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 23 results. Next

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

A263674 Double interprimes: a(n) = (q+r)/2 = (p+s)/2 with p

Original entry on oeis.org

9, 12, 15, 18, 30, 42, 60, 81, 102, 105, 108, 120, 144, 165, 186, 195, 228, 260, 270, 312, 363, 381, 399, 420, 426, 441, 462, 489, 495, 552, 570, 582, 600, 696, 705, 714, 765, 816, 825, 858, 870, 882, 897, 924, 987, 1026, 1050, 1056, 1092, 1113, 1167, 1230
Offset: 1

Views

Author

Antonio Roldán, Oct 23 2015

Keywords

Comments

Values of p (lesser of consecutive primes) are in the sequence A022885.

Examples

			600 is in this sequence because 593, 599, 601 and 607 are consecutive primes, and 600 = (599+601)/2 = (593+607)/2.

Links

Crossrefs

Cf. A022885, A075190, A075277, A075296, A078443, A130178, A263675, A263676.

Programs

  • Mathematica
    (Prime@ # + Prime[# + 3])/2 & /@ Select[Range@ 240, (First@ # + Last@ #)/2 == (#[[2]] + #[[3]])/2 &@ Prime@ Range[#, # + 3] &] (* Michael De Vlieger, Nov 18 2015 *)
Mean/@Select[Partition[Prime[Range[300]],4,1],(#[[2]]+#[[3]])/2==(#[[1]]+#[[4]])/2&] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    {forprime(q=3,2000,p=precprime(q-1); r=nextprime(q+1); s=nextprime(r+1);m=(q+r)/2;if(m==(p+s)/2,print1(m,", ")))}

A263676 Numbers that are both interprime and oblong.

Original entry on oeis.org

6, 12, 30, 42, 56, 72, 240, 342, 420, 462, 506, 552, 600, 650, 870, 1056, 1190, 1482, 1722, 1806, 2550, 2652, 2970, 3540, 4422, 6320, 7140, 8010, 10302, 12656, 13572, 14042, 17292, 18360, 19182, 19460, 20022, 22952, 23562, 24180, 27060, 29070, 29756, 31152, 33306, 35156, 35532, 39006
Offset: 1

Views

Author

Antonio Roldán, Oct 23 2015

Keywords

Examples

			342 is in this sequence because 342 = 18*19 is oblong, and 342 = (337 + 347)/2, with 337 and 347 consecutive primes.

Links

Crossrefs

Intersection of A024675 and A002378. - Omar E. Pol, Oct 24 2015
Lesser of consecutive primes is in the sequence A242383.
Cf. A075190, A075277, A075296, A078443, A130178, A263674, A263675.

Programs

  • Mathematica
    lim = 40000; Intersection[Plus @@@ Partition[Table[Prime@ n, {n, 2, PrimePi@ lim}], 2, 1]/2, Table[n (n + 1), {n, 0, lim}]] (* Michael De Vlieger, Nov 18 2015, after Clark Kimberling at A024675 and Robert G. Wilson v at A002378 *)
obQ[n_]:=With[{divs=Partition[Divisors[n],2,1]},Length[Select[divs,#[[2]]-#[[1]]== 1 && Times@@#==n&]]>0]; Select[Mean/@Partition[Prime[ Range[ 2,40000]],2,1],obQ] (* Harvey P. Dale, Nov 01 2022 *)
  • PARI
    {for(i=1,500,n=i*(i+1);if(n==(precprime(n-1)+nextprime(n+1))/2, print1(n,", ")))}

A263675 Numbers that are both averages of consecutive primes and nontrivial prime powers.

Original entry on oeis.org

4, 9, 64, 81, 625, 1681, 4096, 822649, 1324801, 2411809, 2588881, 2778889, 3243601, 3636649, 3736489, 5527201, 6115729, 6405961, 8720209, 9006001, 12752041, 16056049, 16589329, 18088009, 21743569, 25230529, 29343889, 34586161, 37736449, 39150049
Offset: 1

Views

Author

Antonio Roldán, Oct 23 2015

Keywords

Comments

Intersection of A024675 and A025475.
Lesser of consecutive primes is in the sequence A084289.

Examples

			625 is in this sequence because 625 = 5^4, nontrivial prime power, and 625 = (619+631)/2, with 619 and 631 consecutive primes.

Links

Crossrefs

Cf. A075190, A075277, A075296, A078443, A084289, A130178, A263674, A263676.

Programs

  • Maple
    N:= 10^10: # to get all terms <= N
Primes:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
S:= select(t -> t - prevprime(t) = nextprime(t)-t, {seq(seq(p^j, j=2..floor(log[p](N))),p=Primes)}):
sort(convert(S,list)); # Robert Israel, Dec 27 2015
  • Mathematica
    (* version >= 6 *)(#/2 + NextPrime[#]/2) & /@
Select[Prime[Range[5000000]], PrimePowerQ[#/2 + NextPrime[#]/2] &]
(* Wouter Meeussen, Oct 26 2015 *)
  • PARI
    {for(i=1,10^8,if(isprimepower(i)>1&&i==(precprime(i-1)+nextprime(i+1))/2,print1(i,", ")))}

A075279 Interprimes which are of the form s*prime, s=4.

