A137576 -id:A137576 - cp's OEIS frontend

A138535 Odd primes p_n for which A140140(n) < (p_n-1)/2, where p_n = n-th odd prime (A065091).

13, 17, 31, 41, 61, 73, 89, 109, 113, 131, 151, 163, 193, 199, 211, 229, 233, 241, 281, 307, 313, 337, 379, 397, 401, 421, 461, 463, 521, 523, 541, 577, 593, 601, 613, 617, 631, 661, 673, 701, 727, 757, 761, 769, 811, 829, 859, 881, 883, 911, 937, 941, 953

Offset: 1

Vladimir Shevelev, May 10 2008

nonn

Cf. A140140, A137576, A065091, A138536.

Corrected and extended by Ray Chandler, May 19 2008

A138536 Odd primes p_n for which A140140(n)=(p_n-1)/2, where p_n = n-th odd prime (A065091).

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 29, 37, 43, 47, 53, 59, 67, 71, 79, 83, 97, 101, 103, 107, 127, 137, 139, 149, 157, 167, 173, 179, 181, 191, 197, 223, 227, 239, 251, 257, 263, 269, 271, 277, 283, 293, 311, 317, 331, 347, 349, 353, 359, 367, 373, 383, 389, 409, 419, 431

Offset: 1

Vladimir Shevelev, May 10 2008

nonn

Cf. A140140, A137576, A065091, A138535.

Corrected and extended by Ray Chandler, May 19 2008

A141350 Overpseudoprimes to base 3.

Original entry on oeis.org

121, 703, 3281, 8401, 12403, 31621, 44287, 47197, 55969, 74593, 79003, 88573, 97567, 105163, 112141, 211411, 221761, 226801, 228073, 293401, 313447, 320167, 328021, 340033, 359341, 432821, 443713, 453259, 478297, 497503, 504913, 679057, 709873, 801139, 867043, 894781, 973241, 1042417

Offset: 1

Vladimir Shevelev, Jun 27 2008, corrected Sep 07 2008

nonn

If h_3(n) is the multiplicative order of 3 modulo n, r_3(n) is the number of cyclotomic cosets of 3 modulo n then, by the definition, n is an overpseudoprime to base 3 if h_3(n)*r_3(n)+1=n. These numbers are in A020229.

In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime of base 3 iff h_3(p_1)=...=h_3(p_k).

Amiram Eldar, Table of n, a(n) for n = 1..10798 (calculated using the b-file at A020229)
V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.
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. 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.

Cf. A141232, A137576, A001262, A020229, A062117, A006694.

ops3Q[n_] := CompositeQ[n] && GCD[n, 3] == 1 && MultiplicativeOrder[3, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[3, #] &] - 1) + 1 == n; Select[Range[10^6], ops3Q] (* Amiram Eldar, Jun 24 2019 *)

isok(n) = (n!=1) && !isprime(n) && (gcd(n,3)==1) && (znorder(Mod(3,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(3, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018

a(10)-a(38) from Gilberto Garcia-Pulgarin added by Vladimir Shevelev, Feb 06 2012

A122929 Multiplicative order of 2 mod A141232(n).

Original entry on oeis.org

11, 28, 36, 52, 48, 60, 52, 148, 76, 68, 51, 52, 29, 92, 156, 92, 29, 179, 166, 100, 44, 102, 239, 156, 50, 25, 51, 364, 224, 204, 244, 166, 66, 346, 194, 388, 618, 92, 388, 102, 660, 371, 388, 29, 772, 828, 239, 460, 55, 292, 431, 166, 882, 1060, 532, 155, 68

Offset: 1

Vladimir Shevelev, Jul 05 2008, Jul 12 2008, Jul 23 2008

nonn

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

Cf. A141232, A002326, A001262, A141216.

a137576(n)=my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1;
lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && (a137576((n-1)/2) == n), print1(znorder(Mod(2, n)), ", ");););} \\ Michel Marcus, Feb 09 2015

More terms from Michel Marcus, Feb 09 2015

A140667 Odd composite numbers k for which k = A140607((k-1)/2).

