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.

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A113157 Primes such that the sum of the predecessor and successor primes is divisible by 41.

Original entry on oeis.org

283, 409, 739, 983, 1021, 2213, 2251, 2339, 2663, 2749, 3079, 3821, 3931, 4219, 4463, 4799, 4919, 5413, 5741, 6271, 6917, 7703, 7753, 7873, 8287, 8861, 9013, 10091, 10427, 10709, 11317, 11483, 12421, 12917, 13037, 13693, 13781, 14029, 14759
Offset: 1

Views

Author

Jonathan Vos Post, Jan 05 2006

Keywords

Comments

A112681 is mod 3 analogy. A112794 is mod 5 analogy. A112731 is mod 7 analogy. A112789 is mod 11 analogy. A112795 is mod 13 analogy. A112796 is mod 17 analogy. A112804 is mod 19 analogy. A112847 is mod 23 analogy. A112859 is mod 29 analogy.

Examples

			a(1) = 283 since prevprime(283) + nextprime(283) = 281 + 293 = 574 = 41 * 14.
a(2) = 409 since prevprime(409) + nextprime(409) = 401 + 419 = 820 = 41 * 20.
a(3) = 739 since prevprime(739) + nextprime(739) = 733 + 743 = 1476 = 41 * 36.
a(4) = 983 since prevprime(983) + nextprime(983) = 977 + 991 = 1968 = 41 * 48.

Crossrefs

Cf. A000040, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.

Programs

  • Mathematica
    Prime@Select[Range[2, 1766], Mod[Prime[ # - 1] + Prime[ # + 1], 41] == 0 &] (* Robert G. Wilson v *)
Transpose[Select[Partition[Prime[Range[2000]],3,1],Divisible[First[#] + Last[#], 41]&]][[2]] (* Harvey P. Dale, Jul 25 2012 *)

Formula

a(n) = prime(i) is in this sequence iff prime(i-1)+prime(i+1) = 0 mod 41. a(n) = A000040(i) is in this sequence iff A000040(i-1)+A000040(i+1) = 0 mod 41.

Extensions

More terms from Robert G. Wilson v, Jan 11 2006

A113158 Primes such that the sum of the predecessor and successor primes is divisible by 43.

Original entry on oeis.org

521, 821, 859, 1069, 1459, 1549, 2203, 2411, 2539, 2837, 2969, 3011, 3089, 3359, 3613, 3823, 4259, 4339, 4561, 4643, 4783, 5503, 5557, 6067, 6619, 6967, 7481, 7699, 7741, 8263, 8779, 9419, 10103, 12041, 12379, 12641, 12899, 13417, 13721, 13759
Offset: 1

Views

Author

Jonathan Vos Post, Jan 05 2006

Keywords

Comments

A112681 is mod 3 analogy. A112794 is mod 5 analogy. A112731 is mod 7 analogy. A112789 is mod 11 analogy. A112795 is mod 13 analogy. A112796 is mod 17 analogy. A112804 is mod 19 analogy. A112847 is mod 23 analogy. A112859 is mod 29 analogy.

Examples

			a(1) = 521 since prevprime(521) + nextprime(521) = 509 + 523 = 1032 = 43 * 24.
a(2) = 821 since prevprime(821) + nextprime(821) = 811 + 823 = 1634 = 43 * 38.
a(3) = 859 since prevprime(859) + nextprime(859) = 857 + 863 = 1720 = 43 * 40.
a(4) = 1069 since prevprime(1069)+nextprime(1069) = 1063+1087 = 2150 = 43 * 50.

Crossrefs

Cf. A000040, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.

Programs

  • Mathematica
    Prime@Select[Range[2, 1657], Mod[Prime[ # - 1] + Prime[ # + 1], 43] == 0 &] (* Robert G. Wilson v *)

Formula

a(n) = prime(i) is in this sequence iff prime(i-1)+prime(i+1) = 0 mod 43. a(n) = A000040(i) is in this sequence iff A000040(i-1)+A000040(i+1) = 0 mod 43.

Extensions

More terms from Robert G. Wilson v, Jan 11 2006

A112855 Numbers n such that (2*n)!/n!+1 is prime.

