A002326 -id:A002326 - cp's OEIS frontend

A212953 Minimal order of degree-n irreducible polynomials over GF(2).

1, 3, 7, 5, 31, 9, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 25, 49, 69, 47, 119, 601, 2731, 262657, 29, 233, 77, 2147483647, 65537, 161, 43691, 71, 37, 223, 174763, 79, 187, 13367, 147, 431, 115, 631, 141, 2351, 97, 4432676798593, 251

Offset: 1

Alois P. Heinz, Jun 01 2012

nonn

a(n) = smallest odd m such that A002326((m-1)/2) = n. - Thomas Ordowski, Feb 04 2014

For n > 1; n < a(n) < 2^n, wherein a(n) = n+1 iff n+1 is A001122 a prime with primitive root 2, or a(n) = 2^n-1 iff n is a Mersenne exponent A000043. - Thomas Ordowski, Feb 08 2014

			For n=4 the degree-4 irreducible polynomials p over GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15. (Example: (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2)). Thus the minimal order is 5 and a(4) = 5.

W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer 2004, Third Edition, 4.3 Factorization of Prime Ideals in Extensions. More About the Class Group (Theorem 4.33), 4.4 Notes to Chapter 4 (Theorem 4.40). - Regarding the first comment.

Max Alekseyev, Table of n, a(n) for n = 1..1236 (first 179 terms from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A003060, A059912, A213224.

with(numtheory):
M:= proc(n) option remember;
      divisors(2^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..50);

M[n_] := M[n] = Divisors[2^n-1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
a[n_] := Min[M[n]];
Array[a, 50] (* Jean-François Alcover, Mar 22 2017, translated from Maple *)

a(n) = min(M(n)) with M(n) = {d : d|(2^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A059912(n,1) = A213224(n,1).

A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 10, 2, 1, 1, 1, 6, 3, 2, 1, 9, 2, 3, 3, 2, 2, 6, 1, 13, 9, 1, 1, 10, 5, 1, 3, 2, 8, 3, 2, 2, 1, 1, 10, 3, 8, 7, 9, 2, 2, 3, 1, 2, 26, 1, 3, 9, 4, 2, 9, 4, 1, 6, 1, 18, 9, 1, 7, 3, 2, 1, 3, 2, 5, 10, 1, 10, 6, 38, 3, 3, 4, 1, 41, 2

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

a(n) is the number of integers k in range 1 .. A016754(n-1)-1 such that A000225(k) is an integral multiple of A016754(n-1). - Antti Karttunen, Nov 15 2014
Conjecture: the positions of 1's, a(k)=1, are exactly given by the 2k-1 which are elements of A167791. - Antti Karttunen, Nov 15 2014
From Charlie Neder, Oct 18 2018: (Start)
It would appear that, if 2k-1 is in A167791, then so is (2k-1)^2, and so a(k) = 1 would follow by definition.
Conjecture: Let B be the first value such that (2k-1)^2 divides 2^B - 1. Then either 2k-1 divides B, or 2k-1 is a Wieferich prime (A001220). (End)

			a(2) = 1 because (2^6 - 1)/(2*2 - 1)^2 = 7 is an integer and 6 < 9.
a(3) = 1 because (2^20 - 1)/(2*3 - 1)^2 = 41943 is an integer and 20 < 25.
a(3) = 2 because (2^21 - 1)/(2*4 - 1)^2 = 42799 is an integer and 21 < 49; and also (2^42 - 1)/(2*4 - 1)^2 = 89756051247 is an integer and 42 < 49.

Kevin P. Thompson, Table of n, a(n) for n = 1..1560 (terms 1..128 from Antti Karttunen)

A246703 gives the positions of records.

Cf. A000225, A002326, A016754, A049094, A237043.

