A049492 -id:A049492 - cp's OEIS frontend

A049498 a(n) and a(n)+4^k are primes at least for k=1,2,3,4,5,6,7,8.

163, 15667, 607093, 671353, 1457857, 5772097, 9139453, 11170933, 13243063, 18116473, 19433863, 21960577, 32380177, 52896517, 115831753, 154146133, 165609217, 191489677, 361241743, 394845313, 518774953, 613615423, 705676717, 742403797, 786242293, 945170293

Offset: 1

Labos Elemer

nonn

			163, 163+4 = 167, 163+16 = 179, 163+64 = 227, 163+256 = 419, 163+1024 = 1187, 163+4096 = 4259, 163+16384 = 16547, 163+65536 = 65699 are all primes; the smallest such a 9-chain of primes is {163, 167, 178, 227, 419, 1187, 4259, 16547, 65699}

Cf. A023200, A049492-A049497, A049499-A049500.

  With[{c=4^Range[8]},Select[Prime[Range[500000]],And@@PrimeQ[#+c]&]] (* Harvey P. Dale, May 22 2012 *)

More terms from Michel Marcus, Dec 22 2013

A049499 A finite sequence of primes: the primes 671353+4^k for k=1, 2, 3, 4, 5, 6, 7, 8, 9.

Original entry on oeis.org

671353, 671357, 671369, 671417, 671609, 672377, 675449, 687737, 736889, 933497

Offset: 1

Labos Elemer

Below prime(1000000) just three initial primes give sets of 10 primes like this set: 671353, 5772097 and 13243063

Cf. A023200, A049492-A049498.

Edited by N. J. A. Sloane, Oct 31 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

A145897 Starting prime (and 1): where number of consecutive squares m^2 satisfy r=p+4*m^2, prime.

Original entry on oeis.org

1, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483

Offset: 1

Enoch Haga, Oct 25 2008

Suggested by Farideh Firoozbakht's Puzzle 464 in Carlos Rivera's The Prime Puzzles & Problems Connection. In this sequence Haga accepts 1 as a prime because then m^2 begins the first run of consecutive primes.

This looks like (apparent from the ad-hoc introduced leading 1) an erroneous version of A023200, because the definition says that it registers prime chains p+4*m^2, m=1,2,3,.. but apparently does not consider whether m is actually larger than 1. So 3 should be in the sequence because 3+4*1^2 is prime. - R. J. Mathar, Mar 25 2024

			a(1)=1 because when there are 3 consecutive m^2, first prime is 5 and ending prime is 37: r=1+4*1^1=5, prime; and r=1+4*2^2=17, prime; and r=1+4*3^2=37, prime (and the next value of r does not produce a prime).

Cf. A145896, A145898, A145741, A049492.

10 'p464
20 N=1
30 A=3:S=sqrt(N)
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 M=M+1:R=N+4*M^2:if R=prmdiv(R) and M<100 then print N;R;M:goto 80
90 if M>=1 then stop
100 M=0:N=N+2:goto 30

A092474 a(n) is the first term in a sequence of primes such that a(n)+4m^2 is also prime for m = 1 to n.

Original entry on oeis.org

3, 3, 7, 7, 7, 7, 37, 37, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163

Offset: 1

Ray G. Opao, Mar 25 2004

nonn

a(1) starts sequence A023200; a(2) starts sequence A049492. (I've searched up to 2000000 but haven't yet found the terms from a(20) onward.)

			a(5) = 7.
7+4(1^2) = 7+4(1) = 7+4 = 11 which is prime.
7+4(2^2) = 7+4(4) = 7+16 = 23 which is prime.
7+4(3^2) = 7+4(9) = 7+36 = 43 which is prime.
7+4(4^2) = 7+4(16) = 7+64 = 71 which is prime.
7+4(5^2) = 7+4(25) = 7+100 = 107 which is prime.

Flatten[Table[Select[Prime[Range[100]],And@@PrimeQ[Table[#+4n^2,{n,i}]]&,1],{i,19}]] (* Harvey P. Dale, Oct 04 2012 *)

A247275 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.

Original entry on oeis.org

163, 409333, 376040154163, 1822896797857, 9871431850597, 13491637509487, 19802478368863

Offset: 1

Zak Seidov, Sep 11 2014

All terms are == {7, 13} mod 30.

Subsequence of A247273.

Cf. A049492, A092120, A246842, A247269, A247273, A145741.

forprime(p=1, oo, c=0; for(i=1, 12, if(ispseudoprime(p+(2*i)^2), c++)); if(c==12, print1(p, ", "))) \\ Derek Orr, Sep 11 2014

A247276 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26}.

Original entry on oeis.org

163, 409333, 13491637509487, 19802478368863

Offset: 1

Zak Seidov, Sep 11 2014

All terms are == {7, 13} mod 30.

Subsequence of A247275.

Cf. A049492, A092120, A246842, A247269, A145741, A247273, A247275.

Select[Prime[Range[35000]],AllTrue[#+{4,16,36,64,100,144,196,256,324,400,484,576,676},PrimeQ]&] (* The program generates the first two terms of the sequence. To generate a(3) and a(4), increase the Range constant to 67*10^10 but the program will take a very long time to run. *) (* Harvey P. Dale, Mar 05 2025 *)

forprime(p=1, 10^12, c=0; for(i=1, 13, if(ispseudoprime(p+(2*i)^2), c++);if(!ispseudoprime(p+(2*i)^2),break)); if(c==13, print1(p, ", "))) \\ Derek Orr, Sep 11 2014

cp's OEIS Frontend

A049498 a(n) and a(n)+4^k are primes at least for k=1,2,3,4,5,6,7,8.

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A049499 A finite sequence of primes: the primes 671353+4^k for k=1, 2, 3, 4, 5, 6, 7, 8, 9.

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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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A145897 Starting prime (and 1): where number of consecutive squares m^2 satisfy r=p+4*m^2, prime.

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UBASIC

A092474 a(n) is the first term in a sequence of primes such that a(n)+4m^2 is also prime for m = 1 to n.

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A247275 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.

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PARI

A247276 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26}.

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