A136003 -id:A136003 - cp's OEIS frontend

A136005 Numbers of the form 2^p - 1, where p is a prime number that is not the sum, minus 1, of a Pythagorean triple.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 137438953471, 2199023255551, 8796093022207, 9007199254740991, 2305843009213693951, 147573952589676412927, 9444732965739290427391, 158456325028528675187087900671, 2535301200456458802993406410751

Offset: 1

Omar E. Pol, Dec 17 2007

nonn

See A136003 for the values of p.

			a(3) = 31 because A136003(3) = 5 and 2^5 = 32 and 32-1 = 31.

Amiram Eldar, Table of n, a(n) for n = 1..236

Cf. A000079, A000668, A136002, A136003.

q[n_] := PrimeQ[n] && (n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]); 2^Select[Range[100], q] - 1 (* Amiram Eldar, Oct 20 2024 *)

a(n) = 2^A136003(n) - 1.

a(16)-a(17) from Amiram Eldar, Oct 20 2024

A183071 Positive integers k such that each prime divisor of 2^k - 1 has the form 4j + 3.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 26, 30, 31, 34, 38, 43, 51, 61, 62, 65, 79, 85, 86, 89, 93, 95, 107, 122, 127, 129, 130, 133, 158, 170, 193, 254, 255, 301, 311, 331, 349, 389, 445, 521, 557, 577, 579, 607, 631, 647, 1103, 1167

Offset: 1

Stuart Clary, Dec 23 2010

nonn

The exponents of the Mersenne primes (A000043) are contained in this sequence.

Needed factorizations are in the Cunningham Project.

			6 is in this sequence because 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000043, A136003, A183072, A183073, A183074.

Cf. A000668, A136005, A183075, A183076, A183077, A183078.

A183075(n) = 2^a(n) - 1.

a(53)-a(54) from Amiram Eldar, Feb 18 2019

A183075 Numbers of the form 2^k - 1 for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

1, 3, 7, 31, 63, 127, 1023, 8191, 16383, 32767, 131071, 524287, 67108863, 1073741823, 2147483647, 17179869183, 274877906943, 8796093022207, 2251799813685247, 2305843009213693951

Offset: 1

Stuart Clary, Dec 23 2010

The Mersenne primes (A000668) are contained in this sequence.

Needed factorizations are in the Cunningham Project.

			63 = 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

Amiram Eldar, Table of n, a(n) for n = 1..54
S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000668, A136005, A183076, A183077, A183078.

Cf. A000043, A136003, A183071, A183072, A183073, A183074.

a(n) = 2^A183071(n) - 1.

A183076 Composite numbers of the form 2^k - 1 for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

63, 1023, 16383, 32767, 67108863, 1073741823, 17179869183, 274877906943, 8796093022207, 2251799813685247, 4611686018427387903, 36893488147419103231

Offset: 1

Stuart Clary, Dec 23 2010

Needed factorizations are in the Cunningham Project.

			63 = 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

Amiram Eldar, Table of n, a(n) for n = 1..39
S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000668, A136005, A183075, A183077, A183078.

Cf. A000043, A136003, A183071, A183072, A183073, A183074.

a(n) = 2^A183072(n) - 1.

A183077 Numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 8796093022207, 2305843009213693951, 604462909807314587353087, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Stuart Clary, Dec 23 2010

The Mersenne primes (A000668) are contained in this sequence.

Needed factorizations are in the Cunningham Project.

			8796093022207 = 2^43 - 1 = 431 * 9719 * 2099863.

Amiram Eldar, Table of n, a(n) for n = 1..26
S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000668, A136005, A183075, A183076, A183078.

Cf. A000043, A136003, A183071, A183072, A183073, A183074.

a(n) = 2^A183073(n) - 1.

A183078 Composite numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

Original entry on oeis.org

8796093022207, 604462909807314587353087, 12554203470773361527671578846415332832204710888928069025791, 4171849679533027504677776769862406473833407270227837441302815640277772901915313574263597826047

Offset: 1

Stuart Clary, Dec 23 2010

Needed factorizations are in the Cunningham Project.

			8796093022207 = 2^43 - 1 = 431 * 9719 * 2099863.

Amiram Eldar, Table of n, a(n) for n = 1..12
S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000668, A136005, A183075, A183076, A183077.

Cf. A000043, A136003, A183071, A183072, A183073, A183074.

a(n) = 2^A183074(n) - 1.

A183072 Positive integers k such that 2^k - 1 is composite and each of its prime divisors has the form 4j + 3.