A246702 := proc(n)
    local a,klim,k ;
    a := 0 ;
    klim := (2*n-1)^2 ;
    for k from 1 to klim-1 do
        if modp(2^k-1,klim) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A246702(n),n=1..80) ; # R. J. Mathar, Nov 15 2014

A246702[n_] := Module[{a, klim, k}, a = 0; klim = (2*n-1)^2; For[k = 1, k <= klim-1, k++, If[Mod[2^k-1, klim] == 0, a = a+1]]; a];
Table[A246702[n], {n, 1, 84}] (* Jean-François Alcover, Oct 04 2017, translated from R. J. Mathar's Maple code *)

a(n)=my(t=(2*n-1)^2,m=Mod(1,t)); sum(k=1,t-1,m*=2;m==1) \\ Charles R Greathouse IV, Nov 16 2014

a246702(n) = my(m=(2*n-1)^2); (m-1)\znorder(Mod(2,m)); \\ Max Alekseyev, Oct 11 2023

(define (A246702 n) (let ((u (A016754 (- n 1)))) (let loop ((k (- u 1)) (s 0)) (cond ((zero? k) s) ((zero? (modulo (A000225 k) u)) (loop (- k 1) (+ s 1))) (else (loop (- k 1) s)))))) ;; Antti Karttunen, Nov 15 2014

a(n) = floor( 4*n*(n-1) / A002326(2*n*(n-1)) ). - Max Alekseyev, Oct 11 2023

Corrected by R. J. Mathar, Nov 15 2014

A256608 Longest eventual period of a^(2^k) mod n for all a.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 2, 6, 1, 2, 4, 10, 1, 4, 2, 6, 2, 3, 1, 4, 1, 4, 1, 2, 2, 6, 6, 2, 1, 4, 2, 6, 4, 2, 10, 11, 1, 6, 4, 1, 2, 12, 6, 4, 2, 6, 3, 28, 1, 4, 4, 2, 1, 2, 4, 10, 1, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 1, 18, 4, 20, 2, 1, 6

Offset: 1

Ivan Neretin, Apr 04 2015

nonn

a(n) is a divisor of phi(phi(n)) (A010554).

			In other words, eventual period of {0..n-1} under the map x -> x^2 mod n.
For example, with n=10 the said map acts as follows. Read down the columns: the column headed 2 for example means that (repeatedly squaring mod 10), 2 goes to 4 goes to 16 = 6 (mod 10) goes to 36 = 6 mod 10 --- and has reached a fixed point.
0 1 2 3 4 5 6 7 8 9
0 1 4 9 6 5 6 9 4 1
0 1 6 1 6 5 6 1 6 1
0 1 6 1 6 5 6 1 6 1
and thus every number reaches a fixed point. This means the eventual common period is 1, hence a(10)=1.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A001146, A002322, A002326, A007733, A010554, A037178.
First differs from A256607 at n=43.
LCM of entries in row n of A279185.

a[n_] := With[{lambda = CarmichaelLambda[n]}, MultiplicativeOrder[2, lambda / (2^IntegerExponent[lambda, 2])]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2016 *)

rpsi(n) = lcm(znstar(n)[2]); \\ A002322
pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733
a(n) = pb(rpsi(n)); \\ Michel Marcus, Jan 28 2016

a(n) = A007733(A002322(n)).

a(prime(n)) = A037178(n). - Michel Marcus, Jan 27 2016

Name changed by Jianing Song, Feb 02 2025

A268923 All odd primes a(n) such that for all odd primes q smaller than a(n) the order of 2 modulo a(n)q is a proper divisor of phi(a(n)q)/2. The totient function phi is given in A000010.

