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.

A024675 Average of two consecutive odd primes.

Original entry on oeis.org

4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279, 282, 288, 300
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Sometimes called interprimes.
Where local maxima of A072681 occur: A072681(a(n))=A074927(n+1). - Reinhard Zumkeller, Mar 04 2009
Never prime, for that would contradict the definition. - Jon Perry, Dec 05 2012
A subset of A145025, obtained from that sequence by omitting the primes (which are barycenter of their neighboring primes). - M. F. Hasler, Jun 01 2013
Conjecture: Product_{k=1..n} a(k)/A028334(k+1) is an integer for every natural n. Cf. A352743. - Thomas Ordowski, Mar 31 2022
In contrast to the arithmetic mean, the geometric and the harmonic mean of two consecutive primes is never an integer. What is the first case where either of the two would differ from the arithmetic mean, i.e., this sequence? The existence of such a pair of primes is related to Legendre's conjecture, cf. link to discussion on the math-fun mailing list. - M. F. Hasler, Apr 07 2025

Links

Crossrefs

Cf. A072568, A072569. Bisections give A058296, A079424.
Cf. A373699 (partial sums).

Programs

  • Maple
    seq( ( (ithprime(x)+ithprime(x+1))/2 ),x=2..40);
  • Mathematica
    Plus @@@ Partition[Table[Prime[n], {n, 2, 100}], 2, 1]/2
ListConvolve[{1, 1}/2, Prime /@ Range[2, 70]] (* Jean-François Alcover, Jun 25 2013 *)
Mean/@Partition[Prime[Range[2,70]],2,1] (* Harvey P. Dale, Jul 28 2020 *)
  • PARI
    for(X=2,50,print((prime(X)+prime(X+1))/2)) \\ Hauke Worpel (thebigh(AT)outgun.com), May 08 2008
  • PARI
    first(n)=my(v=primes(n+2)); vector(n,i,v[i+1]+v[i+2])/2 \\ Charles R Greathouse IV, Jun 25 2013
  • Python
    from sympy import prime
def a(n): return (prime(n + 1) + prime(n + 2)) // 2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

Formula

a(n) = (prime(n+1)+prime(n+2))/2 = A001043(n+1)/2. - Omar E. Pol, Feb 02 2012
Conjecture: a(n) = ceiling(sqrt(prime(n+1)*prime(n+2))). - Thomas Ordowski, Mar 22 2013 [This requires gaps to be smaller than approximately sqrt(8p) and hence requires a result on prime gaps slightly stronger than that provided by the Riemann hypothesis. - Charles R Greathouse IV, Jul 13 2022]
Equals A145025 \ A006562 = A145025 \ A000040. - M. F. Hasler, Jun 01 2013

A058296 Average of consecutive primes.

Original entry on oeis.org

2, 4, 9, 15, 21, 30, 39, 45, 56, 64, 72, 81, 93, 102, 108, 120, 134, 144, 154, 165, 176, 186, 195, 205, 225, 231, 240, 254, 266, 274, 282, 300, 312, 324, 342, 351, 363, 376, 386, 399, 414, 426, 436, 446, 459, 465, 483, 495, 506, 522, 544, 560, 570, 582, 596
Offset: 1

Views

Author

Donald Mills (dmills(AT)math.siu.edu), Feb 16 2003

Keywords

Comments

2 together with average of odd primes taken two at a time without overlaps, i.e., 2 together with average of (3,5), (7,11), (13,17), etc. - Harvey P. Dale, Apr 09 2018

Links

Crossrefs

A bisection of A024675.
Cf. A000040, A079424.

Programs

  • Maple
    with(linalg): v := linalg[vector](100): v[1] := 2: for j from 2 to 100 do v[j] := (ithprime(2*j-2)+ithprime(2*j-1))/2: od: print(v);
  • Mathematica
    Join[{2},Mean/@Partition[Prime[Range[2,121]],2]] (* Harvey P. Dale, Apr 09 2018 *)
  • PARI
    { write("b058296.txt", 1, " ", 2); p2=2; for (n=2, 20000, p1=nextprime(p2+1); p2=nextprime(p1+1); a=(p1+p2)/2; write("b058296.txt", n, " ", a); ); } \\ Harry J. Smith, May 30 2009

Formula

a(1)=2, a(n) = (prime(2n-2) + prime(2n-1))/2 for n>1, where prime(i) is the i-th prime.

A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279
Offset: 0

Views

Author

Omar E. Pol, Jul 16 2009

Keywords

Comments

Also {0, 2} together the numbers A024675.
See A162345 for the first differences.

Links

Crossrefs

Cf. A000040, A006005, A024675, A058296, A079424, A162345, A162801, A162802.

Programs

  • Mathematica
    Join[{0, 2}, Most[#] + Differences[#]/2] & [Prime[Range[2, 100]]] (* Paolo Xausa, Jun 17 2024 *)

Extensions

Edited by Omar E. Pol, Jul 18 2009

A162801 Bisection of A162800.

Original entry on oeis.org

0, 4, 9, 15, 21, 30, 39, 45, 56, 64, 72, 81, 93, 102, 108, 120, 134, 144, 154, 165, 176, 186, 195, 205, 225, 231, 240, 254, 266, 274, 282, 300, 312, 324, 342, 351, 363, 376, 386, 399, 414, 426, 436, 446, 459, 465, 483, 495, 506, 522, 544, 560, 570, 582, 596
Offset: 1

Views

Author

Omar E. Pol, Jul 16 2009

Keywords

Comments

Essentially the same as A058296.

Crossrefs

Cf. A058296, A079424, A162345, A162802.

A162802 Bisection of A162800.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 50, 60, 69, 76, 86, 99, 105, 111, 129, 138, 150, 160, 170, 180, 192, 198, 217, 228, 236, 246, 260, 270, 279, 288, 309, 315, 334, 348, 356, 370, 381, 393, 405, 420, 432, 441, 453, 462, 473, 489, 501, 515, 532, 552, 566, 574, 590, 600
Offset: 1

Views

Author

Omar E. Pol, Jul 16 2009

Keywords

Comments

Also, 2 together with the numbers A079424.

Crossrefs

Cf. A079424, A162345, A162800, A162801.
Showing 1-5 of 5 results.

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