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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A123993 Primes p such that p^2 is an interprime = average of two successive primes.

Original entry on oeis.org

2, 3, 41, 907, 1151, 1553, 1609, 1667, 1801, 1907, 1933, 2351, 2473, 2531, 2953, 3001, 3571, 4007, 4073, 4253, 4663, 5023, 5417, 5881, 6143, 6257, 6329, 6343, 7879, 8461, 8521, 8563, 9041, 9067, 10103, 10781, 11243, 11251, 11257, 12097, 12413, 13217
Offset: 1

Views

Author

Alexander Adamchuk, Oct 30 2006

Keywords

Comments

Primes in A075190 (numbers n such that n^2 is an interprime).

Crossrefs

Cf. A075190, A024675 (interprimes).

Programs

  • Mathematica
    Select[PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Select[ Range[25000], 2#^2 == PrevPrim[ #^2] + NextPrim[ #^2] &],PrimeQ]
atsp[n_]:=Module[{n2=n^2},(NextPrime[n2]+NextPrime[n2,-1])/2==n2]; Select[Prime[Range[2000]],atsp]  (* Harvey P. Dale, Jan 05 2011 *)
  • PARI
    isok(p) = isprime(p) && ((nextprime(p^2) + precprime(p^2)) / 2 - p^2 == 0); \\ Michel Marcus, Dec 11 2020

A133450 Difference between 4*n^2 and the average of the two prime numbers which bracket this number.

Original entry on oeis.org

0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 1, 2, 1, 4, 3, -2, -2, 2, 1, 1, -4, -5, -5, 1, 10, 1, 3, 7, -2, 0, 4, 0, 3, -5, 4, 0, 2, 12, 0, -9, -2, 6, -6, -3, 3, 0, 2, 1, -3, 10, -9, 1, 10, -3, 1, 0, 4, 2, -2, 5, 1, 1, 8, -12, 5, -1, 8, -2, 0, 0, -3, -1, 1, 2, 8, -4, 12, 3, 4, 5, 1, -2, -10, 0, 10
Offset: 1

Views

Author

Alexander R. Povolotsky, Dec 22 2007

Keywords

Examples

			a(1)=0 because 4 - (3 + 5)/2 = 0
a(2)=1 because 16 - (13 + 17)/2 = 1
a(3)=2 because 36 - (31 + 37)/2 = 2
a(4)=0 because 64 - (61 + 67)/2 = 0
a(5)=1 because 100 - (97 + 101)/2 = 1

Links

Crossrefs

Cf. A056929, A101593, A075190.

Programs

  • Mathematica
    Table[n^2-(Prime[PrimePi[n^2]]+Prime[PrimePi[n^2]+1])/2,{n,2,200,2}] (* Zak Seidov *)
diff4[n_]:=Module[{x=4n^2},x-(NextPrime[x]+NextPrime[x,-1])/2]; Array[ diff4,90] (* Harvey P. Dale, Aug 31 2017 *)
  • PARI
    A133450(n)=4*n^2-(precprime(4*n^2)+nextprime(4*n^2))/2 \\ M. F. Hasler, Dec 26 2007

Formula

a(n) = A056929(2n). - M. F. Hasler, Dec 26 2007

Extensions

Corrected and extended by Zak Seidov, Dec 23 2007
Edited by N. J. A. Sloane, Dec 23 2007

A205881 Second pentagonal numbers that are interprime.

Original entry on oeis.org

15, 26, 260, 495, 610, 950, 1190, 1962, 2420, 2667, 3197, 3337, 3480, 6370, 7597, 14455, 15965, 18205, 39447, 42926, 43947, 50325, 57135, 63140, 64377, 65000, 66255, 74037, 78090, 82251, 86520, 87967, 106267, 107870, 121980, 125426, 136957, 140607, 153760
Offset: 1

Views

Author

Zak Seidov, Feb 01 2012

Keywords

Comments

Intersection of A005449 and A024675. Also see references in A075190 about interprimes of different forms.

Examples

			a(1) = 15 = A024675(5) = A005449(3).
a(2) = 26 = A024675(8) = A005449(4).

Links

Crossrefs

Cf. A005449, A024675, A075190.

Programs

  • Mathematica
    Select[Mean/@Partition[Prime[Range[15000]],2,1],IntegerQ[(Sqrt[1+24#]-1)/6]&] (* Harvey P. Dale, Feb 02 2020 *)

A248790 Numbers n with the property that p = n^2 - 11 and q = n^2 + 11 are consecutive primes.

Original entry on oeis.org

510, 720, 1200, 2190, 4350, 4980, 5040, 5250, 5670, 6810, 8280, 8490, 9150, 10140, 10650, 11430, 12510, 13800, 13980, 14160, 14640, 14700, 14820, 15000, 15750, 16890, 17220, 18180, 18270, 18750, 19110, 20940, 21270, 22050, 24000, 24570, 24720, 24990, 25620, 25920, 26520
Offset: 1

Views

Author

Zak Seidov, Oct 14 2014

Keywords

Comments

All terms are == 0 (mod 30).

Examples

			n=510, p=260089=prime(22845), q=260111=prime(22846).

Links

Crossrefs

Subsequence of A176683 and of A075190. E.g., a(1)=510=A075190(62)=A176683(6).
Cf. A248785.

Programs

  • Mathematica
    Select[Range[30000],With[{c=#^2-11},PrimeQ[c]&&NextPrime[c]==c+22&]] (* Harvey P. Dale, Apr 03 2025 *)
  • PARI
    isok(n) = isprime(p=n^2-11) && isprime(q=n^2+11) && (q==nextprime(p+1)); \\ Michel Marcus, Oct 14 2014
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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