Original entry on oeis.org

91, 1581, 2465, 8481, 25761, 31609, 33355, 34945, 118405, 146611, 319507, 736291, 994507, 3270403, 3375487, 5176153, 6186403, 6228685, 8650951, 10679131, 22028203, 26017291, 31470211, 33796531, 41710411, 42149971, 42474547, 46672291, 48316969, 49019851, 58986091, 68182003, 69885649

Offset: 1

Ray Chandler, May 20 2008

nonn

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A140608, A138786.

f(n) = (eulerphi(2*n+1) + 1 + g(n))/2; \\ A140607
g(n) = sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(c) = if (!isprime(c) && (c%2), f((c-1)/2) == c); \\ Michel Marcus, Jan 31 2023

More terms from Michel Marcus, Jan 31 2023

A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 170

Offset: 1

Vladimir Shevelev, May 21 2008, May 24 2008

nonn

The sequence properly contains A005097. 170 is the first number which is not in A005097. One can prove that A002326(2^(2t-1)) = 4t. Thus if n=2^(2t-1), where, for any m>0, t=2^(m-1) then 2n is a multiple of A002326(n) while 2n+1 is a Fermat number which, as well known, is not always a prime.

The sequence is the union of A005097 and (A001567 - 1)/2. [Conjectured by Vladimir Shevelev, proved by Ray Chandler, May 26 2008]

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.

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

Cf. A002326, A005097, A001262, A001567, A137576.

Select[Range[160], Divisible[2#, MultiplicativeOrder[2, 2#+1]] &] (* Amiram Eldar, Jun 28 2019 *)

isok(n) = !(2*n % znorder(Mod(2, 2*n+1))); \\ Michel Marcus, Nov 02 2017

Data extended up to a(68) = 170 to clarify distinction from A005097 and essentially identical sequences A130290 and A102781, by M. F. Hasler, Dec 13 2019

A195468 Lesser of overpseudo-twin-primes to base 2 defined in Comment.

Original entry on oeis.org

85487, 104651, 253241, 280601, 458987, 580337, 1082399, 1207361, 1251947, 1678541, 2811269, 3090089, 5044031, 5173601, 5590619, 9567671, 10323767, 12263129, 16324001, 18073817, 20647619, 21303341, 22849481, 25080101, 28527047, 33627299, 36307979, 43363601, 45414431

Offset: 1

Vladimir Shevelev, Oct 12 2011

nonn

If h_2(n) is the multiplicative order of 2 modulo n, r_2(n) is the number of cyclotomic cosets of 2 modulo n then, by the definition, n is an overpseudoprime to base 2 if h_2(n)*r_2(n)+1=n. These numbers are in A141232.

We call numbers {n,n+2} overpseudo-twin-primes to base 2 if each of them either prime or overpseudoprime to base 2, but no two are primes.

Amiram Eldar, Table of n, a(n) for n = 1..20863 (terms below 10^15)
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.

Cf. A141232, A002326, A006694, A137576, A001567, A002997, A194231, A192297.

More terms from Amiram Eldar, Sep 21 2019

A140452 2^(a(n))-1 contains an overpseudoprime divisor.

Original entry on oeis.org

11, 22, 23, 25, 28, 29, 33, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109

Offset: 1

Vladimir Shevelev, Jun 26 2008

nonn

If p is a prime then p is in the sequence iff 2^p-1 is a composite number.

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.

Cf. A141232, A137576, A005420, A049479.

f(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isopp(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ A141232
isok(n) = {fordiv(2^n-1, d, if (isopp(d), return (1));); return (0);} \\ Michel Marcus, Dec 09 2018

More terms from Michel Marcus, Dec 09 2018

cp's OEIS Frontend

A138535 Odd primes p_n for which A140140(n) < (p_n-1)/2, where p_n = n-th odd prime (A065091).

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A138536 Odd primes p_n for which A140140(n)=(p_n-1)/2, where p_n = n-th odd prime (A065091).

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A141350 Overpseudoprimes to base 3.

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A122929 Multiplicative order of 2 mod A141232(n).

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A140667 Odd composite numbers k for which k = A140607((k-1)/2).

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A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

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A195468 Lesser of overpseudo-twin-primes to base 2 defined in Comment.

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A140452 2^(a(n))-1 contains an overpseudoprime divisor.

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