A156104 -id:A156104 - cp's OEIS frontend

A252089 Primes p such that p + 26 is prime.

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A156112 Primes p such that p+36 and p+72 are both prime.

Original entry on oeis.org

7, 11, 17, 31, 37, 67, 101, 127, 157, 191, 197, 241, 277, 281, 311, 317, 337, 347, 431, 521, 541, 571, 647, 751, 787, 911, 941, 947, 977, 997, 1051, 1151, 1187, 1451, 1487, 1621, 1627, 1877, 2017, 2027, 2237, 2311, 2467, 2521, 2621, 2647, 2657, 2677, 2731

Offset: 1

Vincenzo Librandi, Feb 08 2009

A156105 INTERSECT A156104. [R. J. Mathar, Feb 09 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156104, A156105.

[p: p in PrimesUpTo(3000)|IsPrime(p + 36) and IsPrime (p + 72)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[3000]], And @@ PrimeQ[{# + 36,# + 72}]&] (* Vincenzo Librandi, Oct 31 2012 *)
Select[Prime[Range[3000]],AllTrue[#+{36,72},PrimeQ]&] (* Harvey P. Dale, Jul 12 2023 *)

More terms from R. J. Mathar, Feb 09 2009

A092475 Primes p such that p + 2^2, p + 4^2 and p + 6^2 are also primes.

Original entry on oeis.org

7, 37, 43, 67, 163, 277, 463, 487, 823, 1087, 1093, 1213, 1423, 2683, 3907, 4447, 5653, 7687, 8677, 8803, 11467, 11923, 13147, 13693, 15787, 16417, 16657, 16927, 18253, 18397, 19387, 20113, 20353, 21487, 27763, 28627, 30493, 34483, 38917, 39103, 40483, 41227

Offset: 1

Ray G. Opao, Mar 25 2004

nonn

			a(3) = 43.
43 + 2^2 = 43 +  4 = 47, which is prime.
43 + 4^2 = 43 + 16 = 59, which is prime.
43 + 6^2 = 43 + 36 = 79, which is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Subsequence of A049492.

Select[Prime[Range[5000]],And@@PrimeQ[{#+4,#+16,#+36}]&] (* Harvey P. Dale, Jun 09 2011 *)

A049492 INTERSECT A156104. - R. J. Mathar, Mar 26 2024

More terms from Harvey P. Dale, Jun 09 2011

A156109 Primes p such that p+18 and p+36 are both prime.

Original entry on oeis.org

5, 11, 23, 43, 53, 61, 71, 113, 131, 163, 193, 233, 313, 331, 383, 421, 431, 443, 641, 673, 683, 691, 733, 751, 821, 911, 1013, 1033, 1051, 1213, 1283, 1453, 1531, 1583, 1601, 1723, 1741, 1871, 1913, 1993, 2063, 2143, 2203, 2251, 2381, 2423, 2441, 2503

Offset: 1

Vincenzo Librandi, Feb 08 2009

A153418 INTERSECT A156104. [Bruno Berselli, Nov 01 2012]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A153418, A156104.

[p: p in PrimesUpTo(3000)|IsPrime(p + 18) and IsPrime (p + 36)]; // Vincenzo Librandi, Oct 31 2012

a := proc (n) if isprime(ithprime(n)+18) = true and isprime(ithprime(n)+36) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 400); # Emeric Deutsch, Mar 02 2009

Select[Prime[Range[3000]], And @@ PrimeQ[{# + 18, # + 36}]&] (* Vincenzo Librandi, Oct 31 2012 *)

More terms from Emeric Deutsch, Mar 02 2009

A156123 Primes p such that p+36 and p+144 are both prime.

Original entry on oeis.org

5, 7, 23, 37, 47, 53, 67, 113, 127, 137, 163, 193, 257, 277, 313, 317, 347, 397, 443, 463, 487, 557, 607, 617, 683, 733, 823, 827, 947, 1087, 1093, 1223, 1283, 1423, 1453, 1523, 1657, 1723, 1787, 1993, 2017, 2063, 2143, 2203, 2207, 2237, 2273, 2297, 2503, 2543

Offset: 1

Vincenzo Librandi, Feb 08 2009

A156104 INTERSECT A156107. - Vincenzo Librandi, Nov 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156104, A156107.

[p: p in PrimesUpTo(3000)|IsPrime(p + 36) and IsPrime (p + 144)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[3000]], And @@ PrimeQ[{# + 36, # + 144}]&] (* Vincenzo Librandi, Oct 31 2012 *)

137 inserted and sequence extended by R. J. Mathar, Feb 19 2009

A268305 Numbers k such that k - 37, k - 1, k + 1, k + 37 are consecutive primes.

Original entry on oeis.org

1524180, 3264930, 3970530, 5438310, 5642910, 6764940, 8176410, 10040880, 10413900, 10894320, 11639520, 12352980, 13556340, 15900720, 16897590, 17283360, 18168150, 18209100, 18686910, 19340220, 20099940, 20359020, 20483340, 21028290, 21846360

Offset: 1

Karl V. Keller, Jr., Apr 17 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs), A249674 (divisible by 30) and A256753.
The numbers k - 37 and k + 1 belong to A156104 (p and p + 36 are primes) and A134117 (p where p + 36 is the next prime).
The numbers k - 37 and k - 1 belong to A271347 (p and p + 38 are primes).

