A077800 -id:A077800 - cp's OEIS frontend

A256620 Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the arithmetic mean of the four numbers consisting of the two primes before p and the two primes after q.

12, 30, 42, 312, 600, 858, 1032, 1290, 1698, 2112, 2688, 3768, 4218, 4230, 4260, 5850, 6132, 6552, 6702, 7212, 7308, 8292, 9420, 9930, 11970, 12042, 12378, 15972, 17190, 17598, 17922, 19470, 19890, 21600, 24180, 26862, 30012, 30852, 32118

Offset: 1

Karl V. Keller, Jr., Apr 24 2015

nonn

This sequence is a subsequence A014574 (average of twin prime pairs).

All terms are multiples of 6. - Zak Seidov, Apr 25 2015

			For n=12: 5,7,11,13,17,19 are six consecutive primes with 13 = 11 + 2 and (5+7+17+19)/4=12.
For n=1032: 1019,1021,1031,1033,1039,1049 are six consecutive primes with 1033 = 1031 + 2 and (1019+1021+1039+1049)/4=1032.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..500000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A077800 (twin primes), A014574.

avQ[lst_]:=Module[{td=TakeDrop[lst,{3,4}]},Mean[td[[1]]]==Mean[td[[2]]] && td[[1,2]]-td[[1,1]]==2]; Mean[Take[#,{3,4}]]&/@Select[Partition[ Prime[ Range[ 3500]],6,1],avQ] (* The program uses the TakeDrop function from Mathematica version 10.2 *) (* Harvey P. Dale, Jul 16 2015 *)

from sympy import isprime,prevprime,nextprime
for i in range(5,200001,2):
  if isprime(i) and isprime(i+2):
    a = prevprime(i)
    b = prevprime(a)
    if a+b+nextprime(i,2)+nextprime(i,3) == 4*(i+1): print(i+1,end=', ')
  else: continue

Typo in Name fixed by Zak Seidov, Apr 25 2015

A257529 Prime numbers that have a pentagonal Voronoi cell in the Voronoi diagram of the Ulam prime spiral.

Original entry on oeis.org

2, 3, 11, 13, 17, 19, 29, 37, 53, 83, 97, 101, 103, 107, 109, 113, 137, 149, 151, 163, 191, 197, 211, 223, 227, 241, 257, 271, 281, 293, 307, 337, 347, 373, 401, 419, 431, 433, 461, 521, 523, 541, 563, 569, 571, 577, 593, 619, 653, 659, 673

Offset: 1

Vardan Semerjyan, Apr 28 2015

nonn

Vardan Semerjyan, Voronoi diagram of prime spiral

Cf. A257527, A257528, A000040, A077800.

sz  = 201; % Size of the N x N square matrix
mat = spiral(sz); % MATLAB Function
k = 1;
for i =1:sz
    for j=1:sz
        if isprime(mat(i,j)) % Check if the number is prime
            % saving indices of primes
            y(k) = i; x(k) = j;
            k = k+1;
        end
    end
end
xy = [x',y'];
[v,c] = voronoin(xy); %  Returns Voronoi vertices V and
% the Voronoi cells C
k = 1;
for i = 1:length(c)
  szv = size(v(c{i},1));
  polyN(i) = szv(1);
  if polyN(i) == 5
        A(k) = mat(y(i),x(i));
        k = k+1;
      end
end
% Print terms
A = sort(A);
fprintf('A = ');
fprintf('%i, ',A);
% Note that the last terms might not be correct.
% They correspond to the points on the outer edges of the spiral which might be altered when considering a larger spiral.
% Use larger spiral to get more terms

A257638 Numbers n such that n-25, n-1, n+1 and n+25 are consecutive primes.

Original entry on oeis.org

232962, 311712, 431832, 435948, 473352, 501342, 525492, 596118, 635388, 665922, 699792, 754182, 842448, 1013502, 1017648, 1036002, 1156848, 1255452, 1284738, 1306692, 1479912, 1516128, 1551732, 1560708, 1595928, 1659348, 1690572, 1745112

Offset: 1

Karl V. Keller, Jr., Nov 04 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n-25 and n+1 belong to A033560 (p and p+24 are primes) and A098974 (p where p+24 is the next prime).
The numbers n-25 and n-1 belong to A252089 (p and p+26 are primes).

