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 1-4 of 4 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

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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)).

A075590 Smallest number with n distinct prime divisors which is the average of a twin prime pair.

Original entry on oeis.org

4, 6, 30, 420, 2310, 43890, 1138830, 17160990, 300690390, 15651726090, 239378649510, 12234189897930, 461282657605770, 19835154277048110, 693386350578511590, 37508276737897976010, 3338236629672919864890, 209580878166809177658630, 11465419967969569966774410
Offset: 1

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Author

Amarnath Murthy, Sep 26 2002

Keywords

Examples

			a(4) = 420 = 2^2*3*5*7, (419,421) is a twin prime pair. (210 = 2*3*5*7, 211 is prime but 209 is composite).
a(8) = 17160990 = 2*3*5*7*11*17*19*23 and 17160989 = p[1100977], 17160991 = p[1100978].
		

Crossrefs

Programs

  • Mathematica
    t=Table[0, {10}]; Do[s=Length[FactorInteger[Prime[n]+1]]; If[PrimeQ[Prime[n]+2] && s<11 && t[[s]]==0, t[[s]]=Prime[n]+1], {n, 1, 1200000}]; t

Extensions

More terms from Labos Elemer, Sep 27 2002
Corrected and extended by T. D. Noe, Nov 30 2004. a(9)-a(18) were found by testing all the numbers x with n distinct prime factors, x < 3 prime(n)# and both x-1 and x+1 prime.

A294730 Smallest average >= 6 of a twin prime pair that has exactly 2*n divisors, 0 if no such pair exists.

Original entry on oeis.org

6, 12, 30, 0, 60, 192, 270, 180, 240, 0, 420, 0, 2112, 1620, 1320, 0, 2340, 786432, 3120, 4800, 15360, 0, 3360, 388962, 724992, 6300, 29760, 0, 12240, 0, 7560, 617472, 47382528, 81648, 21600, 0, 651952128, 995328, 21840, 0, 33600, 0, 138240
Offset: 2

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Author

Hugo Pfoertner, Nov 08 2017

Keywords

Comments

It is conjectured that a(n)=0 for prime n with the only exceptions given by n=A181490(k)+1, i.e. a(2)=6, a(3)=12, a(7)=192 and a(19)=786432 are the only currently known exceptions.

Crossrefs

A145031 Smallest odd n-almost prime m such that m-2 and m+2 are both prime (cousin primes).

Original entry on oeis.org

5, 9, 45, 81, 675, 1215, 14175, 28431, 72171, 597051, 2679075, 885735, 10333575, 89813529, 286446699, 390609135, 789189885, 7274895849, 55142849601, 204945438681, 154580775111, 1049522104701, 596240132571, 1307544150375
Offset: 1

Views

Author

Donovan Johnson, Sep 30 2008

Keywords

Examples

			a(5) = 675 = 3^3*5^2 is a 5-almost prime and 675-2 and 675+2 are both prime (cousin primes). 675 is the smallest such 5-almost prime.
		

Crossrefs

Showing 1-4 of 4 results.