A167791 -id:A167791 - cp's OEIS frontend

A240104 Numbers with primitive root -19.

2, 3, 6, 13, 26, 29, 31, 37, 41, 53, 58, 59, 62, 67, 71, 74, 79, 82, 89, 103, 106, 107, 113, 118, 134, 142, 158, 167, 173, 178, 179, 193, 206, 214, 223, 226, 227, 257, 269, 281, 293, 317, 331, 334, 337, 346, 358, 379, 383, 386, 401, 431, 433, 439, 446, 449

Offset: 1

Vincenzo Librandi, Apr 01 2014

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

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240101 (r=17), A240103 (r=18), A240106 (r=19).

Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17), A240102 (r=-18).

pr = -19; Select[Range[2, 500], MultiplicativeOrder[pr, #] == EulerPhi[#] &]

A240106 Numbers with primitive root 19.

Original entry on oeis.org

2, 4, 7, 11, 13, 14, 22, 23, 26, 29, 37, 41, 43, 46, 47, 53, 58, 74, 82, 83, 86, 89, 94, 106, 113, 121, 139, 163, 166, 173, 178, 191, 193, 226, 239, 242, 251, 257, 263, 269, 278, 281, 293, 311, 317, 326, 337, 346, 347, 359, 367, 382, 386, 401, 419, 433, 443

Offset: 1

Vincenzo Librandi, Apr 01 2014

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

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240100 (r=17), A240103 (r=18).

Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17), A240102 (r=-18), A240104 (r=-19).

pr = 19; Select[Range[2, 500], MultiplicativeOrder[pr, #] == EulerPhi[#] &]

A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

Original entry on oeis.org

7, 17, 23, 35, 41, 47, 49, 71, 77, 79, 95, 97, 103, 115, 137, 143, 167, 175, 191, 193, 199, 209, 235, 239, 245, 263, 271, 289, 295, 299, 311, 313, 319, 335, 343, 359, 367, 371, 383, 395, 401, 407, 409, 413, 415, 437, 449, 463, 475, 479, 487, 503, 515, 517, 521, 529, 535, 539, 551, 569, 575, 581, 583, 599, 607, 611, 647, 649, 667, 695, 707

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

From Antti Karttunen, Nov 15 2014: (Start)
Equally: Odd numbers n for which A246702((n+1)/2) = 2.
Primes in this sequence: 7, 17, 23, 41, 47, 71, 79, ... seem to be A115591.
A249819 gives the composite terms.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..105

Cf. A002326, A115591, A249819, A167791, A246702.

isA246717(n) = { if(!(n%2), return(0), my(u, s=0); u = n^2; for(k=1, u-1, if(!(((2^k)-1)%u), s++;if(s > 2, return(0)))); return(2==s)); }
n = 0; i = 0; while(i < 105, n++; if(isA246717(n), i++; write("b246717.txt", i, " ", n))); \\ From Antti Karttunen, Nov 15 2014
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246717 (MATCHING-POS 1 1 (lambda (n) (and (odd? n) (= 2 (A246702 (/ (+ 1 n) 2)))))))

Terms corrected by Antti Karttunen, Nov 15 2014

A219030 Powers of odd primes (exponent > 1) for which 2 is not a primitive root.

Original entry on oeis.org

49, 289, 343, 529, 961, 1681, 1849, 2209, 2401, 4913, 5041, 5329, 6241, 7921, 9409, 10609, 11881, 12167, 12769, 16129, 16807, 18769, 22801, 24649, 27889, 29791, 36481, 37249, 39601, 49729, 52441, 54289, 57121, 58081, 63001, 66049, 68921, 69169, 73441

Offset: 1

V. Raman, Nov 10 2012

nonn

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

Cf. A167791, A216848, A108989.

for(n=3,100000,if(n%2==1&&isprime(n)==0&&znorder(Mod(2,n))!=eulerphi(n)&&matsize(factor(n))[1]==1,print1(n",")))

list(lim)=my(v=List(),L=log(lim+.5));forprime(p=3,sqrtint(lim\1), for(e=2,L\log(p), if(znorder(Mod(2,p^e))Charles R Greathouse IV, Nov 12 2012

A240107 Numbers with primitive root -20.