			232962 is the average of the four consecutive primes 232937, 232961, 232963, 232987.
311712 is the average of the four consecutive primes 311687, 311711, 311713, 311737.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-25 and nextprime(i+1) == i+25: print (i,end=', ')

A258088 Numbers n such that n is the average of four consecutive primes n-5, n-1, n+1 and n+5.

Original entry on oeis.org

12, 18, 42, 102, 108, 228, 312, 462, 858, 882, 1092, 1302, 1428, 1488, 1872, 1998, 2688, 3462, 4518, 4788, 5232, 5652, 6828, 7878, 8292, 10458, 13692, 13878, 15732, 16062, 16068, 16188, 17388, 19422, 19428, 20748, 21018, 21318, 22278, 23058

Offset: 1

Karl V. Keller, Jr., May 19 2015

nonn

Previous name was: Numbers n such that n is the average of some twin prime pair p, q (q=p+2) (i.e., n=p+1=q-1) where p-4, p, q, and q+4 are consecutive primes.

This is a subsequence of A014574 (average of twin prime pairs) and A256753.

			12 is the average of the four consecutive primes 7, 11, 13, 17.
18 is the average of the four consecutive primes 13, 17, 19, 23.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A052378, A077800 (twin primes), A256753.

a={};Do[If[Prime[x + 3] - Prime[x]==10, AppendTo[a, Prime[x]+ 5]], {x, 1, 4000}]; a (* Vincenzo Librandi, Jul 18 2015 *)
Mean/@Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&] (* Harvey P. Dale, Sep 18 2018 *)

is(n)=isprime(n-5)&&isprime(n-1)&&isprime(n+1)&&isprime(n+5) \\ Charles R Greathouse IV, Aug 28 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,50001,2):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-5 and nextprime(i+1) == i+5: print (i,end=', ')

a(n) = A052378(n) + 5. - Karl V. Keller, Jr., Jul 17 2015

New name from Karl V. Keller, Jr., Jul 21 2015

A258879 Numbers k such that k is the average of four consecutive primes k-7, k-1, k+1 and k+7.

Original entry on oeis.org

30, 60, 270, 570, 600, 1230, 1290, 1620, 2340, 2550, 3540, 4020, 4650, 5850, 6270, 6360, 6570, 10860, 11490, 14550, 15270, 17490, 19080, 19380, 19470, 23670, 26730, 29130, 32370, 34260, 41610, 48480, 49200, 49530, 51420, 51480

Offset: 1

Karl V. Keller, Jr., Jun 13 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs), A256753 and A249674 (30*n).

			For k=30: 23, 29, 31, 37 are consecutive primes (k-7=23, k-1=29, k+1=31, k+7=37).
For k=60: 53, 59, 61, 67 are consecutive primes (k-7=53, k-1=59, k+1=61, k+7=67).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A078854, A249674, A256753.

[n: n in [13..2*10^5] | IsPrime(n-7) and IsPrime(n-1) and IsPrime(n+1) and IsPrime(n+7)]; // Vincenzo Librandi Jul 16 2015

Select[ 5 Range@ 11000, PrimeQ[# - 7] && PrimeQ[# - 1] && PrimeQ[# + 1] && PrimeQ[# + 7] &] (* Robert G. Wilson v, Jun 28 2015 *)

main(size)={my(v=vector(size),i,t=8);for(i=1,size,while(1,if(isprime(t-7)&&isprime(t-1)&&isprime(t+1)&&isprime(t+7),v[i]=t;break,t++));t++);return(v);} /* Anders Hellström, Jul 17 2015 */

from sympy import isprime, prevprime, nextprime
for i in range(0, 10001, 2):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-7 and nextprime(i+1) == i+7: print(i, end=', ')

a(n) = A078854(n) + 7.

A260959 Numbers n such that n is the average of four consecutive primes n-13, n-1, n+1 and n+13.