Original entry on oeis.org

1, 2, 5, 10, 24, 30, 72, 340, 379, 712, 1020, 1647, 3654, 7923
Offset: 1

Views

Author

Hugo Pfoertner, Sep 25 2005

Keywords

Comments

Next term is > 10000.

Crossrefs

Cf. A112856 primes of the form (2*n)!/n!+1, A112853 (2*n)!/n!-1 is prime, A066726 (2*n)!/(n!)^2-1 is prime, A112859 (2*n)!/(n!)^2+1 is prime, A112861 (2*n)!/(2*(n!)^2)-1 is prime, A112863 (2*n)!/(2*(n!)^2)+1 is prime.

Programs

  • Mathematica
    Select[Range[8000],PrimeQ[(2#)!/#!+1]&] (* Harvey P. Dale, Mar 23 2012 *)

A112545 Least odd number k greater than 1 such that the sum of the predecessor and successor primes of the n-th prime is divisible by k or if no such odd k exists then 2.

Original entry on oeis.org

7, 5, 2, 5, 7, 2, 5, 3, 3, 3, 3, 5, 11, 3, 53, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 13, 53, 5, 59, 61, 3, 3, 11, 5, 3, 157, 3, 3, 173, 3, 5, 11, 97, 7, 3, 211, 3, 113, 5, 3, 3, 5, 3, 257, 263, 3, 3, 3, 5, 7, 5, 151, 5, 157, 7, 3, 3, 7, 5, 3, 3, 3, 373, 3, 3, 3, 5, 13, 5, 5, 5, 7, 3, 3, 3, 3, 5, 5, 29, 3, 3
Offset: 2

Views

Author

Robert G. Wilson v, Jan 11 2006

Keywords

Comments

From Robert Israel, Apr 20 2017: (Start)
a(n) = A078701(prime(n-1)+prime(n+1)) unless that is 1, in which case a(n)=2.
a(n) = 2 if and only if for some m, A007053(m) = n or n-1 with prime(n-1)+prime(n+1) = 2^(m+1). The first m for which this occurs are 3,4,9,379,593, corresponding to n = 4,7,97 and approximately 3*10^116 and 1*10^181. Are there infinitely many? (End)

Links

Crossrefs

Cf. A000040, A112686, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.
Cf. A007053, A078701.

Programs

  • Maple
    f:= proc(n) local t; t:= min(numtheory:-factorset(ithprime(n-1)+ithprime(n+1)) minus {2}); if t::integer then t else 2 fi end proc:
map(f, [$2..200]); # Robert Israel, Apr 20 2017
  • Mathematica
    f[n_] := Block[{k = 3, s = Prime[n - 1] + Prime[n + 1]}, While[Mod[s, k] != 0 && k <= s, k += 2]; If[k > s, 2, k]]; Table[ f[n], {n, 2, 92}]
  • PARI
    a(n) = {p = prime(n); s = precprime(p-1) + nextprime(p+1); f = factor(s); if (#f~ > 1, f[2,1], f[1,1]);} \\ Michel Marcus, Apr 22 2017

A112686 Smallest prime p such that the sum of the predecessor and successor primes is divisible by n.

Original entry on oeis.org

3, 5, 23, 7, 5, 23, 3, 7, 29, 5, 31, 23, 79, 13, 73, 7, 151, 29, 59, 11, 61, 31, 229, 23, 73, 79, 29, 13, 149, 73, 311, 17, 31, 151, 71, 37, 181, 59, 79, 19, 283, 61, 521, 43, 89, 229, 1277, 23, 197, 73, 151, 79, 53, 29, 109, 83, 59, 149, 113, 89, 127, 311, 61, 383, 389, 31
Offset: 1

Views

Author

Robert G. Wilson v, Jan 11 2006

Keywords

Crossrefs

Cf. A000040, A112545, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.

Programs

  • Mathematica
    f[n_] := Block[{k = 2}, While[ Mod[ Prime[k - 1] + Prime[k + 1], n] != 0, k++ ]; Prime[k]]; Array[f, 66]
With[{prs=Partition[Prime[Range[250]],3,1]},Transpose[Flatten[ Table[ Select[ prs, Divisible[ First[#]+Last[#],n]&,1],{n,70}],1]][[2]]] (* Harvey P. Dale, Jan 16 2014 *)
Previous Showing 11-15 of 15 results.

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