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.

Previous Showing 21-29 of 29 results.

A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

Original entry on oeis.org

9, 25, 65, 161, 2049, 4609, 22529, 1048577, 44040193, 283467841537, 1202590842881, 256065421246102339102334047485953, 4259306016766850789028922770063361, 356615920533143509709616588588493085605889, 57729314674570665269045550892293179276409335447553
Offset: 1

Views

Author

K. D. Bajpai, May 04 2014

Keywords

Comments

The k-th Cullen number Cullen(k) = k*2^k + 1.
If Cullen(k) is semiprime, it is in the sequence.
The next term, a(16), has 52 digits.

Examples

			a(4) = 161 = (5*2^5+1) is 5th Cullen number and 161 = 7 * 23 is semiprime.
a(5) = 2049 = (8*2^8+1) is 8th Cullen number and 2049 = 3 * 683 is semiprime.

Links

Crossrefs

Cf. A001358, A002064, A003261, A005849, A242175.

Programs

  • Magma
    IsSemiprime:=func; [s: n in [1..200] | IsSemiprime(s) where s is n*2^n+1]; // // Vincenzo Librandi, May 07 2014
  • Maple
    with(numtheory): A242116:= proc(); if bigomega(x*2^x+1) = 2 then RETURN (x*2^x+1);  fi; end: seq(A242116 (), x=1..200);
  • Mathematica
    cullen[n_] := n * 2^n + 1; Select[cullen[Range[35]], PrimeOmega[#] == 2 &] (* Amiram Eldar, Nov 27 2019 *)
  • PARI
    select(n->bigomega(n)==2, vector(90,n,n<Charles R Greathouse IV, May 06 2014

Formula

a(n) = A002064(A242175(n)). - Amiram Eldar, Nov 27 2019

A383473 Integers k such that d*2^k + 1 is prime for some divisor of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 15, 16, 18, 25, 30, 36, 51, 55, 63, 66, 69, 75, 81, 85, 134, 141, 162, 189, 201, 209, 220, 245, 276, 324, 408, 438, 446, 456, 534, 616, 675, 693, 726, 892, 900, 1305, 1326, 1494, 1824, 2208, 2394, 2766, 2826, 3024, 3168, 3189, 3690, 3703, 3880, 3912, 3927, 4410, 4543, 4713
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 27 2025

Keywords

Examples

			6 is in the sequence a term because 3*2^6 + 1 = 193 prime for divisor 3 of k = 6.

Crossrefs

Supersequence of A005849.
Cf. A002064, A027750, A382811.

Programs

  • Magma
    [k: k in [1..900] | not #[d: d in Divisors(k) | IsPrime(d*2^k+1)] eq 0];
  • Mathematica
    q[k_] := AnyTrue[Divisors[k], PrimeQ[# * 2^k +1] &]; Select[Range[4000], q] (* Amiram Eldar, Apr 28 2025 *)
  • PARI
    isok(k) = fordiv(k, d, if (ispseudoprime(d*2^k+1), return(1))); return(0); \\ Michel Marcus, Apr 28 2025

A137716 Number of digits in the decimal expansion of the n-th Cullen prime.

Original entry on oeis.org

1, 45, 1423, 1749, 1994, 5573, 9726, 9779, 17964, 27347, 79002, 108761, 145072, 407850, 1905090, 2010852
Offset: 1

Views

Author

Ant King, Feb 09 2008

Keywords

Comments

Cullen primes are prime numbers of the form k*2^k+1. This sequence is complete for all values of n up to 3500000.

Examples

			As the sixth Cullen prime, 18496*2^18496 + 1 = 1.311...*10^5572, is a 5573-digit number, we have a(6) = 5573.

Links

Crossrefs

Cf. A055642, A005849, A002064.

Formula

a(n) = A055642(A050920(n)). [Corrected by Georg Fischer, Nov 18 2023]

Extensions

a(15)-a(16) from Amiram Eldar, Oct 27 2024

A195418 a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number.

Original entry on oeis.org

1, 1, 3, 5, 3, 33, 5, 33, 341, 1045, 189, 1299, 891, 4437, 9477, 581, 3855, 105525, 27825, 23751, 173043, 10531345, 56511, 2386125, 380955, 256861, 24926139, 5108467, 32397379, 930343095, 930291, 36512775
Offset: 0

Views

Author

Alonso del Arte, Sep 20 2011

Keywords

Comments

When C(n) is prime (or 1), then a(n) = 1; that is, n is in A005849.
On the penultimate page of their paper, Grau and Luca ask for "a good (large) lower bound on this quantity which is valid for all n and which tends to infinity with n."

Examples

			a(2) = 3 because the second Cullen number is 9; phi(9) = 6, therefore 6/gcd(8, 6) = 6/2 = 3.

Links

Crossrefs

Cf. A000010, A002064, A005849, A160595.

Programs

  • Mathematica
    cullen[n_] := n(2^n) + 1; Table[EulerPhi[cullen[n]]/GCD[cullen[n] - 1, EulerPhi[cullen[n]]], {n, 0, 39}]
  • PARI
    a(n)=my(C=n<Charles R Greathouse IV, Feb 05 2013

A265013 Numbers n such that n*9^n + 1 is prime.

