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 1-10 of 101 results. Next

A075277 Interprimes (A024675) which are of the form s*prime, s=2.

Original entry on oeis.org

4, 6, 26, 34, 86, 134, 254, 274, 334, 386, 446, 566, 974, 1126, 1226, 1234, 1286, 1294, 1546, 2066, 2374, 2386, 2554, 2854, 2906, 2966, 3086, 3326, 3694, 3898, 4054, 4286, 4594, 4742, 4846, 4874, 4954, 5006, 5218, 5366, 5686, 5714, 5854, 6238, 6274, 6326
Offset: 1

Views

Author

Zak Seidov, Sep 12 2002

Keywords

Comments

Interprimes which are of the form s*prime are in A075277-A075296 (s = 2-21). Case s = 1 is impossible.

Examples

			7646 is an interprime and 7646/2 = 3823 is prime.

Links

Crossrefs

Cf. A024675, A075277-A075296.

Programs

  • Mathematica
    s=2; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 1000}], PrimeQ[ #/s]&]
  • PARI
    first(n, {m=2}) = {my(res = List(), p); forprime(p=2, oo, if(precprime(m*p) + nextprime(m*p) == 2*m*p, listput(res, m*p); if(#res>=n,return(res))))} \\ David A. Corneth, Jul 26 2017

A075296 Interprimes (A024675) which are of the form s*prime, s=21.

Original entry on oeis.org

42, 105, 231, 399, 483, 861, 987, 1113, 1281, 1491, 1869, 2121, 2247, 2667, 2751, 3129, 3423, 5649, 5691, 5817, 7539, 8169, 8421, 8589, 9807, 10563, 10689, 10983, 11361, 13881, 14511, 14889, 15519, 17031, 17409, 18627, 19761, 20391, 21189
Offset: 1

Views

Author

Zak Seidov, Sep 12 2002

Keywords

Comments

Interprimes of the form s*prime are in A075277-A075296 ( s = 2 - 21 ). Case s=1 is impossible.

Examples

			231 is an interprime and 231/21 = 11 is prime.

Links

Crossrefs

Cf. A024675, A075277-A075296.

Programs

  • Mathematica
    s=21; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 14000}], PrimeQ[ #/s]&]
  • PARI
    first(n, {m=21}) = {my(res = List(), p); forprime(p=2, oo, if(precprime(m*p) + nextprime(m*p) == 2*m*p, listput(res, m*p); if(#res>=n,return(res))))} \\ David A. Corneth, Jul 26 2017

A052288 First differences of the average of two consecutive primes (A024675).

Original entry on oeis.org

2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6, 9, 6, 6, 6, 4, 5, 5, 7, 6, 3, 3, 8, 10, 6, 6, 6, 5, 9, 7, 10, 12, 8, 8, 6
Offset: 1

Views

Author

Labos Elemer, Feb 08 2000

Keywords

Examples

			a(30) = ((113 + 127)/2) - ((127 + 131)/2) = (131 - 113)/2 = 9;
a(31) = ((127 + 131)/2) - ((137 + 131)/2) = (137 - 127)/2 = 5.

Links

Programs

Formula

a(n) = (prime(n+3) - prime(n+1))/2.
a(n) = A115061(n+2) = A162345(n+2). - Nathaniel Johnston, Jun 25 2011

A145025 Numbers which are the average of two consecutive odd primes (A024675) together with primes which are the average of the previous prime and the following prime (A006562).

Original entry on oeis.org

4, 5, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 53, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 157, 160, 165, 170, 173, 176, 180, 186, 192, 195, 198, 205, 211, 217, 225, 228, 231, 236, 240, 246, 254, 257, 260
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Sep 29 2008

Keywords

Comments

Numbers n such that prevprime(n) + nextprime(n) = 2n. - Wesley Ivan Hurt, May 13 2017

Links

Crossrefs

Equals A024675 U A006562. - M. F. Hasler, Jun 01 2013

Programs

  • Maple
    Primes:= select(isprime, [seq(i,i=3..1000,2)]):
nprimes:= nops(Primes):
A024675:= {seq((Primes[i]+Primes[i+1])/2, i=1..nprimes-1)}:
L:= Primes[1..-3]+Primes[3..-1]:
A006562:=zip((s,t) -> if 2*s=t then s else NULL fi, Primes[2..-2],L):
sort(convert(convert(A006562,set) union A024675, list)); # Robert Israel, Nov 20 2016
  • Mathematica
    Union[Select[Map[Mean@ {First@ #, Last@ #} &, Partition[#, 3, 1]], PrimeQ], Map[Mean, Partition[#, 2, 1]]] &@ Prime@ Range[2, 56] (* Michael De Vlieger, Jan 31 2019 *)
  • PARI
    for(n=2,999,n-precprime(n-1)==nextprime(n+1)-n&&print1(n",")) \\ M. F. Hasler, Jun 01 2013

Extensions

Entry revised by N. J. A. Sloane, Mar 24 2017, replacing old definition with definition from M. F. Hasler

A127556 Decimal expansion of the number 4.1636635147332912770473687837946011358... having continued fraction expansion 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, ... (arithmetical average of two consecutive odd primes A024675).