Original entry on oeis.org

12, 76, 236, 356, 436, 596, 604, 1268, 1324, 1436, 1556, 1604, 1756, 2284, 2396, 3316, 3764, 3812, 4076, 4612, 4996, 5116, 5276, 5492, 5524, 5804, 6628, 6676, 6932, 6964, 7468, 7484, 7892, 8524, 8644, 8716, 9004, 9836, 11276, 12476, 14156, 14636
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

			236 is an interprime and 236/4 = 59 is prime.

Links

Crossrefs

Cf. A075277-A075296.

Programs

  • Mathematica
    s=4; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 1000}], PrimeQ[ #/s]&]
Select[Mean/@Partition[Prime[Range[2000]],2,1],PrimeQ[#/4]&] (* Harvey P. Dale, Jul 31 2018 *)

A075281 Interprimes which are of the form s*prime, s=6.

Original entry on oeis.org

12, 18, 30, 42, 102, 138, 186, 246, 282, 426, 582, 618, 642, 822, 834, 1158, 1698, 1878, 2022, 2082, 2094, 2238, 2382, 2526, 2658, 2802, 2922, 2946, 3462, 3522, 3558, 3714, 3786, 3858, 3918, 4038, 4146, 4206, 4638, 4722, 4866, 4962, 5442, 5946, 6126
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

			138 is an interprime and 138/6 = 23 is prime.

Links

Crossrefs

Cf. A075277--A075296.

Programs

  • Mathematica
    s=6; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 1000}], PrimeQ[ #/s]&]
Select[Total[#]/2&/@Partition[Prime[Range[1000]],2,1],PrimeQ[#/6]&] (* Harvey P. Dale, Dec 18 2021 *)

A075282 Interprimes which are of the form s*prime, s=7.

Original entry on oeis.org

21, 217, 2191, 2933, 3073, 3353, 7063, 7091, 8813, 9079, 9233, 9527, 9569, 10493, 10717, 11851, 12131, 16667, 17857, 18263, 18347, 19243, 19733, 22421, 23093, 24703, 24787, 25417, 27349, 28637, 32347, 32473, 33607, 33691, 35273, 35413
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

			2191 is an interprime and 2191/7 = 317 is prime.

Links

Crossrefs

Cf. A075277-A075296.

Programs

  • Mathematica
    s=7; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 1000}], PrimeQ[ #/s]&]
Select[Mean/@Partition[Prime[Range[4000]],2,1],PrimeQ[#/7]&] (* Harvey P. Dale, Jul 13 2025 *)

A075284 Interprimes which are of the form s*prime, s=9.

Original entry on oeis.org

18, 45, 99, 279, 747, 909, 1611, 1737, 2007, 2259, 2439, 2799, 3879, 5193, 5571, 5787, 6147, 6219, 6471, 6813, 6849, 7677, 8271, 8577, 8703, 8739, 8793, 9279, 9549, 9621, 10107, 10161, 10629, 10953, 11241, 11511, 11619, 11709, 13329, 14031
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

			279 is an interprime and 279/9 = 31 is prime.

Links

Crossrefs

Cf. A075277-A075296.

Programs

  • Mathematica
    s=9; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 1000}], PrimeQ[ #/s]&]
Select[Mean/@Partition[Prime[Range[2000]],2,1],PrimeQ[#/9]&] (* Harvey P. Dale, Apr 27 2024 *)

A075285 Interprimes which are of the form s*prime, s=10.

Original entry on oeis.org

30, 50, 170, 370, 590, 610, 730, 1370, 1390, 1910, 1990, 2290, 3310, 3730, 4010, 4990, 5410, 6070, 6590, 7190, 7870, 8770, 9470, 9970, 10610, 11090, 11810, 11930, 12970, 14470, 15230, 15310, 16270, 16570, 16990, 17330, 18790, 19130, 20110
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

			370 is an interprime and 370/10 = 37 is prime.

Links

Crossrefs

Cf. A075277-A075296.

Programs

  • Mathematica
    s=10; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 4000}], PrimeQ[ #/s]&]
Select[Mean/@Partition[Prime[Range[2500]],2,1],PrimeQ[#/10]&] (* Harvey P. Dale, Jun 24 2017 *)

A075287 Interprimes which are of the form s*prime, s=12.

Original entry on oeis.org

60, 228, 348, 636, 1668, 1788, 1884, 2148, 2364, 2724, 2892, 3252, 3372, 3684, 4236, 4548, 4668, 5316, 6252, 6684, 6828, 7212, 8292, 8628, 9012, 9708, 10068, 10308, 11892, 11964, 12108, 12252, 12396, 12612, 13836, 14676, 15324, 15396, 15564
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

			348 is an interprime and 348/12 = 29 is prime.

Links

Crossrefs

Cf. A075277-A075296.

Programs

  • Mathematica
    s=12; Select[Table[(Prime[n+1]+Prime[n])/2, {n, 2, 4000}], PrimeQ[ #/s]&]
Select[Mean/@Partition[Prime[Range[2,1900]],2,1],PrimeQ[#/12]&] (* Harvey P. Dale, May 08 2012 *)
Showing 1-10 of 23 results. Next

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