Original entry on oeis.org

2, 12382, 27608, 31330, 117852
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Nov 30 2015

Keywords

Comments

All terms are even. - Robert Israel, Jan 18 2016

Crossrefs

Cf. A005849, A006552, A007646, A242176, A242177, A242178, A007647, A242196, A242197, A242198, A242199, A007648.

Programs

  • Magma
    [n: n in [0..100000] | IsPrime(n*9^n+1)];
  • Mathematica
    Select[Range[100000], PrimeQ[# 9^# + 1] &]
  • PARI
    for(n=1,100000, if(isprime(n*9^n+1), print1(n,", ")))

A367287 Numbers k such that k^6*2^k + 1 is a prime.

Original entry on oeis.org

1, 2, 4, 62, 80, 122, 136, 658, 1918, 2998, 3404, 4042, 5678, 8378, 10438, 23530, 24610, 29090, 41650, 120818
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 21 2023

Keywords

Comments

No further terms <= 100000. - Michael S. Branicky, Nov 22 2023

Crossrefs

Numbers k such that k^m*2^k + 1 is a prime: 0, 1, 2, 4, 8, 16, .. (m = 0), A005849 (m = 1), A058780 (m = 2), A357612 (m = 3), A366422 (m = 4), A367421 (m = 5), this sequence (m = 6).
Cf. A367478.

Programs

  • Magma
    [k: k in [1..1000] | IsPrime(k^6*2^k + 1)];

Extensions

a(16)-a(19) from Michael S. Branicky, Nov 21 2023
a(20) from Michael S. Branicky, Aug 30 2024

A367560 Numbers k such that k^7*2^k + 1 is a prime.

Original entry on oeis.org

1, 3, 11, 51, 76, 123, 149, 274, 311, 328, 381, 639, 737, 898, 1156, 9017, 13200, 18348, 26388, 30081
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 22 2023

Keywords

Comments

No further terms <= 100000. - Michael S. Branicky, Aug 28 2024

Crossrefs

Numbers k such that k^m*2^k + 1 is a prime: 0, 1, 2, 4, 8, 16, .. (m = 0), A005849 (m = 1), A058780 (m = 2), A357612 (m = 3), A366422 (m = 4), A367421 (m = 5), A367287 (m = 6), this sequence (m = 7).
Cf. A092506.

Programs

  • Magma
    [k: k in [1..2000] | IsPrime(k^7*2^k+1)];

Extensions

a(18)-a(20) from Michael S. Branicky, Nov 22 2023

A382887 Numbers k such that (k*2^d + 1)*(d*2^k + 1) is semiprime for some divisor d of k.

Original entry on oeis.org

1, 2, 8, 12, 30, 51, 63, 141, 201, 209, 534, 4713, 5795, 6611, 7050, 18496, 24105, 32292, 32469, 52782, 59656, 80190, 90825
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 07 2025

Keywords

Comments

a(24) > 10^5. - Michael S. Branicky, Apr 08 2025

Examples

			12 is in this sequence because (12*2^3 + 1)*(3*2^12 + 1) = 97*12289 is semiprime for divisor 3 of 12.

Crossrefs

Supersequence of A005849.
Cf. A001358, A002064, A382646.

Programs

  • Magma
    [n: n in [1..1000] | not #[d: d in Divisors(n) | IsPrime(d*2^n+1) and IsPrime(n*2^d+1)] eq 0];
  • PARI
    isok(k) = fordiv(k, d, if (ispseudoprime(k*2^d+1) && ispseudoprime(d*2^k+1), return(1))); \\ Michel Marcus, Apr 16 2025
  • Python
    from itertools import count, islice
from sympy import isprime, divisors
def A382887_gen(): # generator of terms
    yield from filter(lambda k:any(isprime((k<A382887_list = list(islice(A382887_gen(),10)) # Chai Wah Wu, Apr 15 2025

Extensions

a(10) inserted and a(15)-a(23) from Michael S. Branicky, Apr 08 2025

A338412 Numbers k such that k * 20^k + 1 is prime.

Original entry on oeis.org

3, 6207, 8076, 22356, 151456
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 25 2020

Keywords

Comments

a(6) > 219976.

Links

Crossrefs

Numbers k such that k * b^k + 1 is prime: A006093 (b=1), A005849 (b=2), A006552 (b=3), A007646 (b=4), A242176 (b=6), A242177 (b=7), A242178 (b=8), A265013 (b=9), A007647(b=10), A242196(b=12), A242197 (b=14), A242198 (b=15), A242199 (b=16), A007648 (b=18), this sequence (b=20).

Programs

  • Magma
    [n: n in [1..10000] |IsPrime(n*20^n+1)]
  • Mathematica
    Select[Range[1, 10000], PrimeQ[n*20^n+1] &]
  • PARI
    for(n=1, 10000, if(isprime(n*20^n+1), print1(n, ", ")))
Previous Showing 21-29 of 29 results.

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