Original entry on oeis.org

4, 1, 6, 3, 6, 6, 3, 5, 1, 4, 7, 3, 3, 2, 9, 1, 2, 7, 7, 0, 4, 7, 3, 6, 8, 7, 8, 3, 7, 9, 4, 6, 0, 1, 1, 3, 5, 8, 0, 5, 7, 6, 4, 4, 9, 7, 4, 6, 3, 7, 4, 3, 9, 6, 9, 1, 5, 9, 0, 3, 6, 9, 5, 1, 4, 8, 8, 9, 8, 3, 6, 6, 8, 4, 4, 8, 0, 3, 1, 3, 7, 5, 7, 8, 0, 5, 3, 7, 9, 7, 1, 6, 5, 3, 8, 4, 7, 2, 6, 7
Offset: 1

Views

Author

Artur Jasinski, Jan 18 2007

Keywords

Crossrefs

Cf. A064442, A127549, A127550, A127551, A127552, A127553, A127555, A127556, A127557, A127558, A127559, A024675.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, (Prime[n] + Prime[n + 1])/2], {n, 2, 500}]; RealDigits[N[FromContinuedFraction[a], 100]][[1]]

Extensions

a(100) corrected by Sean A. Irvine, Jul 09 2023

A079424 A bisection of A024675. Cf. A058296.

Original entry on oeis.org

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, 610, 618, 636
Offset: 1

Views

Author

N. J. A. Sloane, Feb 16 2003

Keywords

A075673 Sum of next n integer interprimes (cf. A024675).

Original entry on oeis.org

4, 15, 45, 111, 232, 422, 704, 1129, 1667, 2403, 3287, 4470, 5810, 7508, 9414, 11663, 14363, 17454, 20715, 24739, 29214, 33957, 39183, 45540, 52056, 59497, 67181, 75862, 84831, 95697, 106608, 117812, 130356, 143759, 158617, 174312, 190500
Offset: 1

Views

Author

Zak Seidov, Sep 24 2002

Keywords

Comments

Sum of next n primes is A007468. Sum of next n odd interprimes is A075674. Sum of next n even interprimes is A075675.

Examples

			a(1) = (3+5)/2 = 4; a(2) = (5+7)/2+(7+11)/2 = 15; a(3) = (11+13)/2+(13+17)/2 +(17+19)/2 = 45.

Links

Crossrefs

Cf. A024675, A007468, A075673, A075674, A075675.

Programs

  • Mathematica
    (* sum of next n integer interprimes*) i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; Table[Sum[(Prime[i+2]+Prime[i+1])/2, {i, i1, i2}], {n, 1, 40}]
With[{nn=40},Total/@TakeList[Mean/@Partition[Prime[Range[2,(nn(nn+1))/2+2]],2,1],Range[nn]]] (* Harvey P. Dale, Feb 24 2023 *)

A205372 Least s(k) such that n divides s(k)-s(j) for some jA024675.

Original entry on oeis.org

6, 6, 9, 12, 9, 12, 18, 12, 15, 26, 15, 18, 30, 18, 21, 34, 21, 30, 34, 26, 30, 26, 50, 30, 34, 30, 39, 34, 50, 34, 76, 50, 39, 60, 39, 42, 76, 42, 45, 86, 45, 60, 64, 50, 60, 50, 56, 60, 64, 56, 60, 56, 129, 60, 64, 60, 69, 64, 93, 64, 76, 134, 69, 76, 69, 72, 76
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2012

Keywords

Comments

For a guide to related sequences, see A204892.
a(n) >= n+4, with equality if and only if n+4 is in A024675.

Links

Crossrefs

Cf. A024675, A205153, A204892.

Programs

  • Maple
    N:= 200: # for terms before the first > the greatest prime <= N
P:= select(isprime, [seq(i,i=3..N,2)]):
S:= (P[1..-2]+P[2..-1])/2:
f:= proc(n) local T,R,i;
   T:= S mod n;
   R:= {}:
   for i from 1 to nops(T)-1 do
     R:= R union {T[i]};
     if member(T[i+1],R) then return S[i+1] fi;
   od;
   FAIL
end proc:
Res:= NULL:
for n from 1 do
  v:= f(n);
  if v = FAIL then break fi;
  Res:= Res, v
od:
Res; # Robert Israel, Sep 09 2020
  • Mathematica
    (See the program at A205153.)

A024677 Smallest prime divisor of n-th terms of sequence A024675 (averages of two consecutive odd primes).

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 5, 7, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 11, 3, 3, 3, 3, 2, 5, 2, 2, 2
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

From Robert Israel, Nov 03 2019: (Start)
If prime(n+1) and prime(n+2) are twin primes, then a(n)=2.
If prime(n+1)>3 is in A023200, then a(n)=3.
Dickson's conjecture implies that for any prime p>3, there are infinitely many primes q>=p such that pq-6 and pq+6 are consecutive primes, so that a(pi(pq)-1) = p. Thus each prime should occur infinitely many times in the sequence. (End)

Links

Crossrefs

Cf. A023200

Programs

  • Maple
    P:= select(isprime,[seq(i,i=3..104759,2)]):
Q:= (P[2..-1]+P[1..-2])/2:
map(min @ numtheory:-factorset, Q); # Robert Israel, Nov 03 2019
  • Mathematica
    Table[First@First@FactorInteger[(Prime[n+1]+Prime[n])/2],{n,2,150}] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)

A024676 a(n) is the number of prime divisors (counted by multiplicity) of A024675(n).

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024675, A001222.

Programs

  • Maple
    seq(numtheory:-bigomega((ithprime(n+1)+ithprime(n+2))/2), n=1..100); # Robert Israel, Jan 22 2024

Formula

a(n) = A001222(A024675(n)).

Extensions

Edited by Robert Israel, Jan 22 2024
Showing 1-10 of 101 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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