A263676 -id:A263676 - cp's OEIS frontend

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

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

Antonio Roldán, Oct 23 2015

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Interprime

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

(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 *)

{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,", ")))}

A242383 Lesser of consecutive primes whose average is an oblong number.

Original entry on oeis.org

5, 11, 29, 41, 53, 71, 239, 337, 419, 461, 503, 547, 599, 647, 863, 1051, 1187, 1481, 1721, 1801, 2549, 2647, 2969, 3539, 4421, 6317, 7129, 8009, 10301, 12653, 13567, 14033, 17291, 18353, 19181, 19457, 20021, 22943, 23561, 24179, 27059, 29063, 29753, 31151, 33301, 35153

Offset: 1

Antonio Roldán, May 12 2014

nonn

			53 is in the sequence because it is prime, nextprime(53) = 59 and (53+59)/2 = 56 = 8*7, an oblong number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002378, A154634, A263676.

f[n_] := NextPrime[n*(n + 1), -1] ; f /@ Select[Range[200], (ob =  #*(# + 1)) == (NextPrime[ob, -1] + NextPrime[ob])/2 &] (* Amiram Eldar, May 06 2020 *)
Select[Partition[Prime[Range[4000]],2,1],OddQ[Sqrt[1+4*Mean[#]]]&][[All,1]] (* Harvey P. Dale, Oct 25 2020 *)

{for(i=3,10^5,if(isprime(i),k=(i+nextprime(i+1))/4;if(issquare(8*k+1),print1(i,", "))))}

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

Antonio Roldán, Oct 23 2015

nonn

Intersection of A024675 and A025475.

Lesser of consecutive primes is in the sequence A084289.

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

Robert Israel, Table of n, a(n) for n = 1..1560
Eric Weisstein's World of Mathematics, Interprime

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

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

(* version >= 6 *)(#/2 + NextPrime[#]/2) & /@
Select[Prime[Range[5000000]], PrimePowerQ[#/2 + NextPrime[#]/2] &]
(* Wouter Meeussen, Oct 26 2015 *)

{for(i=1,10^8,if(isprimepower(i)>1&&i==(precprime(i-1)+nextprime(i+1))/2,print1(i,", ")))}

cp's OEIS Frontend

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

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A242383 Lesser of consecutive primes whose average is an oblong number.

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Mathematica

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

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Maple

Mathematica

PARI