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 1-3 of 3 results.

A078571 Total number of prime factors of the average of n-th twin prime pair.

Original entry on oeis.org

2, 2, 3, 3, 3, 3, 4, 5, 3, 5, 3, 4, 5, 7, 4, 4, 6, 5, 3, 5, 4, 5, 7, 4, 4, 4, 6, 3, 3, 5, 6, 3, 5, 4, 5, 5, 5, 5, 4, 5, 9, 4, 4, 4, 4, 6, 5, 5, 4, 6, 5, 7, 4, 3, 4, 4, 7, 3, 5, 5, 5, 5, 3, 6, 8, 4, 5, 3, 7, 5, 6, 3, 5, 9, 3, 9, 5, 5, 5, 3, 6, 7, 7, 8, 4, 4, 6, 5, 8, 4, 4, 3, 5, 7, 5, 3, 4, 7, 5, 5, 5, 3, 4, 4, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 29 2002

Keywords

Comments

Between every twin prime pair is a composite number. This sequence looks at a characteristic of those numbers. If the number, n, is the average of a twin prime pair, p&q, then n=(p+q)/2 and p*q=n^2 -1. [Robert G. Wilson v, Aug 02 2010]

Examples

			12th twin prime pair = (A001359(12), A006512(12)) = (149,151), hence A014574(12) = 150 = 2*3*5*5, therefore a(12) = 4.
From _Robert G. Wilson v_, Aug 02 2010: (Start)
2) 4, 6 and no others < 10^9.
3) 12, 18, 30, 42, 102, 138, 282, 618, 642, 822, 1698, 1878, 2082, ...
4) 60, 150, 198, 228, 348, 462, 522, 570, 858, 1062, 1230, 1278, ...
5) 72, 108, 180, 270, 312, 420, 660, 828, 882, 1020, 1032, 1050, ...
6) 240, 600, 810, 1320, 1488, 2088, 2340, 2970, 3300, 4158, 4272, ...
7) 192, 432, 1620, 1872, 2268, 3000, 3120, 3528, 3672, 4050, 4128, ...
8) 2112, 3168, 3360, 5280, 7128, 7560, 9000, 12240, 13680, 16632, ...
9) 1152, 2592, 2688, 4800, 7488, 9720, 18048, 29760, 34848, 35280, ...
10) 14592, 21600, 22272, 29568, 32832, 33600, 64152, 71808, 75168, ...
11) 26112, 26880, 49920, 81648, 100800, 102912, 108288, 131712, ...
12) 15360, 23040, 58368, 95232, 133632, 134400, 196992, 219648, ...
13) 139968, 235008, 241920, 279552, 365568, 472392, 617472, 694272, ...
14) 138240, 202752, 345600, 684288, 724992, 783360, 817152, 875520, ...
... (End)
		

Crossrefs

Programs

  • Mathematica
    f[n_] := Plus @@ Last /@ FactorInteger@n; p = 3; lst = {}; While[p < 1000, If[ PrimeQ[p + 2], AppendTo[lst, f[p + 1]]]; p = NextPrime@p]: lst (* Robert G. Wilson v, Aug 02 2010 *)

Formula

a(n) = A001222(A014574(n)).

A078570 Number of distinct prime factors of the average of n-th twin prime pair.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 2, 4, 3, 4, 4, 4, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 4, 4, 3, 3, 5, 4, 3, 4, 2, 3, 3, 3, 5, 4, 3, 4, 3, 4, 3, 3, 3, 4, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 29 2002

Keywords

Examples

			12th twin prime pair = (A001359(12), A006512(12)) = (149,151), hence A014574(12) = 150 = 2*3*5*5, therefore a(12) = 3.
		

Crossrefs

Programs

  • Mathematica
    midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; PrimeNu /@ Select[Range[5000], midQ] (* Amiram Eldar, Nov 03 2019 *)
    PrimeNu[Mean[#]]&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Jul 31 2023 *)

Formula

a(n) = A001221(A014574(n)).

A078573 The maximum exponent in prime factorization of the average of n-th twin prime pair.

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 2, 3, 1, 3, 1, 2, 2, 6, 2, 2, 4, 3, 1, 3, 2, 2, 4, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 2, 2, 3, 2, 2, 2, 7, 1, 2, 1, 1, 3, 2, 2, 1, 4, 3, 4, 2, 1, 1, 2, 4, 1, 2, 2, 3, 2, 1, 3, 6, 1, 2, 1, 4, 1, 2, 1, 2, 5, 1, 7, 3, 1, 2, 1, 3, 3, 4, 5, 2, 2, 2, 2, 5, 2, 1, 1, 2, 3, 2, 1, 2, 3, 3, 2, 2, 1, 1, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 29 2002

Keywords

Examples

			10th twin prime pair = (A001359(10), A006512(10)) = (107,109), hence A014574(10) = 108 = 2^2 * 3^2, therefore a(10) = 3.
		

Crossrefs

Programs

  • Mathematica
    midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; f[n_] := Max @@ FactorInteger[n][[;;,2]]; f /@ Select[Range[5000], midQ] (* Amiram Eldar, Nov 03 2019 *)
    Max[FactorInteger[Mean[#]][[All,2]]]&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]] - #[[1]]==2&] (* Harvey P. Dale, Apr 21 2022 *)
  • PARI
    lista(pmax) = {my(prv = 2); forprime(p = 3, pmax, if(p - prv ==2, print1(vecmax(factor(p-1)[,2]), ", ")); prv = p);} \\ Amiram Eldar, Sep 09 2024

Formula

a(n) = A051903(A014574(n)).
Showing 1-3 of 3 results.