Original entry on oeis.org

17, 31, 41, 43, 73, 89, 97, 109, 113, 127, 137, 151, 157, 193, 223, 229, 233, 241, 251, 257, 277, 281, 283, 307, 313, 331, 337, 353, 397, 401, 409, 431, 433, 439, 449, 457, 499, 521, 569, 571, 577, 593, 601, 617, 631, 641, 643, 673, 683, 691, 727, 733, 739, 761, 769, 809, 811, 857, 881, 911, 919

Offset: 1

Wolfdieter Lang, Apr 01 2016

nonn

This sequence was inspired by A269454 submitted by Marina Ibrishimova.
It seems that if for an odd prime p > 3 the order(2, p*3) < phi(p*3)/2 = p-1 then p is in this sequence.
Note that 2^(phi(p*q)/2) == 1 (mod p*q) for distinct odd primes p and q, due to Nagell's corollary on Theorem 64, p. 106. The products of distinct primes considered in the present sequence have order of 2 modulo p*q smaller than phi(p*q)/2.
Up to and including prime(100) = 541 the only odd primes p such that for all odd primes q smaller than p the order of 2 modulo p*q equals phi(p*q)/2 are 5, 7, and 11.
Complement of A216371 = A001122 U A105874 in the set of odd primes. Composed of the primes modulo which neither 2 nor -2 is a primitive root. Also, prime(n) is a term iff A376010(n) > 2. - Max Alekseyev, Sep 05 2024

			n=1: Order(2, 17*3) = 8, and 8 is a proper divisor of phi(17*3)/2 = 16;
   order(2, 17*5) =  8, and 8 is a proper divisor of phi(17*5)/2 = 32;
   order(2, 17*7) = 24, and 24 is a proper divisor of phi(17*7)/2 = 48;
   order(2, 17*11) = 40, and 40 is a proper divisor of phi(17*11)/2 = 80;
   order(2, 17*13) = 24, and 24 is a proper divisor of phi(17*13)/2 = 96.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A000010, A002326, A001122, A105874, A216371, A269454, A376010.

Select[Prime@ Range[3, 157], Function[p, AllTrue[Prime@ Range[2, PrimePi@ p - 1], Function[q, With[{e = EulerPhi[p q]/2}, And[Divisible[e, #], # != e]] &@ MultiplicativeOrder[2, p q]]]]] (* Michael De Vlieger, Apr 01 2016, Version 10 *)

More terms from Michael De Vlieger, Apr 01 2016

A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 14, 23, 9, 29, 42, 40, 65, 80, 90, 31, 40, 121, 44, 142, 189, 109, 61, 115, 77, 302, 273, 148, 318, 94, 434, 532, 20, 497, 115, 86, 148, 826, 702, 271, 148, 355, 230, 601, 119, 220, 265, 131, 299, 1178, 297, 485, 86, 265, 1430, 838, 320, 328, 271, 556, 1769, 1957, 1890, 50, 142, 2017, 148, 751, 2277, 179, 373, 832, 665, 2932, 54, 856, 485

Offset: 1

Antti Karttunen, May 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A007733, A046523, A286160, A286161.

A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from Michel Marcus, Apr 11 2015
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286573(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n));

from sympy import divisors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a002326(n):
    m=1
    while True:
        if (2**m - 1)%(2*n + 1)==0: return m
        else: m+=1
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a007733(n): return a002326((a000265(n) - 1)/2)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a007733(n), a046523(n)) # Indranil Ghosh, May 26 2017

a(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n)).

A295740 Even pseudoprimes (A006935) that are not squarefree.

Original entry on oeis.org

190213279479817426, 283959621257123566, 301971651496560046, 575203724324614126, 800951203404568126, 849341919686285026, 1118572636403947726, 2080713636347910526, 2270517620327541586, 2767984602684877486, 5013069719001987826, 5133266340887464066, 5252931629341901506, 5743747078662858526

Offset: 1

Max Alekseyev, Nov 26 2017

nonn

For a prime p, if p^2 divides an even pseudoprime, then p is a Wieferich prime (A001220) and A007733(p)=A002326((p-1)/2) is odd. Currently, the only known such prime is p=3511.

