A005123 -id:A005123 - cp's OEIS frontend

A224467 Numbers n such that 27*n+1 is prime.

4, 6, 10, 14, 16, 18, 20, 28, 30, 34, 48, 54, 58, 60, 66, 74, 76, 80, 84, 88, 94, 96, 98, 108, 110, 114, 118, 128, 130, 136, 138, 144, 146, 150, 154, 166, 170, 180, 184, 186, 188, 198, 206, 214, 230, 236, 238, 240, 258, 264, 268, 278, 280, 284, 286, 296, 300

Offset: 1

K. D. Bajpai, Jul 20 2013

			27*6 + 1 = 163 which is prime. Hence 6 is in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..5736

Cf. A005123, A024906.

isA224467 := proc(n)
    isprime(27*n+1) ;
end proc:
for n from 1 to 300 do
    if isA224467(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 20 2013

Select[Range[300],PrimeQ[27#+1]&] (* Harvey P. Dale, Apr 14 2017 *)

is(n)=isprime(27*n+1) \\ Charles R Greathouse IV, Jun 06 2017

A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

Original entry on oeis.org

9, 24, 39, 57, 84, 144, 150, 165, 207, 219, 234, 249, 252, 267, 309, 324, 357, 402, 414, 507, 522, 534, 555, 570, 639, 654, 759, 765, 777, 792, 795, 882, 924, 927, 942, 969, 1044, 1065, 1089, 1155, 1200, 1215, 1227, 1389, 1395, 1437, 1509, 1530, 1554, 1557

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that if k is in this sequence then (4k + 1)*(8k + 1) is a triangular Fermat pseudoprime to base 2 (A293622), and thus under Schinzel's Hypothesis H there are infinitely many triangular Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2701, 18721, 49141, 104653, 226801, 665281, 721801, ...

			9 is in the sequence since 4*9 + 1 = 37 and 8*9 + 1 = 73 are both primes.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A000217, A001567, A293622.

Intersection of A005098 and A005123.

Select[Range[1000], PrimeQ[4#+1] && PrimeQ[8#+1] &]

isok(n) = isprime(4*n+1) && isprime(8*n+1); \\ Michel Marcus, Nov 20 2018

from sympy import isprime
def ok(n): return isprime(4*n + 1) and isprime(8*n + 1)
print(list(filter(ok, range(1558)))) # Michael S. Branicky, Sep 24 2021

A123978 Numbers k for which 8k+1, 8k+3 and 8*k+7 are primes.

Original entry on oeis.org

2, 5, 80, 107, 110, 185, 260, 332, 500, 1067, 1307, 1472, 1625, 1760, 1790, 1955, 2255, 2612, 2627, 2672, 2882, 2945, 3197, 3335, 3467, 3965, 4007, 4037, 4040, 4202, 4355, 4880, 5147, 5252, 5525, 6242, 6812, 6917, 6977, 7430, 7787, 8192, 8612, 8657, 8720

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

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

Cf. A005122, A005123, A005124.

Select[Range[10^4], And @@ PrimeQ /@ ({1, 3, 7} + 8#) &] (* Ray Chandler, Nov 05 2006 *)

Extended by Ray Chandler, Nov 05 2006

A123980 Numbers k for which 8k+1, 8k+5 and 8*k+7 are primes.

Original entry on oeis.org

12, 24, 57, 162, 234, 249, 267, 297, 432, 519, 564, 717, 969, 984, 1167, 1179, 1389, 1734, 2007, 2364, 2427, 2544, 2664, 2769, 2784, 3582, 3627, 3819, 3897, 4089, 4287, 5244, 5307, 5337, 5472, 5577, 5667, 5727, 5967, 6084, 6102, 6399, 6522, 6822, 6987

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

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

Cf. A005122, A005123, A005124, A105133.

Select[Range[7000], And @@ PrimeQ /@ ({1, 5, 7} + 8#) &] (* Ray Chandler, Nov 05 2006 *)

Extended by Ray Chandler, Nov 05 2006

A123983 Numbers k for which 8k+1, 8k+5, 8k+7 and 8k+11 are primes.