Original entry on oeis.org

7950, 10500, 32970, 33330, 34470, 36900, 43050, 66360, 71550, 74610, 87120, 89070, 92400, 94560, 95190, 102000, 104310, 121950, 125790, 133980, 148470, 156900, 160710, 168630, 174930, 182640, 194070, 204600, 206250, 230340, 244380, 246510

Offset: 1

Karl V. Keller, Jr., Aug 06 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs), A256753 and A249674 (30n).

			7950 is the average of the four consecutive primes 7937, 7949, 7951, 7963.
10500 is the average of the four consecutive primes 10487, 10499, 10501, 10513.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

use ntheory ":all"; say join ", ", map { $+1 } grep { next_prime($+2)-$==14 } grep { $-prev_prime($)==12 } @{twin_primes(1e6)}; # _Dana Jacobsen, Oct 03 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,300001,2):
   if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-13 and nextprime(i+1) == i+13 :  print (i,end=', ')

A261731 Initial member of five twin prime pairs with gap 210 between them.

Original entry on oeis.org

1308497, 3042491, 3042701, 7445309, 20031101, 31572521, 44687987, 54266291, 141208619, 182316521, 237416369, 357080021, 448436321, 611641187, 699458411, 761126027, 774997367, 794065967, 836452961, 915215591, 944958941, 1009194617, 1581935939, 1763255561, 1871007371

Offset: 1

K. D. Bajpai, Aug 30 2015

nonn

More precisely, primes p such that p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842 are all primes.

All the terms in this sequence are congruent to 2 (mod 3).

			1308497 appears in this sequence because: (a) {1308497, 1308499}, {1308707, 1308709}, {1308917, 1308919}, {1309127, 1309129}, and {1309337, 1309339} are five twin prime pairs; (b) the gap between each twin prime pair {1308707 - 1308497} = {1308917-1308707} = {1309127 - 1308917} = {1309337 - 1309127} = 210.

K. D. Bajpai and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 46 terms from K. D. Bajpai]

Cf. A077800, A113274, A253624, A261701.

[p: p in PrimesUpTo (100000) | IsPrime(p+2) and IsPrime(p+210) and IsPrime(p+212) and IsPrime(p+420) and IsPrime(p+422) and IsPrime(p+630) and IsPrime(p+632) and IsPrime(p+840) and IsPrime(p+842) ];

select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842]),[seq(p, p=1..2*10^7)]);

k = 210; Select[Prime@Range[6*10^7], PrimeQ[# + 2] && PrimeQ[# + k] && PrimeQ[# + k + 2] && PrimeQ[# + 2 k] && PrimeQ[# + 2 k + 2] && PrimeQ[# + 3 k] &&   PrimeQ[# + 3 k + 2] && PrimeQ[# + 4 k] && PrimeQ[# + 4 k + 2] &]
Select[Prime[Range[93*10^6]],AllTrue[#+{2,210,212,420,422,630,632,840,842},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 05 2018 *)

forprime(p= 1,3*10^9, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632) && isprime(p+840) && isprime(p+842), print1(p,", ")));

use ntheory ":all"; say for sieve_prime_cluster(1,1e10, 2, 210, 212, 420, 422, 630, 632, 840, 842); # Dana Jacobsen, Oct 02 2015

A262176 Numbers k such that k-17, k-1, k+1 and k+17 are consecutive primes.

Original entry on oeis.org

3390, 66570, 70140, 84810, 132330, 136710, 222840, 225750, 242730, 271770, 288930, 320010, 330330, 377370, 390390, 414330, 463890, 489960, 505710, 644670, 758340, 819390, 830310, 857010, 895650, 906540, 908910, 924810, 952380, 968520, 974820

Offset: 1

Karl V. Keller, Jr., Sep 13 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs), A256753 and A249674 (30n).

