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.

A096477 Subscripts for "secondary twin-primes": numbers k such that prime(prime(k)+1) - prime(prime(k)) = A073124(k) = 2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 13, 14, 23, 24, 29, 30, 40, 59, 63, 65, 71, 74, 90, 103, 106, 110, 112, 117, 122, 128, 132, 187, 188, 193, 207, 248, 257, 258, 267, 271, 281, 285, 292, 296, 299, 300, 303, 304, 311, 317, 325, 340, 343, 344, 354, 356, 360, 368, 382, 389, 395, 402
Offset: 1

Views

Author

Labos Elemer, Jun 23 2004

Keywords

Examples

			24 is a term since prime(prime(24)+1) - prime(prime(24)) = prime(89+1) - prime(89) = 463 - 461 = 2.

Links

Crossrefs

Cf. A006450, A072677, A073124, A096478.

Programs

  • Mathematica
    Flatten[Position[Table[Prime[Prime[n]+1]-Prime[Prime[n]], {n, 1, 1000}], 2]]

Formula

a(n) = primepi(A096478(n)). - Amiram Eldar, Aug 13 2024

A096479 "Secondary twin primes": a(n) = A006450(A096477(n)).

Original entry on oeis.org

3, 5, 11, 17, 41, 59, 179, 191, 431, 461, 599, 617, 1031, 1787, 2027, 2081, 2381, 2549, 3299, 4091, 4217, 4421, 4517, 4787, 5021, 5441, 5651, 8999, 9041, 9461, 10457, 13217, 13709, 13757, 14591, 14867, 15641, 16061, 16451, 16901, 17189, 17291
Offset: 1

Views

Author

Labos Elemer, Jun 23 2004

Keywords

Examples

			a(10) = 461 since prime(10) = 89 and prime(89 + 1) - prime(89) = 463 - 461 = 2.

Links

Crossrefs

Subsequence of A001359.
Cf. A072677, A006450, A073124, A094068, A096477, A096478.

Programs

  • Mathematica
    Prime[Prime[Flatten[Position[Table[Prime[Prime[n]+1] -Prime[Prime[n]], {n, 1, 1000}], 2]]]]

A096480 a(n) = Min{x : A073124(x) = 2n}.

Original entry on oeis.org

1, 8, 5, 22, 16, 15, 33, 67, 62, 164, 88, 56, 73, 202, 134, 504, 201, 261, 799, 1461, 289, 282, 1309, 1053, 1143, 939, 527, 3531, 2179, 4751, 2461, 5308, 2837, 3983, 1946, 8622, 9488, 12862, 6377, 4653, 7594, 7646, 19251, 22538, 9561, 32509, 26146, 17568
Offset: 1

Views

Author

Labos Elemer, Jun 23 2004

Keywords

Examples

			For n = 4: a(4) = 22 since A073124(22) = prime(1+prime(22)) - prime(prime(22)) = prime(1+79) - prime(79) = 409 - 401 = 8.
For n = 5: a(5) = 16 since A073124(16) = prime(1+prime(16)) - prime(prime(16)) = prime(54) - prime(53) = 251 - 241 = 10.

Links

Crossrefs

Cf. A000040, A006450, A073124, A072677, A096477, A096478, A096479, A096481, A096482.

Programs

  • Mathematica
    Table[Min[Flatten[Position[Table[Prime[Prime[n]+1]- Prime[Prime[n]], {n, 1, 5000}], 2*j]]], {j, 1, 20}]
seq[max_] := Module[{p = Prime[Range[max + 1]], m = PrimePi[max], ind, t}, ind = Prime[Range[m]]; t = p[[ind + 1]] - p[[ind]]; TakeWhile[FirstPosition[t, 2*#] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]]; seq[10^6] (* Amiram Eldar, Feb 15 2025 *)
  • PARI
    {m=48;for(n=1,m,k=1;while((prime(prime(k)+1)-prime(prime(k)))!=2*n,k++);print1(k,","))} \\ Klaus Brockhaus, Jun 27 2004

Extensions

a(31)-a(48) from Klaus Brockhaus, Jun 27 2004

A096481 a(n) = A000040(A096480(n)).

Original entry on oeis.org

2, 19, 11, 79, 53, 47, 137, 331, 293, 971, 457, 263, 367, 1231, 757, 3607, 1229, 1663, 6131, 12227, 1879, 1831, 10733, 8423, 9221, 7393, 3793, 32941, 19213, 45863, 21961, 51871, 25763, 37591, 16879, 89017, 98867, 138241, 63611, 44773, 77279, 77783
Offset: 1

Views

Author

Labos Elemer, Jun 23 2004

Keywords

Links

Crossrefs

Cf. A000040, A006450, A073124, A072677, A096477, A096478, A096479, A096480, A096482.

