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 1-6 of 6 results.

A131559 Indices of records in A078571.

Original entry on oeis.org

1, 3, 7, 8, 14, 41, 270, 277, 1595, 5899, 6320, 6668, 117221, 180037, 295677, 587152, 703625, 4397877, 12151726, 29573235
Offset: 1

Views

Author

Jonathan Vos Post, Oct 02 2007

Keywords

Comments

Numbers k such that total number of prime factors (with multiplicity) of the average of k-th twin prime pair is a record.
The records are 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, ... and the associated twin primes recorded in A086827. - R. J. Mathar, Oct 24 2007

Examples

			a(1) = 1 because A078571(1) = 2, since the average of twin prime pair (3,5) is 4 = 2^2 is a semiprime with two prime factors (with multiplicity).
a(2) = 3 because A078571(3) = smallomega(12) = 3.
a(3) = 7 because A078571(7) = smallomega(60) = 4.
a(4) = 8 because A078571(8) = smallomega(72) = 5.
a(5) = 14 because A078571(14) = smallomega(192) = 7.
a(6) = 41 because A078571(41) = smallomega(1152) = smallomega(2^7 * 3^2) = 9.

Crossrefs

Cf. A001221, A001359, A006512, A014574, A078571.

Programs

  • PARI
    lista(nn) = {my(r=0, c=0, p=2); forprime(q=3, nn, if(q-p==2, c++; if(bigomega(p+1)>r, r=bigomega(p+1); print1(c, ", "))); p=q); } \\ Jinyuan Wang, Apr 01 2020

Formula

a(n) = {k such that A001222(A014574(k)) is a record}.

Extensions

More terms from R. J. Mathar, Oct 24 2007
a(13)-a(20) from Jinyuan Wang, Apr 01 2020

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.

Links

Crossrefs

Cf. A001221, A014574, A078571, A078572.

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)).

A078572 The minimum exponent in prime factorization of the average of n-th twin prime pair.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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) = 2.

Links

Crossrefs

Cf. A001359, A006512, A014574, A051904, A078571, A078573.

Programs

  • Maple
    p:= 2: count:= 0: Res:= NULL:
while count < 100 do
  q:= nextprime(p);
  if q - p = 2 then
    count:= count+1;
    Res:= Res, min(map(t->t[2],ifactors(p+1)[2]))
  fi;
  p:= q;
od:
Res; # Robert Israel, Oct 26 2018
  • Mathematica
    Min[Transpose[FactorInteger[Mean[#]]][[2]]]&/@Select[Partition[Prime[ Range[ 1000]],2,1], #[[2]]-#[[1]]==2&] (* Harvey P. Dale, Mar 04 2013 *)
  • PARI
    lista(pmax) = {my(prv = 2); forprime(p = 3, pmax, if(p - prv ==2, print1(vecmin(factor(p-1)[,2]), ", ")); prv = p);} \\ Amiram Eldar, Sep 08 2024

Formula

a(n) = A051904(A014574(n)).

A078574 Number of divisors of the average of n-th twin prime pair.

Original entry on oeis.org

3, 4, 6, 6, 8, 8, 12, 12, 8, 12, 8, 12, 18, 14, 12, 12, 20, 16, 8, 16, 12, 24, 20, 16, 12, 16, 24, 8, 8, 24, 20, 8, 18, 16, 18, 24, 16, 24, 12, 24, 24, 16, 12, 16, 16, 32, 24, 18, 16, 20, 16, 30, 12, 8, 16, 12, 30, 8, 24, 24, 16, 18, 8, 24, 28, 16, 24, 8, 30, 32, 36, 8, 24, 30, 8
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 29 2002

Keywords

Examples

			4th twin prime pair = (A001359(4), A006512(4)) = (17,19), hence A014574(4) = 18 with divisors = {1,2,3,6,9,18} therefore a(4) = 6.

Links

Crossrefs

Cf. A000005, A014574, A078570, A078571, A078575.

Programs

  • Mathematica
    midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; DivisorSigma[0, Select[Range[3000], midQ]] (* Amiram Eldar, Nov 03 2019 *)
DivisorSigma[0,#]&/@(Mean/@Select[Partition[Prime[Range[500]],2,1],#[[2]]- #[[1]] == 2&]) (* Harvey P. Dale, Nov 07 2022 *)

Formula

a(n) = A000005(A014574(n)).

A078577 Sum of all prime factors of the average of n-th twin prime pair.

Original entry on oeis.org

4, 5, 7, 8, 10, 12, 12, 12, 22, 13, 28, 15, 15, 15, 19, 26, 16, 16, 52, 22, 36, 19, 17, 23, 37, 29, 19, 108, 112, 23, 19, 142, 33, 29, 22, 29, 52, 22, 67, 27, 20, 51, 79, 53, 43, 25, 31, 29, 37, 42, 76, 21, 146, 288, 53, 156, 27, 318, 37, 28, 48, 33, 352, 41, 26, 81, 32
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) = 2+3+5+5 = 15.

Links

Crossrefs

Cf. A001414, A014574, A078576, A078575, A078571.

Programs

  • Mathematica
    midQ[n_] := PrimeQ[n-1] && PrimeQ[n+1]; f[n_] := Plus @@ Times @@@ FactorInteger[n]; f /@ Select[Range[2500], midQ] (* Amiram Eldar, Nov 03 2019 *)

Formula

a(n) = A001414(A014574(n)).

A176915 Average of n-th twin prime pair minus total number of prime factors of average of n-th twin prime pair.

Original entry on oeis.org

2, 4, 9, 15, 27, 39, 56, 67, 99, 103, 135, 146, 175, 185, 194, 224, 234, 265, 279, 307, 344, 415, 425, 458, 518, 566, 594, 615, 639, 655, 804, 819, 823, 854, 877, 1015, 1027, 1045, 1058, 1087, 1143, 1226, 1274, 1286, 1298, 1314, 1423, 1447, 1478, 1482, 1603
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 28 2010

Keywords

Comments

Prime factors are counted with multiplicity. - Harvey P. Dale, Aug 19 2011

Links

Crossrefs

Cf, A014574, A069345, A078571.

Programs

  • Maple
    TP:= select(t -> isprime(t) and isprime(t+2), [3, seq(i,i=5..2000,6)]):
map(t -> t+1 - numtheory:-bigomega(t+1), P1); # Robert Israel, Dec 31 2024
  • Mathematica
    f[n_]:=Module[{mn=Mean[n]},mn-PrimeOmega[mn]]; f/@Select[Partition[ Prime[ Range[300]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Aug 19 2011 *)

Formula

a(n) = A014574(n)-A078571(n).
a(n) = A069345(A014574(n)). - Jason Yuen, Sep 01 2025

Extensions

Formula index, a(8), a(15) and a(23) corrected by R. J. Mathar, May 01 2010
Showing 1-6 of 6 results.

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