A141350 -id:A141350 - cp's OEIS frontend

A141390 Overpseudoprimes to base 5.

781, 1541, 5461, 13021, 15751, 25351, 29539, 38081, 40501, 79381, 100651, 121463, 133141, 195313, 216457, 315121, 318551, 319507, 326929, 341531, 353827, 375601, 416641, 432821, 453331, 464881, 498451, 555397, 556421, 753667, 764941, 863329, 872101, 886411

Offset: 1

Vladimir Shevelev, Jun 29 2008

nonn

If h_5(n) is the multiplicative order of 5 modulo n, r_5(n) is the number of cyclotomic cosets of 5 modulo n then, by the definition, n is an overpseudoprime of base 5 if h_5(n)*r_5(n)+1=n. These numbers are in A020231. In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime to base 5 iff h_5(p_1)=...=h_5(p_k). E.g., since h_5(101)=h_5(251)=h_5(401)=25, the number 101*251*401=10165751 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..327 (calculated from the b-file at A020231)
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, 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, A141350, A020231, A020229.

ops5Q[n_] := CompositeQ[n] && GCD[n, 5] == 1 && MultiplicativeOrder[5, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[5, #] &] - 1) + 1 == n; Select[Range[6, 10^6], ops5Q] (* Amiram Eldar, Jun 24 2019 *)

isok(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018

Inserted a(2) and a(8) and extended at the suggestion of Gilberto Garcia-Pulgarin by Vladimir Shevelev, Feb 06 2012

A140658 Overpseudoprimes to bases 2 and 3.

Original entry on oeis.org

5173601, 13694761, 16070429, 27509653, 54029741, 66096253, 102690677, 117987841, 193949641, 206304961, 314184487, 390612221, 393611653, 717653129, 960946321, 1157839381, 1236313501, 1481626513, 1860373241, 1921309633, 2217879901, 2412172153, 2626783921

Offset: 1

Vladimir Shevelev, Jul 10 2008

nonn

From the first 19 strong pseudoprimes to bases 2 and 3 (A072276) only 6 are overpseudoprimes to the same bases.

Daniel Suteu, Table of n, a(n) for n = 1..10000 (terms 1..105 from Amiram Eldar)
Index entries for sequences related to pseudoprimes

Intersection of A141232 and A141350; subsequence of A072276.

Cf. A001262, A020229.

More terms from Amiram Eldar, Jun 24 2019

A140507 Numbers k such that 3^k-1 contains a divisor which is an overpseudoprime in base 3.

Original entry on oeis.org

5, 10, 11, 15, 16, 17, 18, 19, 20, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Vladimir Shevelev, Jun 30 2008

nonn

An odd prime p is in the sequence iff p is not in A028491.

Cf. A141350, A020229, A141232, A141390, A003462, A028491.

isokd(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); \\ A141350
isok(n) = {fordiv (3^n-1, d, if (isokd(d), return (1));); return (0);}

More terms from Michel Marcus, Oct 25 2018

A140509 Numbers k such that 5^k-1 contains a divisor which is an overpseudoprime to base 5.

Original entry on oeis.org

5, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Vladimir Shevelev, Jun 30 2008

nonn

An odd prime p is in the sequence iff p is not in A004061.

Cf. A141390, A020231, A141232, A141350, A003463, A004061.

isokd(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ A141390
isok(n) = {fordiv (5^n-1, dd, if (isokd(dd), return (1));); return (0);} \\ Michel Marcus, Oct 25 2018

Corrected and more terms from Michel Marcus, Oct 25 2018

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A141390 Overpseudoprimes to base 5.

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A140658 Overpseudoprimes to bases 2 and 3.

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A140507 Numbers k such that 3^k-1 contains a divisor which is an overpseudoprime in base 3.

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A140509 Numbers k such that 5^k-1 contains a divisor which is an overpseudoprime to base 5.

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