A001567 -id:A001567 - cp's OEIS frontend

A005935 Pseudoprimes to base 3.

91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161

Offset: 1

N. J. A. Sloane

nonn

Theorem: If q>3 and both numbers q and (2q-1) are primes then n=q*(2q-1) is a pseudoprime to base 3 (i.e. n is in the sequence). So for n>2, A005382(n)*(2*A005382(n)-1) is in the sequence (see Comments lines for the sequence A122780). 91,703,1891,2701,12403,18721,38503,49141... are such terms. This sequence is a subsequence of A122780. - Farideh Firoozbakht, Sep 13 2006
Composite numbers n such that 3^(n-1) == 1 (mod n).
Theorem (R. Steuerwald, 1948): if n is a pseudoprime to base b and gcd(n,b-1)=1, then (b^n-1)/(b-1) is a pseudoprime to base b. In particular, if n is an odd pseudoprime to base 3, then (3^n-1)/2 is a pseudoprime to base 3. - Thomas Ordowski, Apr 06 2016
Steuerwald's theorem can be strengthened by weakening his assumption as follows: if n is a weak pseudoprime to base b and gcd(n,b-1)=1, then ... - Thomas Ordowski, Feb 23 2021

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 91, p. 33, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..102839 (terms a(1)-a(798) from R. J. Mathar, a(799)-a(1000) from T. D. Noe)
J. Bernheiden, Pseudoprimes (Text in German)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
F. Richman, Primality testing with Fermat's little theorem
Carlos Rivera, Conjecture 105: conjecture about twin primes and pseudo twin primes to base 3, The Prime Puzzles & Problems Connection.
Rudolf Steuerwald, Über die Kongruenz a^(n-1) == 1 (mod n), Sitzungsber. math.-naturw. Kl. Bayer. Akad. Wiss. München, 1948, pp. 69-70.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Pseudoprimes to other bases: A001567 (2), A005936 (5), A005937 (6), A005938 (7), A005939 (10).
Subsequence of A122780.
Cf. A005382.

base = 3; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *)

is_A005935(n)={Mod(3,n)^(n-1)==1 & !ispseudoprime(n) & n>1}  \\ M. F. Hasler, Jul 19 2012

More terms from David W. Wilson, Aug 15 1996

A164368 Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.

Original entry on oeis.org

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 109, 127, 137, 149, 151, 167, 179, 181, 191, 197, 227, 229, 233, 239, 241, 263, 269, 281, 283, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 521, 569, 571, 587, 593, 599, 601, 607

Offset: 1

Vladimir Shevelev, Aug 14 2009

nonn

The Ramanujan primes possess the following property:
If p = prime(n) > 2, then all numbers (p+1)/2, (p+3)/2, ..., (prime(n+1)-1)/2 are composite.
The sequence equals all primes with this property, whether Ramanujan or not.
All Ramanujan primes A104272 are in the sequence.
Every lesser of twin primes (A001359), beginning with 11, is in the sequence. - Vladimir Shevelev, Aug 31 2009
109 is the first non-Ramanujan prime in this sequence.
A very simple sieve for the generation of the terms is the following: let p_0=1 and, for n>=1, p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=0,1,2,... From every interval containing at least one prime we remove the last one. Then all remaining primes form the sequence. Let us demonstrate this sieve: For p_n=1,2,3,5,7,11,... consider intervals (2,4), (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the last prime of each interval, i.e., 3,5,7,13,19,23,31,... we obtain 2,11,17,29, etc. - Vladimir Shevelev, Aug 30 2011
This sequence and A194598 are the mutually wrapping up sequences:
   A194598(1) <= a(1) <= A194598(2) <= a(2) <= ...
From Peter Munn, Oct 30 2017: (Start)
The sequence is the list of primes p = prime(k) such that there are no primes between prime(k)/2 and prime(k+1)/2. Changing "k" to "k-1" and therefore "k+1" to "k" produces a definition very similar to A164333's: it differs by prefixing an initial term 3. From this we get a(n+1) = prevprime(A164333(n)) = A151799(A164333(n)) for n >= 1.
The sequence is the list of primes that are not the largest prime less than 2*prime(k) for any k, so that - as a set - it is the complement relative to A000040 of the set of numbers in A059788.
{{2}, A166252, A166307} is a partition.
(End)

			2 is in the sequence, since then q=2, and there is a prime 3 between 2 and 4. - _N. J. A. Sloane_, Oct 15 2009

Alois P. Heinz, Table of n, a(n) for n = 1..20000
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
J. Sondow, Ramanujan Prime in MathWorld
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011 (see Section 5 Prime gaps).
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2 (see Section 5 Prime gaps).