			3390 is the average of the four consecutive primes 3373, 3389, 3391, 3407.
66570 is the average of the four consecutive primes 66553, 66569, 66571, 66587.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

Select[Prime@ Range@ 50000, NextPrime[#, {1, 2, 3}] == {16, 18, 34} + # &] + 17 (* Giovanni Resta, Sep 14 2015 *)

list(l)=for(i=1,l,my(p=prime(i)); if(p+16==prime(i+1) && p+18==prime(i+2) && p+34==prime(i+3), print1(p+17,", "))) \\ Anders Hellström, Sep 14 2015

use ntheory ":all"; say $+1 for grep { next_prime($+2)-$ == 18 && $-prev_prime($) == 16} @{twin_primes(1e9)}; # _Dana Jacobsen, Oct 13 2015

use ntheory ":all"; say $+17 for grep { next_prime($+0)-$ == 16 && next_prime($+18)-$ == 34} sieve_prime_cluster(1,1e9,16,18,34); # _Dana Jacobsen, Oct 13 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,3000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1)==i-17 and nextprime(i+1)==i+17 : print (i,end=', ')

A262668 Numbers n such that n-19, n-1, n+1 and n+19 are consecutive primes.

Original entry on oeis.org

20982, 28182, 51768, 57222, 76422, 87720, 90678, 104850, 108108, 110730, 141180, 199602, 227112, 248118, 264600, 268842, 304392, 304458, 320082, 322920, 330018, 382728, 401670, 414432, 429972, 450258, 467082, 489408, 520548, 535608, 540120

Offset: 1

Karl V. Keller, Jr., Sep 26 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are divisible by 30 (cf. A249674).
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.

			20982 is the average of the four consecutive primes 20963, 20981, 20983, 21001.
28182 is the average of the four consecutive primes 28163, 28181, 28183, 28201.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

Select[Range[6, 600000, 6], And[AllTrue[{# - 1, # + 1}, PrimeQ], NextPrime[# - 1, -1] == # - 19, NextPrime[# + 1] == # + 19] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
Select[Prime@Range@60000, NextPrime[#, {1, 2, 3}] == {18, 20, 38} + # &] + 19 (* Vincenzo Librandi, Oct 10 2015 *)
Mean/@Select[Partition[Prime[Range[50000]],4,1],Differences[#]=={18,2,18}&] (* Harvey P. Dale, Jan 16 2019 *)

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-19 and nextprime(i+1) == i+19 : print(i,end=', ')

A263298 Numbers n such that n-23, n-1, n+1 and n+23 are consecutive primes.

Original entry on oeis.org

19890, 43890, 157770, 400680, 436650, 609780, 681090, 797310, 924360, 978180, 1093200, 1116570, 1179150, 1185930, 1313700, 1573110, 1663350, 2001510, 2110290, 2163570, 2336310, 2372370, 2408280, 2415630, 2562690, 2877840, 2896740, 2961900

Offset: 1

Karl V. Keller, Jr., Oct 13 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs), A256753 and A249674 (30n).
From Michel Marcus, Oct 15 2015: (Start)
n-23 and n+1 belong to A242476 (p and p+22 are primes).
n-23 and n-1 belong to A033560 (p and p+24 are primes).
(End)

			19890 is the average of the four consecutive primes 19867, 19889, 19891, 19913.
43890 is the average of the four consecutive primes 43867, 43889, 43891, 43913.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

{p, q, r, s} = {2, 3, 5, 7};lst={}; While[p<5000000, If[Differences[{p, q, r, s}]=={22, 2, 22}, AppendTo[lst, q + 1]]; {p, q, r, s}={q, r, s,NextPrime@s}]; lst (* Vincenzo Librandi, Oct 14 2015 *)

isok(n) = isprime(n-1) && isprime(n+1) && (precprime(n-2) == n-23) && (nextprime(n+2) == n+23); \\ Michel Marcus, Oct 14 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,5000001,6):
  if  isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-23 and nextprime(i+1) == i+23: print (i,end=', ')

cp's OEIS Frontend

A256620 Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the arithmetic mean of the four numbers consisting of the two primes before p and the two primes after q.

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A257529 Prime numbers that have a pentagonal Voronoi cell in the Voronoi diagram of the Ulam prime spiral.

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A257638 Numbers n such that n-25, n-1, n+1 and n+25 are consecutive primes.

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A258088 Numbers n such that n is the average of four consecutive primes n-5, n-1, n+1 and n+5.

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A258879 Numbers k such that k is the average of four consecutive primes k-7, k-1, k+1 and k+7.

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A260959 Numbers n such that n is the average of four consecutive primes n-13, n-1, n+1 and n+13.

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A261731 Initial member of five twin prime pairs with gap 210 between them.

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Magma