So, all known terms are multiples of 3511^2. Furthermore, no term can be a multiple of 3511^3.

			a(1) = 190213279479817426 = 2 * 7 * 79 * 1951 * 3511^2 * 7151.
a(2) = 283959621257123566 = 2 * 599 * 937 * 3511^2 * 20521.
a(3) = 301971651496560046 = 2 * 31 * 71 * 73 * 3511^2 * 76231.

Max Alekseyev, Table of n, a(n) for n = 1..80 (5 wrong terms were removed by _Mauro Fiorentini_, May 31 2020)

Intersection of A006935 and A013929.

The even terms of A158358. Also, unless there is a Wieferich prime greater than 3511, the even terms of A247831.

A003571 Order of 3 mod 3n+1.

Original entry on oeis.org

1, 2, 6, 4, 3, 4, 18, 5, 20, 6, 30, 16, 18, 4, 42, 11, 42, 6, 20, 28, 10, 16, 22, 12, 12, 18, 78, 8, 16, 10, 6, 23, 48, 20, 34, 52, 27, 12, 44, 29, 5, 30, 126, 12, 18, 16, 138, 35, 28, 18, 50, 30, 78, 8, 162, 41, 39, 42, 60, 88, 45, 22, 80, 36, 16, 42, 198, 100, 8

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A002326, A003573, A217469.

List([0..70],n->OrderMod(3,3*n+1)); # Muniru A Asiru, Feb 16 2019

a := n -> `if`(n=0, 1, numtheory:-order(3, 3*n+1)):
seq(a(n), n = 0..68);

Table[MultiplicativeOrder[3, 3*n + 1], {n, 0, 68}] (* Arkadiusz Wesolowski, Nov 27 2012 *)

a(n) = znorder(Mod(3, 3*n+1)); \\ Michel Marcus, Feb 16 2019

def A003571(n):
    s, m, N = 0, 1, 3*n + 1
    while True:
        k = N + m
        v = valuation(k, 3)
        s += v
        m = k // 3^v
        if m == 1: break
    return s
print([A003571(n) for n in (0..68)]) # Peter Luschny, Oct 07 2017

a(0) = 1 added by Peter Luschny, Oct 07 2017

A007346 Order of group generated by perfect shuffles of 2n cards.

Original entry on oeis.org

2, 8, 24, 24, 1920, 7680, 322560, 64, 92897280, 3715891200, 40874803200, 194641920, 25505877196800, 1428329123020800, 21424936845312000, 160, 23310331287699456000, 1678343852714360832000, 31888533201572855808000

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.
P. Diaconis, R. L. Graham and W. M. Kantor, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196.
Index entries for sequences related to groups

Cf. A002326, A024222, A274299.

Bisections give A002671, A274303.

f:=proc(n) local k,i,np;
if n=1 then 2
elif (n mod 2) = 1 then n!*2^(n-1)
elif n=6 then 2^9*3*5
elif n=12 then 2^17*3^3*5*11
elif n=2 then 8
elif (n mod 4)=2 then n!*2^n
else
np:=n; k:=1;
for i while (np mod 2) = 0 do
   np:=np/2; k:=k+1; od;
   if (n=2^(k-1)) then k*2^k else n!*2^(n-2); fi;
fi;
end;
[seq(f(n),n=1..64)]; # N. J. A. Sloane, Jun 20 2016

a[1] = 2; a[2] = 8; a[n_] := With[{m = 2^n*n!}, Which[Mod[n, 4] == 2, If[n == 6, m/6, m], Mod[n, 4] == 1, m/2, Mod[n, 4] == 3, m/2, True, If[n == 2^IntegerExponent[n, 2], 2*n*(IntegerExponent[n, 2] + 1), If[n == 12, m/(2*7!), m/4]]]]; Table[a[n], {n, 1, 19}](* Jean-François Alcover, Feb 17 2012, after Franklin T. Adams-Watters *)

A007346(n) = local(M); M=2^n*n!; if(n%4==2, if(n==2, 8, if(n==6, M/6, M)), if(n%4==1, if(n==1, 2, M/2), if(n%4==3, M/2, if(n==2^valuation(n, 2), 2*n*(valuation(n, 2)+1), if(n==12, M/(7!*2), M/4))))) \\ Franklin T. Adams-Watters, Nov 30 2006

See Maple program. - N. J. A. Sloane, Jun 20 2016

Corrected and extended by Franklin T. Adams-Watters, Nov 30 2006

A056951 Triangle whose rows show the result of flipping the first, first two, ... and finally first n coins when starting with the stack (1,2,3,4,...,n) [starting with all heads up, where signs show whether particular coins end up heads or tails].