Cf. Ramanujan primes, A104272, and related sequences: A164288, A080359, A164294, A193507, A194184, A194186.
Cf. A059788, A164333, A194598.
A001359, A166252, A166307 are subsets.
Cf. A001262, A001567, A062568, A141232 (all relate to pseudoprimes to base 2).

a:= proc(n) option remember; local q, k, p;
      k:= nextprime(`if`(n=1, 1, a(n-1)));
      do q:= nextprime(floor(k/2));
         p:= nextprime(k);
         if p<2*q then break fi;
         k:= p
      od; k
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 30 2011

Reap[Do[q=NextPrime[p/2]; If[PrimePi[2*q] != PrimePi[p], Sow[p]], {p, Prime[Range[100]]}]][[2, 1]]
(* Second program: *)
fQ[n_] := PrimePi[ 2NextPrime[n/2]] != PrimePi[n];
Select[ Prime@ Range@ 105, fQ]

is(n)=nextprime(n+1)<2*nextprime(n/2) && isprime(n) \\ Charles R Greathouse IV, Apr 24 2015

As a set, this sequence = A000040 \ A059788 = A000040 \ prevprime(2*A000040) = A000040 \ A151799(A005843(A000040)). - Peter Munn, Oct 30 2017

Definition clarified and simplified by Jonathan Sondow, Oct 25 2011

A015919 Positive integers k such that 2^k == 2 (mod k).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 341, 347, 349, 353, 359, 367

Offset: 1

Robert G. Wilson v

nonn

Includes 341 which is first pseudoprime to base 2 and distinguishes sequence from A008578.
First composite even term is a(14868) = 161038 = A006935(2). - Max Alekseyev, Feb 11 2015
If k is a term, then so is 2^k - 1. - Max Alekseyev, Sep 22 2016
Terms of the form 2^k - 2 correspond to k in A296104. - Max Alekseyev, Dec 04 2017
If 2^k - 1 is a term, then so is k. - Thomas Ordowski, Apr 27 2018

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Contains A002997 as a subsequence.
The odd terms form A176997.
Cf. A000040, A001567, A008578.

Prepend[ Select[ Range@370, PowerMod[2, #, #] == 2 &], {1, 2}] // Flatten (* Robert G. Wilson v, May 16 2018 *)

is(n)=Mod(2,n)^n==2 \\ Charles R Greathouse IV, Mar 11 2014

def ok(n): return pow(2, n, n) == 2%n
print([k for k in range(1, 400) if ok(k)]) # Michael S. Branicky, Jun 03 2022

Equals {1} U A000040 U A001567 U A006935 = A001567 U A006935 U A008578. - Ray Chandler, Dec 07 2003; corrected by Max Alekseyev, Feb 11 2015

A062173 a(n) = 2^(n-1) mod n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 8, 1, 2, 4, 0, 1, 14, 1, 8, 4, 2, 1, 8, 16, 2, 13, 8, 1, 2, 1, 0, 4, 2, 9, 32, 1, 2, 4, 8, 1, 32, 1, 8, 31, 2, 1, 32, 15, 12, 4, 8, 1, 14, 49, 16, 4, 2, 1, 8, 1, 2, 4, 0, 16, 32, 1, 8, 4, 22, 1, 32, 1, 2, 34, 8, 9, 32, 1, 48, 40, 2, 1, 32, 16, 2, 4, 40, 1, 32, 64, 8, 4, 2, 54, 32, 1, 58, 58, 88, 1, 32, 1, 24, 46

Offset: 1

Henry Bottomley, Jun 12 2001

nonn

If p is an odd prime then a(p)=1. However, a(n) is also 1 for pseudoprimes to base 2 such as 341.

			a(5) = 2^(5-1) mod 5 = 16 mod 5 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..101101 (first 1000 terms from Harry J. Smith)
Index entries for sequences related to pseudoprimes

Cf. A000079, A001567, A015910, A015919, A062172.
Cf. A082495, A106262, A257531, A305890.
Cf. A176997 (after the initial term, gives the positions of ones).

