cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A253851 Mersenne primes (A000668) of the form 2^sigma(n) - 1 for some n.

Original entry on oeis.org

7, 127, 8191, 2147483647, 170141183460469231731687303715884105727
Offset: 1

Views

Author

Jaroslav Krizek, Jan 16 2015

Keywords

Comments

Numbers n such that 2^sigma(n) - 1 is a Mersenne primes are given in A253849.
Sequence of corresponding values of sigma(n) are given in A253850 and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043).
If a(6) exists, it must be bigger than A000668(43) = 2^30402457-1.

Examples

			Mersenne prime 2147483647 is in the sequence because there are two numbers n (16 and 25) with 2^sigma(n) - 1 = 2^31 - 1 = 2147483647.

Crossrefs

Cf. A000043, A000203, A000668, A023195, A253849, A253850.

Programs

  • Magma
    Set(Sort([(2^SumOfDivisors(n))-1: n in[1..10000] | IsPrime((2^SumOfDivisors(n))-1)]));

A368651 Numbers k such that 2^sigma(k) - k is a prime.

Original entry on oeis.org

3, 5, 17, 49, 53, 185, 503, 1301, 1689, 1797, 5929, 14747, 20433, 29903, 42137, 64763
Offset: 1

Views

Author

J.W.L. (Jan) Eerland, Jan 02 2024

Keywords

Comments

If it exists, a(17) > 120000. - Michael S. Branicky, Aug 19 2024

Examples

			5 is in the sequence because 2^sigma(5)-5 = 2^6-5 = 59 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - n)];
  • Mathematica
    a[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[2^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

Extensions

a(16) from J.W.L. (Jan) Eerland, Jan 25 2024

A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

Original entry on oeis.org

2, 4, 9, 16, 25, 64
Offset: 1

Views

Author

Jaroslav Krizek, Jan 16 2015

Keywords

Comments

Also numbers n such that sigma(n) is in A000043, i.e., p = 2^sigma(n) - 1 is a Mersenne prime (A000668). The sequence of corresponding primes p reads: 7, 127, 8191, 2147483647, 2147483647, 170141183460469231731687303715884105727, ..., see A253851.
Subsequence of A023194 (numbers n such that sigma(n) is a prime), see there for an explanation why all terms except the first one are squares.
The sequence of values of sigma(a(n)) is 3, 7, 13, 31, 31, 127, ... and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043). See A253850.
Sequence differs from A023194 because A023194(7) = 289 but if a(7) exists, it must be a number n such that sigma(n) > A000043(43) = 30402457.
a(n) must be an even power of a prime. If it is the square of an odd prime, then this prime must be in A053182. If a(n) is an even power of 2, a(n)=2^(2k), then sigma(a(n))=2^(2k+1)-1. Thus, 2k+1 must be a double Mersenne prime exponent, i.e., such that the corresponding Mersenne prime is again a Mersenne exponent, cf. A103901. Only 4 such primes are known, and a(6)=2^6 (k=3) corresponds to the largest known prime of this type, 2^(2k+1)-1 = 127. - M. F. Hasler, Jan 21 2015

Examples

			4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)];
  • Mathematica
    a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* Michael De Vlieger, Jan 19 2015 *)

A367460 Numbers k such that 3^sigma(k) - k is a prime.

Original entry on oeis.org

1, 10, 52, 400, 2480, 7202, 28222
Offset: 1

Views

Author

J.W.L. (Jan) Eerland, Jan 26 2024

Keywords

Comments

a(8) > 67569.

Examples

			10 is in the sequence because 3^sigma(10) - 10 = 3^18 - 10 = 387420479 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((3^SumOfDivisors(n)) - n)]
  • Mathematica
    a[n_] := Select[Range@ n, PrimeQ[3^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[3^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

Extensions

a(7) from Michael S. Branicky, Jan 27 2024

A377927 Numbers k such that 4^sigma(k) - k is a prime.

Original entry on oeis.org

1, 5, 17, 57, 675, 1329, 1425, 3803, 39617
Offset: 1

Views

Author

J.W.L. (Jan) Eerland, Nov 11 2024

Keywords

Examples

			17 is in the sequence because 4^sigma(17) - 17 = 4^18 - 17 = 68719476719 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((4^SumOfDivisors(n)) - n)];
  • Mathematica
    a[n_] := Select[Range@ n, PrimeQ[4^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[4^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

Extensions

a(9) from Michael S. Branicky, Nov 11 2024

A377786 Numbers k such that 5^sigma(k) - k is a prime.

Original entry on oeis.org

4, 524, 7206, 11156
Offset: 1

Views

Author

J.W.L. (Jan) Eerland, Nov 11 2024

Keywords

Examples

			4 is in the sequence because 5^sigma(4) - 4 = 5^7 - 4 = 78121 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460, A377927.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((5^SumOfDivisors(n)) - n)];
  • Mathematica
    a[n_] := Select[Range@ n, PrimeQ[5^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[5^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

Extensions

a(4) from Michael S. Branicky, Nov 11 2024

A378512 Numbers k such that 6^sigma(k) - k is a prime.

Original entry on oeis.org

1, 7, 13, 77, 395, 2867, 3959, 5023
Offset: 1

Views

Author

J.W.L. (Jan) Eerland, Nov 29 2024

Keywords

Comments

a(9) > 10^5. - Michael S. Branicky, Dec 01 2024

Examples

			7 is in the sequence because 6^sigma(7) - 7 = 6^8 - 7 = 1679609 is prime.

Crossrefs

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460, A377927, A377786.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime((6^SumOfDivisors(n)) - n)];
  • Mathematica
    a[n_] := Select[Range@ n, PrimeQ[6^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[6^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]
  • PARI
    isok(k) = ispseudoprime(6^sigma(k) - k); \\ Michel Marcus, Dec 09 2024
Showing 1-7 of 7 results.

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