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 1-5 of 5 results.

A078866 The quadruples (d1,d2,d3,d4) with elements in {2,4,6} are listed in lexicographic order; for each quadruple, this sequence lists the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), if such a prime exists.

Original entry on oeis.org

5, 17, 41, 29, 71, 149, 3299, 7, 13, 67, 1597, 19, 43, 12637, 1601, 23, 593, 31, 61, 3313, 157, 47, 601, 151, 251, 3301
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

The 26 quadruples for which p exists are listed, in decimal form, in A078868.

Examples

			The term 12637 corresponds to the quadruple (4,6,6,6): 12637, 12641, 12647, 12653 and 12659 are consecutive primes.

Crossrefs

The quadruples are in A078868. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Extensions

Edited by Dean Hickerson, Dec 20 2002

A078867 Sorted version of A078866.

Original entry on oeis.org

5, 7, 13, 17, 19, 23, 29, 31, 41, 43, 47, 61, 67, 71, 149, 151, 157, 251, 593, 601, 1597, 1601, 3299, 3301, 3313, 12637
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

Each term is the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), for some quadruple (d1,d2,d3,d4) with elements in {2,4,6}.

Examples

			The term 3299 corresponds to the quadruple (2,6,6,6): 3299, 3301, 3307, 3313, 3319 are consecutive primes.

Crossrefs

The quadruples are in A078868. The same primes, in lexicographic order of the quadruples, are in A078866. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Extensions

Edited by Dean Hickerson, Dec 20 2002

A078870 Decimal concatenations of the 38 quintuples (d1,d2,d3,d4,d5) with elements in {2,4,6} for which there exists a prime p >= 7 such that the differences between the 6 consecutive primes starting with p are (d1,d2,d3,d4,d5).

Original entry on oeis.org

24246, 24626, 24662, 26424, 26426, 26462, 26466, 26642, 26646, 26664, 42424, 42462, 42466, 42646, 46246, 46264, 46266, 46626, 46662, 62462, 62642, 62646, 62664, 62666, 64242, 64246, 64264, 64624, 64626, 64662, 64666, 66264, 66266, 66424, 66462, 66466, 66626, 66646
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Examples

			66646 is in the sequence because 3301, 3307, 3313, 3319, 3323 and 3329 are consecutive primes with differences (6,6,6,4,6).

Crossrefs

Cf. A078868, A078869.

Programs

  • Mathematica
    With[{k = 5}, FromDigits /@ Select[Tuples[Range[2, 6, 2], k], Function[m, Count[Range[k - 1, 10^4], n_ /; Times @@ Boole@ Map[PrimeQ, Prime@ n + Accumulate@ m] == 1] > 0]]] (* Michael De Vlieger, Mar 25 2017 *) (* or *)
FromDigits /@ Union@ Select[ Partition[ Differences@ Prime[Range[4, 9000]], 5, 1], Max@ # <= 6 &] (* Giovanni Resta, Mar 25 2017 *)

Extensions

Edited by Dean Hickerson, Dec 20 2002

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.
An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Links

Crossrefs

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

Programs

  • Mathematica
    test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Extensions

Edited by Dean Hickerson, Dec 20 2002

A078871 Decimal concatenations of the 6-tuples (d1,d2,d3,d4,d5,d6) with elements in {2,4,6} for which there exists a prime p >= 7 such that the differences between the 7 consecutive primes starting with p are (d1,d2,d3,d4,d5,d6).

Original entry on oeis.org

242462, 246264, 246266, 246626, 264242, 264246, 264264, 264626, 264666, 266424, 266466, 266646, 424626, 424662, 462462, 462642, 462646, 462664, 462666, 466264, 466266, 466626, 624626, 626424, 626426, 626462, 626466, 626642, 626646, 626664, 642462, 642466, 642646, 646246, 646264, 646266, 646626, 646662, 662642, 662646, 662664, 662666, 664246, 664626, 664662, 666264, 666266, 666462
Offset: 1

Views

Author

Labos Elemer, Dec 20 2002

Keywords

Examples

			For 424662, the first 2 primes with the given differences are 37 and 25767877. For 646626, the least start prime is 6268957.

Crossrefs

The least primes corresponding to the 6-tuples are in A078874. The same primes, in increasing order, are in A078875. The similarly defined quadruples and quintuples are in A078868 and A078870. Cf. A001223, A078869.

Extensions

Edited by Dean Hickerson, Dec 21 2002
Showing 1-5 of 5 results.

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