import Math.NumberTheory.Moduli (powerMod)
a062173 n = powerMod 2 (n - 1) n  -- Reinhard Zumkeller, Oct 17 2015

[Modexp(2,n-1,n): n in [1..110]]; // G. C. Greubel, Jan 11 2023

Array[Mod[2^(# - 1), #] &, 105] (* Michael De Vlieger, Jul 01 2018 *)
Array[PowerMod[2,#-1,#]&,120] (* Harvey P. Dale, May 17 2023 *)

A062173(n) = if(1==n, 0, lift(Mod(2, n)^(n-1))); \\ Antti Karttunen, Jul 01 2018

[power_mod(2,n-1,n) for n in range(1,110)] # G. C. Greubel, Jan 11 2023

a(n) = A106262(2*n-3, n-2). - G. C. Greubel, Jan 11 2023

More terms from Antti Karttunen, Jul 01 2018

A050259 Numbers n such that 2^n == 3 (mod n).

Original entry on oeis.org

1, 4700063497, 3468371109448915, 8365386194032363, 10991007971508067

Offset: 1

Eric W. Weisstein

No other terms below 10^18. - Max Alekseyev, Oct 17 2017

Terms were computed: a(2) by the Lehmers, a(3) by Max Alekseyev, a(4) and a(5) by Joe K. Crump, a(?) = 63130707451134435989380140059866138830623361447484274774099906755 by P.-L. Montgomery.

R. Daniel Mauldin and S. M. Ulam, Mathematical problems and games. Adv. in Appl. Math. 8 (1987), pp. 281-344.

Joe K. Crump, 2^n mod n
R. K. Guy, The Strong Law of Small Numbers, Amer. Math. Monthly 95 (1988), pp. 697-712. See example 13.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, 2
OEIS Wiki, 2^n mod n

Cf. A001567, A036236, A015940, A015910.

m = 2; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

is(n)=Mod(2,n)^n==3 \\ Charles R Greathouse IV, Jun 11 2015

A141232 Overpseudoprimes to base 2: composite k such that k = A137576((k-1)/2).

Original entry on oeis.org

2047, 3277, 4033, 8321, 65281, 80581, 85489, 88357, 104653, 130561, 220729, 253241, 256999, 280601, 390937, 458989, 486737, 514447, 580337, 818201, 838861, 877099, 916327, 976873, 1016801, 1082401, 1145257, 1194649, 1207361, 1251949, 1252697, 1325843

Offset: 1

Vladimir Shevelev, Jun 16 2008

nonn

Numbers are found by prime factorization of numbers from A001262 and a simple testing of the conditions indicated in comment to A141216.
All composite Mersenne numbers (A001348), Fermat numbers (A000215) and squares of Wieferich primes (A001220) are in this sequence. - Vladimir Shevelev, Jul 15 2008
C. Pomerance proved that this sequence is infinite (see Theorem 4 in the third reference). - Vladimir Shevelev, Oct 29 2011
Odd composite numbers k such that ord(2,k) * ((Sum_{d|k} phi(d)/ord(2,d)) - 1) = k-1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..664 from Michel Marcus)
J. H. Castillo, G. García-Pulgarín and J. M. Velásquez-Soto, q-pseudoprimality: A natural generalization of strong pseudoprimality, arXiv:1412.5226 [math.NT], 2014.
Vladimir Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich primes, arXiv:0806.3412 [math.NT], 2008-2012.
Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.
Vladimir Shevelev, On upper estimate for the overpseudoprime counting function, arXiv:0807.1975 [math.NT], 2008.
Vladimir 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.
Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv:1206.0606 [math.NT], 2012.

Cf. A000010, A001262, A141216, A001567, A002326.

A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #]&] - t + 1];
okQ[n_] := n > 1 && CompositeQ[n] && n == A137576[(n-1)/2];
Reap[For[k = 2, k < 2*10^6, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jan 11 2019, from PARI *)

f(n)=my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ Michel Marcus, Oct 05 2018

Sum_{n:a(n)<=x} 1 <= x^(3/4)(1+o(1)).

Name edited by Michel Marcus, Oct 05 2018

A164288 Members of A164368 which are not Ramanujan primes.