Original entry on oeis.org

-1, -2, 1, -3, -1, 2, -4, -2, 1, 3, -5, -3, -1, 2, 4, -6, -4, -2, 1, 3, 5, -7, -5, -3, -1, 2, 4, 6, -8, -6, -4, -2, 1, 3, 5, 7, -9, -7, -5, -3, -1, 2, 4, 6, 8, -10, -8, -6, -4, -2, 1, 3, 5, 7, 9, -11, -9, -7, -5, -3, -1, 2, 4, 6, 8, 10, -12, -10, -8, -6, -4, -2, 1, 3, 5, 7, 9, 11, -13, -11, -9, -7, -5, -3, -1, 2, 4, 6, 8, 10, 12, -14, -12, -10

Offset: 1

Henry Bottomley, Sep 05 2000

			Third row is constructed by starting from (1, 2, 3), going to (-1, 2, 3), then going to (-2, 1, 3) and finally going to (-3, -1, 2). Rows are: (-1), (-2, 1), (-3, -1, 2), (-4, -2, 1, 3) etc. as each row is reverse of previous row, with signs changed and -n added as the first term in the row.

A003558 is the number of times the operation needs to be repeated to return to the starting point, taking no account of heads/tails (i.e., signs). A002326 is the number required if heads/tails (i.e., signs) are also required to return to their original position.

Cf. A130517 (unsigned version).

t[n_, 1] := -n; t[n_, n_] := n - 1; t[n_, k_] := 2 * k - n - If[2 * k <= n + 1, 2, 1]; Table[t[n, k], {n, 14}, {k, n}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

T(n, k) = 2k - n - b with 1 <= k <= n (where b = 2 if 2k <= n + 1, b = 1 otherwise).

A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

Original entry on oeis.org

0, 3, 10, 16, 37, 10, 42, 24, 58, 53, 164, 26, 68, 38, 32, 68, 169, 22, 222, 38, 42, 50, 328, 40, 180, 219, 108, 26, 334, 82, 460, 82, 92, 72, 220, 108, 449, 86, 128, 80, 192, 22, 336, 110, 222, 218, 540, 84, 778, 129, 150, 80, 270, 54, 328, 356, 132, 68, 348, 22

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{9,21,63}| = 3, a(3) = |{7,14,28,52,56,91,104,182,364,728}| = 10, a(4) = |{13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095}| = 16,...

Cf. A059907-A059910, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{6, 3, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025*)

a(n) = if(n == 1, 0, numdiv(n^6-1) - numdiv(n^3-1) - numdiv(n^2-1) + numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^6-1)-tau(n^3-1)-tau(n^2-1)+tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

cp's OEIS Frontend

A212953 Minimal order of degree-n irreducible polynomials over GF(2).

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A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

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A256608 Longest eventual period of a^(2^k) mod n for all a.

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A268923 All odd primes a(n) such that for all odd primes q smaller than a(n) the order of 2 modulo a(n)*q is a proper divisor of phi(a(n)*q)/2. The totient function phi is given in A000010.

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A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A295740 Even pseudoprimes (A006935) that are not squarefree.

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A003571 Order of 3 mod 3n+1.

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A268923 All odd primes a(n) such that for all odd primes q smaller than a(n) the order of 2 modulo a(n)q is a proper divisor of phi(a(n)q)/2. The totient function phi is given in A000010.