Original entry on oeis.org

6, 10, 14, 15, 26, 30, 34, 38, 43, 51, 62, 65, 79, 85, 86, 93, 95, 122, 129, 130, 133, 158, 170, 193, 254, 255, 301, 311, 331, 349, 389, 445, 557, 577, 579, 631, 647, 1103, 1167

Offset: 1

Stuart Clary, Dec 23 2010

Needed factorizations are in the Cunningham Project.

			6 is in this sequence because 2^6 - 1 = 3^2 * 7, and 3 and 7 have the form 4j + 3.

S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000043, A136003, A183071, A183073, A183074.

Cf. A000668, A136005, A183075, A183076, A183077, A183078.

c4j3Q[n_]:=Module[{c=2^n-1},CompositeQ[c]&&AllTrue[(#-3)/4&/@ FactorInteger[ c] [[All,1]],IntegerQ]]; Select[Range[650],c4j3Q] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run *) (* Harvey P. Dale, Sep 23 2018 *)

A183076(n) = 2^a(n) - 1.

a(38)-a(39) from Amiram Eldar, Feb 18 2019

A183073 Prime numbers p such that each prime divisor of 2^p - 1 has the form 4j + 3.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 43, 61, 79, 89, 107, 127, 193, 311, 331, 349, 389, 521, 557, 577, 607, 631, 647, 1103

Offset: 1

Stuart Clary, Dec 23 2010

The exponents of the Mersenne primes (A000043) are contained in this sequence.
Needed factorizations are in the Cunningham Project.
Also in the sequence are 1279, 2203, 2281, 2909, 3217, 4253. - Amiram Eldar, Feb 18 2019

			43 is in this sequence because 2^43 - 1 = 431 * 9719 * 2099863, and each of those primes has the form 4j + 3.

S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000043, A136003, A183071, A183072, A183074.

Cf. A000668, A136005, A183075, A183076, A183077, A183078.

Select[Prime[Range[30]],And@@(IntegerQ[(#-3)/4]&/@Transpose[ FactorInteger[ 2^#-1]][[1]])&] (* Increase the value of Range to increase the number of terms generated, but processing times grow very quickly as the value increases. *)(* Harvey P. Dale, Jan 01 2013 *)

A183077(n) = 2^a(n) - 1.

a(26) from Amiram Eldar, Feb 18 2019

A183074 Prime numbers p such that 2^p - 1 is composite and each of its prime divisors has the form 4j + 3.

Original entry on oeis.org

43, 79, 193, 311, 331, 349, 389, 557, 577, 631, 647, 1103

Offset: 1

Stuart Clary, Dec 23 2010

Needed factorizations are in the Cunningham Project.

			43 is in this sequence because 2^43 - 1 = 431 * 9719 * 2099863, and each of those primes has the form 4j + 3.

S. S. Wagstaff, Jr., The Cunningham Project.

Cf. A000043, A136003, A183071, A183072, A183073.

Cf. A000668, A136005, A183075, A183076, A183077, A183078.

cQ[n_]:=Module[{x=2^n-1},!PrimeQ[x]&&Union[Mod[Transpose[ FactorInteger[ x]][[1]],4]]=={3}]; Select[Prime[Range[120]],cQ] (* Harvey P. Dale, Jun 17 2014 *)

A183078(n) = 2^a(n) - 1.

a(12) from Amiram Eldar, Feb 18 2019

A136007 Primes of the form 2^p - 1, where p is a prime number that is not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151

Offset: 1

Omar E. Pol, Dec 17 2007

nonn

Primes in A136005.

Amiram Eldar, Table of n, a(n) for n = 1..13

Cf. A000079, A000668, A136002, A136003, A136005, A152961.

q[n_] := n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]; 2^Select[MersennePrimeExponent[Range[13]], q] - 1 (* Amiram Eldar, Oct 20 2024 *)

a(n) = 2^A152961(n) - 1. - Amiram Eldar, Oct 20 2024

Extended by Ray Chandler, Dec 13 2008

cp's OEIS Frontend

A136005 Numbers of the form 2^p - 1, where p is a prime number that is not the sum, minus 1, of a Pythagorean triple.

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A183071 Positive integers k such that each prime divisor of 2^k - 1 has the form 4j + 3.

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A183075 Numbers of the form 2^k - 1 for which each prime divisor has the form 4j + 3.

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A183076 Composite numbers of the form 2^k - 1 for which each prime divisor has the form 4j + 3.

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A183077 Numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

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A183078 Composite numbers of the form 2^p - 1, with p prime, for which each prime divisor has the form 4j + 3.

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A183072 Positive integers k such that 2^k - 1 is composite and each of its prime divisors has the form 4j + 3.

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A183073 Prime numbers p such that each prime divisor of 2^p - 1 has the form 4j + 3.

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