)) \\ Jianing Song, Feb 07 2019

Name changed by Jianing Song, Feb 07 2019 to include the even pseudoprimes to base 2 (A006935) at the suggestion of Max Alekseyev.

A270698 Composite numbers k == 1 (mod 4) such that (1 + i)^k == 1 + i (mod k), where i = sqrt(-1).

Original entry on oeis.org

561, 1105, 1729, 1905, 2465, 3277, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 29341, 30121, 33153, 34945, 41041, 46657, 49141, 52633, 62745, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 104653, 113201, 115921, 126217, 129921

Offset: 1

José María Grau Ribas, Mar 21 2016

nonn

From Jianing Song, Sep 05 2018: (Start)
Numbers in A047713 that are congruent to 1 mod 4. Most terms are congruent to 1 mod 8. For terms congruent to 5 mod 8, see A244626.
Also composite k == 1 (mod 4) such that (-4)^((k-1)/4) == 1 (mod k). Note that this is satisfied by all primes == 1 (mod 4), see A318898. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..889 from Jianing Song using data from A047713)

Subsequence of A001567 and A047713.
A244626 is a proper subsequence.
Cf. A006971, A270697, A318898.

Select[1 + 4*Range[100000], PrimeQ[#] == False && PowerMod[1 + I, #, #] == 1 + I &]

forstep(n=5, 10^5, 4, if(Mod(2, n)^((n-1)/2)==kronecker(2, n) && !isprime(n), print1(n, ", "))) \\ Jianing Song, Sep 06 2018

A293622 Fermat pseudoprimes to base 2 that are triangular.

Original entry on oeis.org

561, 2701, 4371, 8911, 10585, 18721, 33153, 41041, 49141, 93961, 104653, 115921, 157641, 226801, 289941, 314821, 334153, 534061, 665281, 721801, 831405, 873181, 915981, 1004653, 1373653, 1537381, 1755001, 1815465, 1987021, 2035153, 2233441, 2284453, 3059101

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Rotkiewicz proved that this sequence is infinite.
Intersection of A001567 and A000217.
Supersequence of A290945 (triangular Carmichael numbers).
All values of A098025(n)*(2*A098025(n)-1) are terms in this sequence.
The corresponding indices of the triangular numbers are 33, 73, 93, 133, 145, 193, 257, 286, 313, 433, 457, 481, 561, 673, 761, 793, 817, ...

			2701 = 73 * 74 / 2 = 37 * 73 is in the sequence since it is triangular and composite, and 2^2700 == 1 (mod 2701).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.
Andrzej Rotkiewicz, Sur les nombres pseudopremiers triangulaires, Elemente der Mathematik, Vol. 19 (1964), pp. 82-83.

Cf. A000217, A001567, A098025, A290945.

t[n_]:=n(n+1)/2; Select[t[Range[3, 10^4]], PowerMod[2, (# - 1), # ] == 1 &]

forcomposite(c=1, 31*10^5, if(Mod(2, c)^(c-1)==1 && ispolygonal(c, 3), print1(c, ", "))) \\ Felix Fröhlich, Oct 14 2017

cp's OEIS Frontend

A178997 Super-Poulet numbers having more than two different prime factors.

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A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

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A085012 For p = prime(n), a(n) is the smallest prime q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq; a(n) is 0 if no such prime exists.

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A129492 Composite numbers k such that 2^k mod k is a power of 2.

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A163209 Catalan pseudoprimes: odd composite integers n=2*m+1 satisfying A000108(m) == (-1)^m * 2 (mod n).

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A175865 Numbers k with property that 2^(k-1) == 1 (mod k) and 2^((3*k-1)/2) - 2^((k-1)/2) + 1 == 0 (mod k).

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A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

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A163209 Catalan pseudoprimes: odd composite integers n=2m+1 satisfying A000108(m) == (-1)^m 2 (mod n).