A014662 -id:A014662 - cp's OEIS frontend

A287145 Smallest k such that both of the consecutive Woodall numbers A003261(k) and A003261(k+1) are divisible by A014662(n), the n-th prime p with even order of 2 mod p.

4, 13, 64, 89, 83, 188, 433, 701, 449, 342, 1429, 1768, 1889, 2276, 3484, 2423, 5149, 5776, 2069, 1693, 8644, 4793, 9728, 11173, 4237, 13364, 15049, 16108, 16469, 9455, 19501, 22364, 25876, 8929, 3131, 6524, 2311, 36313, 13017, 10114, 13582, 43069, 15962

Offset: 1

Amiram Eldar, May 20 2017

nonn

Keller proved that the occurrence of 2 consecutive Woodall numbers that are divisible by the same prime is restricted to primes p with even h(p), the order of 2 mod p, and that there are an infinity of such pairs.

			11 is the 3rd prime p with even order of 2 mod p. A003261(k)=k*2^k-1 is divisible by 11 for k = 16,48,61,64,65,73,79,100,... The first occurrence of 2 consecutive numbers is 64 and 65, thus a(3) = 64.

Amiram Eldar, Table of n, a(n) for n = 1..350
Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (October 1995), pp. 1733-1741.

Cf. A003261, A014662, A014664.

a = {}; For[p=0, p<=11699, p++; If[!PrimeQ[p], Continue[]]; h=MultiplicativeOrder[2, p]; If[!EvenQ[h], Continue[]]; n=(h/2+1)*p-2; a = AppendTo[a, n]]; a

a(n) = (h(p)/2 + 1)*p - 2, where p=A014662(n), and h(p) is the order of 2 modulo p (A014664).

A059349 Primes p such that x^32 = 2 has no solution mod p.

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419

Offset: 1

Klaus Brockhaus, Jan 27 2001

Complement of A049564 relative to A000040.
Differs from A014662 first at p=6529, then at p=21569. [R. J. Mathar, Oct 05 2008]
Differs from A045316 (x^8 == 2 (mod p) has no solution) first at a(37) = 257 which is not a term of A045316. See A070184 for all such terms. - M. F. Hasler, Jun 21 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A049564, A216747.

Cf. A070184 = (this sequence) \ A045316.

[p: p in PrimesUpTo(450) | not exists{x : x in ResidueClassRing(p) | x^32 eq 2 }]; // Vincenzo Librandi, Sep 20 2012

ok[p_] := Reduce[Mod[x^32 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[100]], ok ] (* Vincenzo Librandi, Sep 20 2012  *)

A091317 Primes p that divide 2^n+1 for some n.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419, 421, 433

Offset: 1

N. J. A. Sloane, Feb 21 2004

nonn

From Charles R Greathouse IV, Feb 13 2009: (Start)
Essentially the same as A014662.
Also primes p for which p^2 divides 2^n+1 for some n. If p | 2^g + 1, then 2^g = kp - 1 for some k, so 2^gp = (kp - 1)^p = (-1)^p + (-1)^(p-1) * kp * (p choose 1) + ... and so 2^gp = -1 (mod p^2). (End)

T. D. Noe, Table of n, a(n) for n=1..1000
Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse, The density of primes dividing a particular non-linear recurrence sequence, arXiv:1508.02464 [math.NT], 2015 (see Introduction).
H. H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a != 0 von durch eine vorgegebene Primzahl l != 2 teilbarer bzw. unteilbarer Ordnung mod. p ist, Math. Ann., 162 (1965), 74-76.
H. H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a != 0 von gerader bzw. ungerader Ordnung mod. p ist, Math. Ann., 166 (1966), 19-23.
Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Complement in primes of A014663.

Cf. A014662. - Charles R Greathouse IV, Feb 13 2009

2, op(select(t -> isprime(t) and numtheory:-order(2,t)::even, [seq(2*i+1, i=1..1000)])); # Robert Israel, Aug 12 2015

Join[{2}, Select[Prime[Range[100]], EvenQ[MultiplicativeOrder[2, #/ (2^IntegerExponent[#, 2])]]&]] (* Jean-François Alcover, Sep 02 2018 *)

isA091317(p)=!bitand(znorder(Mod(2,p)),1) \\ Charles R Greathouse IV, Feb 13 2009

Has density 17/24 (Hasse 1966).