Original entry on oeis.org

109, 137, 191, 197, 283, 521, 617, 683, 907, 991, 1033, 1117, 1319, 1493, 1619, 1627, 1697, 1741, 1747, 1801, 1931, 1949, 2011, 2111, 2143, 2153, 2293, 2417, 2539, 2543, 2549, 2591, 2621, 2837, 2927, 2953, 2969, 3079, 3119, 3187, 3203, 3329, 3389, 3407

Offset: 1

Vladimir Shevelev, Aug 12 2009

nonn

Every lesser of twin primes (A001359), beginning with 137, which is not in A104272, is in the sequence. [From Vladimir Shevelev, Aug 31 2009]

			p=137 is the least lesser of twin primes which is not a Ramanujan prime. Therefore it is in the sequence. [From _Vladimir Shevelev_, Aug 31 2009]

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, J. Int. Seq. 15 (2012) # 12.5.4.

Cf. A104272, A001262, A001567, A062568, A141232.

nn = 250;
A164368 = Select[Prime[Range[2 nn]], PrimePi[2 NextPrime[#/2]] != PrimePi[#]&];
Rama = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, Rama[[s+1]] = k], {k, Prime[3 nn]}];
A104272 = Rama+1;
Complement[A164368, A104272] (* Jean-François Alcover, Oct 27 2018, after T. D. Noe in A104272 *)

A164368 \ A104272.

I added 521. - Vladimir Shevelev, Aug 17 2009

Redefined in terms of A164368 and extended by R. J. Mathar, Aug 18 2009

A164294 Primes prime(k) such that all integers in [(prime(k-1)+1)/2,(prime(k)-1)/2] are composite, excluding those primes in A080359.

Original entry on oeis.org

131, 151, 229, 233, 311, 571, 643, 727, 941, 1013, 1051, 1153, 1373, 1531, 1667, 1669, 1723, 1783, 1787, 1831, 1951, 1979, 2029, 2131, 2213, 2239, 2311, 2441, 2593, 2621, 2633, 2659, 2663, 2887, 3001, 3011, 3019, 3121, 3169, 3209, 3253, 3347, 3413, 3457

Offset: 1

Vladimir Shevelev, Aug 12 2009

nonn

The primes of A080359 larger than 3 all have the property that the integers in the interval selected by halving the value of the preceding prime and halving their own value are all composite. This sequence here collects the primes that are not in A080359 but still share this property of the prime-free subinterval.

			For the prime 1531=A000040(242), the preceding prime is A000040(241)=1523, and the integers from (1523+1)/2 = 762 up to (1531-1)/2 = 765 are all composite, as they fall in the gap between A000040(135) and A000040(136). In addition, 1531 is not in A080359, which adds 1531 to this sequence here.

Jean-François Alcover, Table of n, a(n) for n = 1..1106
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009. [From _Vladimir Shevelev_, Aug 20 2009]
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.

Cf. A080359, A104272, A164288, A001262, A001567, A062568, A141232, A164368.

maxPrime = 3500;
kmax = PrimePi[maxPrime];
A164333 = Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ] &][[All, 2]]*2 + 1;
b[1] = 2; b[n_] := b[n] = Module[{k = b[n - 1]}, While[(PrimePi[k] - PrimePi[Quotient[k, 2]]) != n, k++]; k];
A080359 = Reap[For[n = 1, b[n] <= maxPrime, n++, Sow[b[n]]]][[2, 1]];
Complement[A164333, A080359] (* Jean-François Alcover, Sep 14 2018 *)

okprime(p) = { my(k = primepi(p)); for (i = (prime(k-1)+1)/2, (prime(k)-1)/2, if (isprime(i), return (0));); return (1);}
lista(nn) = {vlp = readvec("b080359.txt"); forprime (p=2, nn, if (! vecsearch(vlp, p) && okprime(p), print1(p, ", ")););} \\ Michel Marcus, Jan 15 2014

A164333 \ A080359.

Extended beyond 571 by R. J. Mathar, Oct 02 2009

A244626 Composite numbers k congruent to 5 (mod 8) such that 2^((k-1)/2) mod k = k-1.