Original entry on oeis.org

11, 13, 17, 31, 37, 53, 59, 73, 79, 113, 121, 131, 137, 139, 157, 169, 173, 179, 191, 199, 211, 233, 239, 257, 271, 277, 289, 293, 313, 317, 331, 337, 353, 359, 379, 397, 419, 431, 433, 439, 479, 499, 557, 593, 599, 613, 631, 653, 659, 673, 677, 719, 751

Offset: 1

Vincenzo Librandi, Apr 01 2014

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

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240100 (r=17), A240103 (r=18), A240106 (r=19), A240108 (r=20).

Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17), A240102 (r=-18), A240104 (r=-19).

pr = -20; Select[Range[2, 800], MultiplicativeOrder[pr, #] == EulerPhi[#] &]

A240108 Numbers with primitive root 20.

Original entry on oeis.org

3, 9, 13, 17, 23, 27, 37, 43, 47, 53, 67, 73, 81, 83, 103, 107, 113, 137, 157, 163, 167, 169, 173, 223, 227, 233, 243, 257, 263, 277, 283, 289, 293, 313, 317, 337, 347, 353, 367, 383, 397, 433, 443, 463, 467, 487, 503, 529, 547, 557, 563, 587, 593, 607

Offset: 1

Vincenzo Librandi, Apr 01 2014

nonn

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

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240100 (r=17), A240103 (r=18), A240106 (r=19).

Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17), A240102 (r=-18), A240104 (r=-19), A240107 (r=-20).

pr = 20; Select[Range[2, 700], MultiplicativeOrder[pr, #] == EulerPhi[#] &]
Join[{3,9,13,17},Select[Range[610],MemberQ[PrimitiveRootList[#],20]&]] (* Harvey P. Dale, Jul 17 2025 *)

A246719 Smallest natural number m for which there are exactly n distinct values k such that 0 < k < m^2 and 2^k - 1 is divisible by m^2.

Original entry on oeis.org

1, 3, 7, 15, 113, 65, 31, 91, 73, 39, 21, 331, 267, 55, 217, 435, 203, 697, 127, 703, 565, 429, 451, 231, 595, 253, 105, 327, 171, 1045, 1335, 255, 385, 497, 341, 1295, 219, 455, 155, 1417, 969, 165, 2143, 861, 357, 453, 555, 2821, 195, 1477, 301, 205, 2091

Offset: 0

Juri-Stepan Gerasimov, Nov 15 2014

nonn

Smallest odd number of the form 2q - 1 such that A246702(q) = n.

Additional terms include: a(426) = 1705, a(451) = 903, a(516) = 2067, a(536) = 2145, a(563) = 2255, a(566) = 2265, a(593) = 2373, a(761) = 3045, a(770) = 3081, a(786) = 2359, a(1333) = 2667, and a(3282) = 1093. - Kevin P. Thompson, Nov 26 2021

			The first occurrence of 3 in the sequence A246702 occurs at n = 8. Therefore, a(3) = 2n - 1 = 2*8 - 1 = 15.

Kevin P. Thompson, Table of n, a(n) for n = 0..350, with missing terms.

Cf. Numbers of the form 2n - 1 such that A246702(n) = i: number 1 (i = 0), A167791 (i = 1), A246717 (i = 2), A246755 (i = 3).

NumK[m_]:=NumK[m]=(m2=m^2;nk=0;Do[If[Mod[2^i,m2]==1,nk++],{i,m2-1}];nk)
nterms=10;Table[m=0;While[NumK[++m]!=n];m,{n,0,nterms-1}] (* Paolo Xausa, Nov 30 2021 *)

isok(m, n) = {my(v = vector(m^2-1, k, Mod(2, m^2)^k == 1)); vecsum(v) == n;}
a(n) = {my(m=1); while (!isok(m, n), m++); m;} \\ Michel Marcus, Nov 27 2021

Name corrected by Antti Karttunen, Nov 18 2014

Multiple corrections and new terms a(17)-a(52) from Kevin P. Thompson, Nov 26 2021

A246755 Numbers of the form 2k - 1 such that A246702(k) = 3.