Programs

  • Mathematica
    Prime[Table[Min[Flatten[Position[Table[Prime[Prime[n]+1]- Prime[Prime[n]], {n, 1, 10000}], 2*j]]], {j, 1, 100}]]
  • PARI
    {m=42;for(n=1,m,k=1;while((prime(prime(k)+1)-prime(prime(k)))!=2*n,k++);print1(prime(k),","))} \\ Klaus Brockhaus, Jun 27 2004

Extensions

a(31)-a(42) from Klaus Brockhaus, Jun 27 2004

A096482 a(n) = prime(prime(A096480(n))).

Original entry on oeis.org

3, 67, 31, 401, 241, 211, 773, 2221, 1913, 7649, 3229, 1669, 2477, 10009, 5749, 33647, 9973, 14107, 60821, 130729, 16141, 15683, 113233, 86629, 95651, 74959, 35617, 388403, 214993, 557093, 248909, 637003, 296843, 448451, 186481, 1145899, 1283603, 1845637, 795349, 542603
Offset: 1

Views

Author

Labos Elemer, Jun 23 2004

Keywords

Comments

a(n) = prime(p) where p is the smallest prime such that prime(p+1) - prime(p) = 2*n.
Both a(n) and a(n) + 2*n are primes while pi(a(n)) = A096481(n) and pi(pi(a(n))) = A096480(n).

Examples

			a(2) = 67 = prime(19) since prime(19+1) - prime(19) = 71 - 67 = 2*2 and 19 is the smallest prime with this property.

Links

Crossrefs

Cf. A000040, A006450, A073124, A072677, A096477, A096478, A096479, A096480, A096481.

Programs

  • Mathematica
    Prime[Prime[Table[Min[Flatten[Position[Table[Prime[Prime[n]+1]- Prime[Prime[n]], {n, 1, 5000}], 2*j]]], {j, 1, 100}]]]
  • PARI
    a(n) = {my(p=2); while((prime(p+1)-prime(p))!=2*n, p=nextprime(p+1)); prime(p)} \\ Klaus Brockhaus, Jun 27 2004
  • PARI
    a(n) = {my(p=2,k=1); forprime(q=3, oo, if(q==p+2*n && isprime(k), return(p)); p=q; k++)} \\ Andrew Howroyd, Dec 16 2024

Formula

a(n) = A006450(A096480(n)) = prime(A096481(n)).
a(n) + 2*n = prime(1 + prime(A096480(n))).

Extensions

a(31)-a(36) from Klaus Brockhaus, Jun 27 2004
a(37) onwards from Andrew Howroyd, Dec 16 2024

A237283 Primes p with prime(prime(p)) + 2 also prime.

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 59, 71, 103, 193, 257, 271, 281, 311, 317, 389, 433, 439, 463, 569, 577, 619, 673, 683, 691, 797, 811, 857, 859, 887, 1031, 1069, 1109, 1129, 1153, 1229, 1307, 1597, 1613, 1867, 1949, 1951, 2069, 2297, 2477, 2551, 2621, 2657, 2699, 2753
Offset: 1

Views

Author

Zhi-Wei Sun, Feb 05 2014

Keywords

Comments

This sequence is interesting because of the conjecture in A237253.
A236481, A236482 and A236484 are subsequences of the sequence.

Examples

			a(1) = 2 since 2 and prime(prime(2)) + 2 = prime(3) + 2 = 7 are both prime.

Links

Crossrefs

Cf. A000040, A001359, A006512, A096478, A218829, A236457, A236458, A236467, A236481, A236482, A236484, A236568, A237253.

Programs

  • Mathematica
    n=0;Do[If[PrimeQ[Prime[Prime[Prime[k]]]+2],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[500]],PrimeQ[Prime[Prime[#]]+2]&] (* Harvey P. Dale, May 30 2018 *)

A167761 a(n) = sqrt(A167657(n)).

Original entry on oeis.org

0, 1, 2, 3, 16, 5, 36, 7, 64, 729, 10, 1331, 144, 13, 196, 3375, 4096, 17, 5832, 361, 20, 9261, 484, 12167, 331776, 625, 26, 729, 28, 841, 21870000000, 961, 32768, 33, 45435424, 35, 46656, 50653, 1444, 59319, 64000, 41, 130691232, 43, 1936, 45, 9474296896
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 11 2009, corrected Nov 12 2009

Keywords

Comments

a(n) = prime => A096478(n).
If a(n) = prime => A096478(n). - Giovanni Teofilatto, Nov 12 2009

Crossrefs

Cf. A167657.
Cf. A096478. - Giovanni Teofilatto, Nov 12 2009

Programs

  • Maple
    A000720 := proc(n) numtheory[pi](n) ; end proc: A167761 := proc(nmax) local ol,a ; ol := 0 ; a := 0 ; for n from 1 to nmax do if A000720(n) = ol then a := a*ol ; else printf("%d,",sqrt(a)) ; a := A000720(n) ; end if; ol := A000720(n) ; end do ; return ; end proc: A167761(260) ; # R. J. Mathar, Jan 30 2010

Extensions

Previous a(15) removed, a(35) and a(43) corrected by R. J. Mathar, Jan 30 2010
a(47) from Kritsada Moomuang, Apr 19 2025
Showing 1-7 of 7 results.

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