Original entry on oeis.org

3277, 29341, 49141, 80581, 88357, 104653, 196093, 314821, 458989, 489997, 800605, 838861, 873181, 1004653, 1251949, 1373653, 1509709, 1678541, 1811573, 1987021, 2269093, 2284453, 2387797, 2746477, 2909197, 3400013, 3429037, 3539101, 3605429, 4360621, 4502485, 5590621, 5599765

Offset: 1

Gary Detlefs, Jul 02 2014

nonn

This sequence contains the n mod 8 = 5 pseudoprimes to the following modified Fermat primality criterion:
Conjecture 1: if p is an odd prime congruent to {3,5} (mod 8) then 2^((p-1)/2) mod p = p-1.
This conjecture has been tested to 10^8.
This criterion produces far fewer pseudoprimes than the 2^(n-1) mod n = 1 test and thus has a higher probability of success. The number of pseudoprimes for the two tests up to 10^k are:
   10^5    5   26     19.23%
   10^6   13   78     16.66%
   10^7   40  228     17.54%
There are 40 terms < 10^7. If an additional constraint 3^(n-1) mod n = 1 and 5^(n-1) mod n = 1 is added, only 4 terms remain: (29341, 314821, 873181, 9863461).
This sequence appears to be a subset of A175865, A001262, A047713, A020230.
Number of terms below 10^k for k = 5..15: 5, 13, 40, 132, 369, 975, 2534, 6592, 17403, 45801, 122473. The corresponding numbers for 2^(n-1) mod n = 1: 26, 78, 228, 637, 1718, 4505, 11645, 29902, 76587, 197455, 513601. - Jens Kruse Andersen, Jul 13 2014
Also composite numbers 2n+1 with n even such that 2n+1 | 2^n+1. - Hilko Koning, Jan 27 2022
Conjecture 1 is true. With p = 2k+1 then 2^k mod (2k+1) == 2k. So 2k+1 | 2k-2^k. Prime numbers 2k+1 == +-3 (mod 8) are the prime numbers such that 2k+1 | 2^k+1 (Comments A007520). A reflection across the x-axis and +1 translation across the y-axis of the graph (2k-2^k) / (2k+1) gives the graph (2^k+1) / (2k+1). So the k values of both 2k+1 | 2k-2^k and 2k+1 | 2^k+1 are identical. - Hilko Koning, Feb 04 2022

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A001567, A003629, A070179.

for n from 5 to 10^7 by 8 do if 2^((n-1)/2) mod n = n-1 and not isprime(n) then print(n) fi od;

a(18) corrected by Jens Kruse Andersen, Jul 13 2014

A047713 Euler-Jacobi pseudoprimes: 2^((n-1)/2) == (2 / n) mod n, where (2 / n) is a Jacobi symbol.

Original entry on oeis.org

561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 29341, 30121, 33153, 34945, 41041, 42799, 46657, 49141, 52633, 62745, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 90751, 104653

Offset: 1

N. J. A. Sloane, Richard Pinch and Robert G. Wilson v

Odd composite numbers n such that 2^((n-1)/2) == (-1)^((n^2-1)/8) mod n. - Thomas Ordowski, Dec 21 2013

Most terms are congruent to 1 mod 8 (cf. A006971). Among the first 1000 terms, the number of terms congruent to 1, 3, 5 and 7 mod 8 are 764, 47, 125 and 64, respectively. - Jianing Song, Sep 05 2018

R. K. Guy, Unsolved Problems in Number Theory, A12.
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the subsequence A006971).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.
Eric Weisstein's World of Mathematics, Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A002997, A001567, A048950.

Terms in this sequence satisfying certain congruence: A270698 (congruent to 1 mod 4), A270697 (congruent to 3 mod 4), A006971 (congruent to +-1 mod 8), A244628 (congruent to 3 mod 8), A244626 (congruent to 5 mod 8).

Select[ Range[ 3, 105000, 2 ], Mod[ 2^((# - 1)/2) - JacobiSymbol[ 2, # ], # ] == 0 && ! PrimeQ[ # ] & ]

is(n)=n%2 && Mod(2,n)^(n\2)==kronecker(2,n) && !isprime(n) \\ Charles R Greathouse IV, Dec 20 2013

Corrected by Eric W. Weisstein; more terms from David W. Wilson

cp's OEIS Frontend

A005935 Pseudoprimes to base 3.

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A164368 Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.

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Maple

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A015919 Positive integers k such that 2^k == 2 (mod k).

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A062173 a(n) = 2^(n-1) mod n.

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A050259 Numbers n such that 2^n == 3 (mod n).

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A141232 Overpseudoprimes to base 2: composite k such that k = A137576((k-1)/2).

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A164288 Members of A164368 which are not Ramanujan primes.

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