Original entry on oeis.org

15, 33, 43, 45, 69, 75, 87, 99, 109, 135, 141, 157, 159, 177, 207, 213, 225, 229, 249, 261, 277, 283, 297, 303, 307, 321, 363, 375, 393, 405, 423, 447, 477, 499, 501, 519, 531, 537, 573, 591, 621, 639, 643, 675, 681, 691, 717, 733, 739, 747, 783, 789, 807, 811

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

Composites in this sequence: 15, 33, 45, 69, 75, 87, 99, 135, 141, 159, 177, 207, 213, 225, 249, 261, 297, 303, 321, 363, 375, 393, 405, 423, 447, 477, ...

			A246702(8) = 3 for the first time, hence a(1) = 2*8 - 1 = 15.

Cf. Numbers of the form 2k - 1 such that A246702(k) = m: number 1 (m = 0), A167791 (m = 1), A246717 (m = 2), this sequence (m = 3), A001133 (primes in this sequence).

is(k) = (m=Mod(k%2, k*k)) && sum(i=1, k*k-1, m*=2; m==1) == 3; \\ Jinyuan Wang, May 15 2020

More terms from and terms corrected by Jinyuan Wang, May 15 2020

A319009 Numbers k such that the multiplicative order of 2 modulo k is psi(k), psi = A002322.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 19, 21, 25, 27, 29, 33, 35, 37, 39, 45, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 87, 91, 95, 99, 101, 105, 107, 111, 115, 117, 121, 125, 131, 133, 135, 139, 141, 143, 145, 147, 149, 159, 163, 165, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Jianing Song, Sep 07 2018

nonn

Numbers k such that the multiplicative order of 2 modulo k is at its maximum possible value.
Numbers k such that the binary expansion of 1/k has period psi(n).
Numbers k such that A002326((k-1)/2) = A002322(k).
This is a generalization of A167791, so A167791 is a proper subsequence.
Write k as k = Product_{i=1..t} (p_i)^(e_i) where p_i are distinct primes. If (p_i)^(e_i) belongs to A167791 (and thus here) for 1 <= i <= t, then k is also here, but the converse is not true. In fact, this sequence has terms such that none of (p_i)^(e_i) belongs to A167791, the smallest of which is 301 = 7*43. The multiplicative order of 2 modulo 7 and 43 are 3 (< psi(7) = 6) and 14 (< psi(43) = 42), so the multiplicative order of 2 modulo 301 is lcm(3, 14) = 42 = psi(301).

			The multiplicative order of 2 modulo 15 is 4 = A002322(15), so 15 is a term.
The multiplicative order of 2 modulo 21 is 6 = A002322(21), so 21 is a term.
The multiplicative order of 2 modulo 51 is 8, but A002322(51) = 16, so 51 is not a term.

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

Cf. A001122, A002322, A002326, A167791.

select(n -> numtheory:-order(2,n)=numtheory:-lambda(n), [seq(i,i=1..1000,2)]); # Robert Israel, Sep 12 2018

forstep(n=1, 200, 2, if(znorder(Mod(2, n))==lcm(znstar(n)[2]), print1(n, ", ")))

cp's OEIS Frontend

A240104 Numbers with primitive root -19.

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A240106 Numbers with primitive root 19.

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A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

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A219030 Powers of odd primes (exponent > 1) for which 2 is not a primitive root.

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A240107 Numbers with primitive root -20.

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A240108 Numbers with primitive root 20.

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A246719 Smallest natural number m for which there are exactly n distinct values k such that 0 < k < m^2 and 2^k - 1 is divisible by m^2.

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A246755 Numbers of the form 2k - 1 such that A246702(k) = 3.

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A319009 Numbers k such that the multiplicative order of 2 modulo k is psi(k), psi = A002322.

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