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.

A177961 a(1)=2. Otherwise the average of the smallest prime divisors of 2n-1 and 2n+1.

Original entry on oeis.org

2, 4, 6, 5, 7, 12, 8, 10, 18, 11, 13, 14, 4, 16, 30, 17, 4, 21, 20, 22, 42, 23, 25, 27, 5, 28, 29, 4, 31, 60, 32, 4, 36, 35, 37, 72, 38, 5, 43, 41, 43, 44, 4, 46, 48, 5, 4, 51, 50, 52, 102, 53, 55, 108, 56, 58, 59, 4, 5, 9, 7, 4, 66, 65, 67, 69, 5, 70, 138, 71, 7, 8, 4, 76, 150, 77, 4, 81
Offset: 1

Views

Author

Vladimir Shevelev, May 16 2010, May 22 2010

Keywords

Comments

As n tends to infinity, we have 1) lim inf (a(n)/n)=0; 2) if there exist infinitely many twin primes, then lim sup (a(n)/n)=2, otherwise, lim sup (a(n)/n)=1.

Links

Crossrefs

Cf. A177930, A177964, A177965, A177966.

Programs

  • Magma
    [2] cat [1/2*(Min(PrimeFactors(2*n-1))+ Min(PrimeFactors(2*n+1))):n in [2..80]]; // Vincenzo Librandi, Feb 07 2016
  • Maple
    N:= 100: # to get a(1) to a(N)
S:= [1,seq(min(numtheory:-factorset(2*i-1)),i=2..N+1)]:
(S[2..-1]+S[1..-2])/2; # Robert Israel, Jul 31 2015
  • Mathematica
    Table[If[n == 1, 2, Mean[{FactorInteger[2 n - 1][[1, 1]], FactorInteger[2 n + 1][[1, 1]]}]], {n, 78}] (* Michael De Vlieger, Aug 02 2015 *)
  • PARI
    a(n) = if (n==1, 2, (vecmin(factor(2*n-1)[,1]) + vecmin(factor(2*n+1)[,1]))/2); \\ Michel Marcus, Feb 07 2016

Formula

a(n) = (A090368(n)+A090368(n+1))/2. [R. J. Mathar, May 31 2010]

Extensions

More terms from R. J. Mathar, May 31 2010

A177965 Indices m for which A177961(m) - m = 1.

Original entry on oeis.org

1, 4, 7, 10, 16, 19, 22, 31, 34, 37, 40, 49, 52, 55, 64, 70, 76, 79, 82, 91, 97, 100, 106, 112, 115, 121, 136, 139, 142, 154, 157, 166, 169, 175, 184, 187, 190, 199, 205, 211, 217, 220, 229, 232, 244, 250, 262, 271, 274, 286, 289, 301, 304, 307, 310, 316, 322, 331, 337, 346
Offset: 1

Views

Author

Vladimir Shevelev, May 16 2010

Keywords

Comments

1 is the smallest value of |A177961(m) - m|.

Links

Crossrefs

Cf. A002476, A177961, A177964, A177966.

Programs

  • Maple
    1,op(map(t -> 3*t+1, select(t -> isprime(6*t+1),[$1..1000]))); # Robert Israel, Jul 31 2015
  • Mathematica
    Position[Table[If[m == 1, 2, Mean[{FactorInteger[2 m - 1][[1, 1]], FactorInteger[2 m + 1][[1, 1]]}]] - m, {m, 346}], n_ /; n == 1] // Flatten (* Michael De Vlieger, Aug 02 2015 *)

Formula

a(n) = (A002476(n-1) + 1)/2, n > 1.

Extensions

More terms from R. J. Mathar, Oct 25 2010

A177966 Indices m for which A177961(m) = 2 + m.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 20, 23, 26, 27, 29, 35, 41, 42, 44, 50, 53, 56, 57, 65, 68, 74, 83, 86, 87, 89, 95, 98, 113, 116, 117, 119, 125, 128, 131, 132, 134, 140, 146, 147, 155, 158, 173, 176, 177, 179, 191, 192, 194, 200, 209, 215, 221, 222, 224, 230, 233, 239, 245, 251, 252, 254
Offset: 1

Views

Author

Vladimir Shevelev, May 16 2010

Keywords

Comments

All m for which 2*m+1 is in A003627 are in the sequence:
This concerns m=2, 5, 8, 11, 14, 20, 23, 26, 29, 35,...
Union of (A003627-1)/2 and (A132235+1)/2. - Robert Israel, Jul 31 2015

Links

Crossrefs

Cf. A003627, A132235, A177961, A177964, A177965.

Programs

  • Maple
    A090368 := proc(n) A020639(2*n-1) ; end proc:
A177961 := proc(n) (A090368(n)+A090368(n+1)) /2 ; end proc:
isA177966 := proc(n) A177961(m) = m+2 ; end proc:
for m from 1 to 800 do if isA177966(m) then printf("%d,",m) ; end if; end do:
# R. J. Mathar, Oct 25 2010
N:= 1000: # to get all terms <= N
A1:= map(t -> (t-1)/2, select(isprime, {seq(6*i-1, i=1..(N+1)/3)})):
A2:= map(t -> (t+1)/2, select(isprime, {seq(23+30*i,i=0..(N-12)/15)})):
sort(convert(A1 union A2,list));
# Robert Israel, Jul 31 2015
  • Mathematica
    M = 1000; (* to get all terms <= M *)
A1 = (Select[Table[6 i - 1, {i, 1, (M + 1)/3}], PrimeQ] - 1)/2;
A2 = (Select[Table[23 + 30 i, {i, 0, (M - 12)/15}], PrimeQ] + 1)/2;
Union[A1, A2] (* Jean-François Alcover, Jul 17 2020, after Robert Israel *)

Extensions

Corrected (11, 23, 27, etc. inserted) and extended by R. J. Mathar, Oct 25 2010
Showing 1-3 of 3 results.

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