A183071 -id:A183071 - cp's OEIS frontend

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

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

A298705 Numbers from the 15-theorem for universal Hermitian lattices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 14, 15

Offset: 1

Ralf Steiner, Jan 29 2018

Unlike the fifteen theorem for universal quadratic forms (A030050), this theorem includes the number 13 arising from the case Q(sqrt(-39)).

Byeong Moon Kim, Ji Young Kim, and Poo-Sung Park, The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields, Math. Comp. 79 (2010), 1123-1144.

Cf. A030050, A183071.

cp's OEIS Frontend

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

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Programs

Mathematica

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Extensions

A298705 Numbers from the 15-theorem for universal Hermitian lattices.

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