			1524180 is the average of the four consecutive primes 1524143, 1524179, 1524181, 1524217.
3264930 is the average of the four consecutive primes 3264893, 3264929, 3264931, 3264967.

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[Partition[Prime[Range[14*10^5]],4,1],Differences[#]=={36,2,36}&][[All,2]]+1 (* Harvey P. Dale, Mar 12 2018 *)

from sympy import isprime,prevprime,nextprime
for i in range(0,30000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-37 and nextprime(i+1) == i+37 : print (i,end=', ')

A271349 Numbers n such that n - 35, n - 1, n + 1 and n + 35 are consecutive primes.

Original entry on oeis.org

276672, 558828, 1050852, 1278288, 1486908, 1625418, 2536308, 2538918, 2690958, 2731242, 3015162, 3252678, 3268338, 3508278, 3711612, 4233708, 4575912, 4717962, 5004402, 5108352, 5404032, 5482782, 5519082, 5525328, 5640918, 5654358, 5995818

Offset: 1

Karl V. Keller, Jr., Apr 04 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 2 (resp. 8) are congruent to 12 (resp. 18) mod 30.
The numbers n - 35 and n + 1 belong to A252091 (p and p + 34 are primes) and A134116 (p such that p + 34 is the next prime).
The numbers n - 35 and n - 1 belong to A156104 (p and p + 36 are primes).

			276672 is the average of the four consecutive primes 276637, 276671, 276673, 276707.
558828 is the average of the four consecutive primes 558793, 558827, 558829, 558863.

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), A256753.

Select[Partition[Prime[Range[500000]],4,1],Differences[#]=={34,2,34}&] [[All, 2]]+1 (* Harvey P. Dale, Oct 11 2017 *)

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-35 and nextprime(i+1) == i+35 :  print (i,end=', ')

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109

Offset: 1

Felix Fröhlich, Jul 28 2019

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Wikipedia, Polignac's conjecture

Cf. A231608.

Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

A365850 Numbers k for which k^2 + (k')^2 is a square, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

0, 1, 12, 15, 35, 81, 143, 323, 400, 441, 899, 1540, 1763, 2700, 3599, 4641, 5183, 5929, 9375, 10395, 10403, 11663, 13585, 18225, 19043, 21952, 22499, 30576, 32399, 35581, 36863, 39203, 48841, 51983, 57599, 72899, 79523, 97343, 121103, 148176, 166375, 175692, 176399

Offset: 1

Marius A. Burtea, Oct 09 2023

nonn

If p and p + 2 are twin primes (A001359) then m = p*(p + 2) is a term. Indeed, m' = p + (p + 2) = 2*p + 2 and m^2 + (m')^2 = p^2*(p + 2)^2 + (2*p + 2)^2 = (p^2 + 2*p + 2)^2.

More generally, if p and p + 2*k, k >= 1, are prime numbers, then m = p^k*(p + 2*k)^k is a term. Indeed, m' = k*p^(k - 1)*(p + 2*k)^k+ k*p^k*(p + 2*k)^(k - 1) = k*p^(k - 1)*(p + 2*k)^(k-1)*(2*p + 2*k). Thus, m^2 + (m')^2 = p^(2*k)*(p + 2*k)^(2*k) + (k^2)*p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*p + 2*k)^2 = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(p^2*(p + 2*k)^2 + k^2*(2*p + 2*k)^2) = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*k^2 + 2*k*p + p^2)^2.

			For k = 12, k' = 16 and 12^2 + 16^2 = 144 + 256 = 400 = 20^2, so 12 is a term.
For k = 15, k' = 8 and 15^2 + 8^2 = 225 + 64 = 289 = 17^2, so 15 is a term.
For k = 143, k' = 24 and 143^2 + 24^2 = 144 + 256 = 21025 = 145^2, so 143 is a term.

Cf. A077800, A153417 - A153419, A156104, A242476, A252089 - A252091, A271347, A271981, A271982, A272176.

f:=func; [n:n in [0..200000] |IsSquare( n^2+( Floor(f(n))^2))];

ader:= proc(n) local f;   n*add(f[2]/f[1], f=ifactors(n)[2]) end proc:
select(t -> issqr(t^2 + ader(t)^2), [$0..10^6]; # Robert Israel, Oct 17 2023

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 180000], IntegerQ[Sqrt[#^2 + d[#]^2]] &] (* Amiram Eldar, Oct 11 2023 *)

cp's OEIS Frontend

A252089 Primes p such that p + 26 is prime.

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Magma

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A156112 Primes p such that p+36 and p+72 are both prime.

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Magma

Mathematica

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A092475 Primes p such that p + 2^2, p + 4^2 and p + 6^2 are also primes.

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Mathematica

Formula

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A156109 Primes p such that p+18 and p+36 are both prime.

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Magma

Maple

Mathematica

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A156123 Primes p such that p+36 and p+144 are both prime.

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Magma

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A268305 Numbers k such that k - 37, k - 1, k + 1, k + 37 are consecutive primes.

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Python

A271349 Numbers n such that n - 35, n - 1, n + 1 and n + 35 are consecutive primes.

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Author

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Programs

Mathematica

Python

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

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