A296243 Numbers k such that the multiplicative order of 2 modulo k is even.

Original entry on oeis.org

3, 5, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 85, 87, 91, 93, 95, 97, 99, 101, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 153

Offset: 1

Max Alekseyev, Dec 09 2017

nonn

Odd numbers k such that A007733(k) = A002326((k-1)/2) is even.

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

Set difference of A005408 and A036259.
Contains A296244 as a subsequence.
The prime terms are given by A014662.
Cf. A002326, A007733.

A036259 = Select[Range[1, 199, 2], OddQ[MultiplicativeOrder[2, #]] &];
Range[1, A036259[[-1]], 2] ~Complement~ A036259 (* Jean-François Alcover, Dec 20 2017 *)
Select[Range[1, 153, 2], EvenQ[MultiplicativeOrder[2, #]] &] (* Amiram Eldar, Jul 30 2020 *)

{ is_A296243(n) = (n%2) && !(znorder(Mod(2,n))%2); }

A213049 Primes p such that the order of 2 mod p is a square.

Original entry on oeis.org

5, 37, 73, 101, 109, 197, 257, 577, 601, 641, 677, 727, 1601, 1801, 2593, 3137, 3389, 3457, 4057, 4357, 5477, 8101, 8837, 10369, 14401, 14407, 16901, 17957, 18253, 18433, 20809, 21317, 22501, 25601, 30977, 33857, 37447, 42437, 44101, 47629, 47653, 50177

Offset: 1

Joerg Arndt, Jun 03 2012

nonn

			The order of 2 mod 601 is 25, which is a square, so 601 is a term.

Vincenzo Librandi, Table of n, a(n) for n = 1..178

Cf. A014662, A014663, A122094.

[NthPrime(n): n in [2..6275] | IsSquare(Modorder(2, NthPrime(n)))]; // Bruno Berselli, Jun 08 2012

{ forprime (p=3, 10^6,
    r = znorder(Mod(2,p));
    if ( issquare(r), print1(p,", ") );
); }

A308732 Primes p such that the smallest possible number of 1's in binary representation of a multiple of p equals 3.

Original entry on oeis.org

7, 23, 47, 71, 73, 79, 103, 151, 167, 191, 199, 239, 263, 271, 311, 337, 359, 367, 383, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 919, 937, 967, 983, 991, 1031, 1039, 1063, 1087, 1151, 1223, 1231, 1279, 1289, 1303

Offset: 1

Jeffrey Shallit, Jun 20 2019

The first few corresponding multipliers that give three 1's are (for the numbers listed above) are 1, 3, 11, 119, 1, 13, 5, 7, 791, 87839, 247, 17970575, 3987, 8048111, 7, 49, 23, 2995944847, 5607007, 7, 2319663.

Robert Israel, Table of n, a(n) for n = 1..10000
Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, Bull. Aust. Math. Soc. 94 (2016), 224-235.
Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
Kenneth B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica 38 (1980), 117-128.

Cf. A014662, which enumerates the same sequence for two 1's instead of three.

filter:= proc(n) local S, r, j;
  if not isprime(n) then return false fi;
  r:= numtheory:-order(2,n);
  if r::even then return false fi;
  S:= {seq(2 &^ j mod n, j=1..r)};
  S intersect map(t -> -t-1 mod n, S) <> {}
end proc:
select(filter, [seq(i,i=3..2000, 2)]); # Robert Israel, Jun 23 2019

cp's OEIS Frontend

A287145 Smallest k such that both of the consecutive Woodall numbers A003261(k) and A003261(k+1) are divisible by A014662(n), the n-th prime p with even order of 2 mod p.

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A059349 Primes p such that x^32 = 2 has no solution mod p.

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Magma

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A091317 Primes p that divide 2^n+1 for some n.

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Maple

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A296243 Numbers k such that the multiplicative order of 2 modulo k is even.

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PARI

A213049 Primes p such that the order of 2 mod p is a square.

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Magma

PARI

A308732 Primes p such that the smallest possible number of 1's in binary representation of a multiple of p equals 3.

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Maple