Original entry on oeis.org

12, 57, 162, 249, 432, 564, 984, 1734, 2007, 2427, 2664, 2784, 3627, 5307, 5472, 5727, 6399, 7614, 11082, 11547, 11607, 11694, 14127, 14274, 14484, 14862, 15117, 17049, 19104, 19422, 20577, 25677, 27612, 27714, 28152, 29307, 32232, 34602, 35592

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

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

Cf. A005122, A005123, A005124, A005125, A105133.

isA123983 := proc(n) RETURN( isprime(8*n+1) and isprime(8*n+5) and isprime(8*n+7) and isprime(8*n+11) ) ; end: for n from 1 to 7000 do if isA123983(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Nov 06 2006

Select[Range[37000], And @@ PrimeQ /@ ({1, 5, 7, 11} + 8#) &] (* Ray Chandler, Nov 05 2006 *)

Edited and extended by Ray Chandler and R. J. Mathar, Nov 05 2006

A365958 Least k such that 8nk+1 is a prime.

Original entry on oeis.org

2, 1, 3, 3, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 2, 2, 1, 3, 3, 4, 2, 2, 7, 1, 2, 6, 2, 2, 1, 1, 6, 1, 5, 5, 1, 2, 2, 4, 1, 2, 7, 1, 3, 1, 5, 9, 3, 2, 8, 1, 1, 3, 4, 1, 2, 1, 1, 2, 4, 7, 2, 3, 2, 15, 1, 4, 3, 10, 3, 5, 1, 1, 3, 1, 1, 2, 1, 2, 6, 1, 2, 12, 3, 1, 2

Offset: 1

Robert Price, Dec 17 2023

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A016014, A005123.

A070852 lists the corresponding primes.

A365958 = {};
Do[k=1; While[!PrimeQ[8 n k+1], k++]; AppendTo[A365958 ,k], {n,85}];
A365958

a(n) = my(k=1); while (!isprime(8*n*k+1), k++); k; \\ Michel Marcus, Dec 17 2023

A329269 Integers k such that 8*k + 1 is a prime or a square.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 14, 15, 17, 21, 24, 28, 29, 30, 32, 35, 36, 39, 42, 44, 45, 50, 51, 54, 55, 56, 57, 65, 66, 71, 72, 74, 75, 77, 78, 80, 84, 91, 95, 96, 101, 105, 107, 110, 116, 117, 119, 120, 122, 126, 129, 131, 136, 137, 141, 144, 149, 150

Offset: 1

Frank Ellermann, Feb 23 2020

All odd squares have the form 8*n + 1.

			8*0 + 1 =  1 = 1^2, so 0 is a term;
8*1 + 1 =  9 = 3^2, so 1 is a term;
8*2 + 1 = 17 = prime(7), so 2 is a term;
8*3 + 1 = 25 = 5^2, so 3 is a term;
8*4 + 1 = 33 is neither prime nor square, so 4 is not a term;
8*5 + 1 = 41 = prime(13), so 5 is a term.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, theorem 14 and ch. 4.5

Union of the triangular numbers A000217 and A005123.

Cf. A000040, A016754 (odd squares).

q:= k-> (t-> isprime(t) or issqr(t))(8*k+1):
select(q, [$0..200])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[0, 150], PrimeQ[(m = 8*# + 1)] || IntegerQ @ Sqrt[m] &] (* Amiram Eldar, Feb 29 2020 *)

isok(k) = my(x=8*k+1); isprime(x) || issquare(x); \\ Michel Marcus, Feb 27 2020

S = 0 ;  U = 1 ;  P = 1
do N = 1 while length( S ) < 256
   C = 8 * N + 1
   do I = U by 2
      K = I * I      ;  if K > C then  leave I
      U = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
   do I = P
      K = PRIME( I ) ;  if K > C then  leave I
      P = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
end N
say S ;  return S

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A224467 Numbers n such that 27*n+1 is prime.

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A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

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A123978 Numbers k for which 8*k+1, 8*k+3 and 8*k+7 are primes.

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A123980 Numbers k for which 8*k+1, 8*k+5 and 8*k+7 are primes.

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A123983 Numbers k for which 8*k+1, 8*k+5, 8*k+7 and 8*k+11 are primes.

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A365958 Least k such that 8*n*k+1 is a prime.

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A329269 Integers k such that 8*k + 1 is a prime or a square.

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Rexx

A123978 Numbers k for which 8k+1, 8k+3 and 8*k+7 are primes.

A123980 Numbers k for which 8k+1, 8k+5 and 8*k+7 are primes.

A123983 Numbers k for which 8k+1, 8k+5, 8k+7 and 8k+11 are primes.

A365958 Least k such that 8nk+1 is a prime.