A141232 -id:A141232 - cp's OEIS frontend

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

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

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

A143012 Numbers of the form (4^p + 2^p + 1)/7, where p > 3 is prime.

Original entry on oeis.org

151, 2359, 599479, 9588151, 2454285751, 39268347319, 10052678938039, 41175768098368951, 658812288653553079, 2698495133088002829751, 690814754065816531725751, 11053036065049294753459639, 2829577232652317876553477559, 11589948344943812957569751412151

Offset: 1

Vladimir Shevelev, Jul 15 2008, Jul 21 2008

nonn

If 8^p-1 is squarefree then the terms of the sequence are either primes (A000040) or overpseudoprimes to base 2 (A141232). In particular, composite numbers of the sequence are strong pseudoprimes to base 2 (A001262). E.g., a(5)=2454285751 is A001262(1828).

V. Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.

Cf. A001262, A141232, A000040, A126614.

p:=ithprime: seq((4^p(n)+2^p(n)+1)*1/7, n=3..14); # Emeric Deutsch, Aug 16 2008

(4^#+2^#+1)/7&/@Prime[Range[3,30]] (* Harvey P. Dale, Feb 19 2013 *)

Extended by Emeric Deutsch, Aug 16 2008

More terms from Harvey P. Dale, Feb 19 2013

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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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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A140658 Overpseudoprimes to bases 2 and 3.

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A143012 Numbers of the form (4^p + 2^p + 1)/7, where p > 3 is prime.

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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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