A005473 -id:A005473 - cp's OEIS frontend

A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

2, 3, 7, 5, 41, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 617, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 383, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73

Offset: 1

Labos Elemer, Feb 26 2001

nonn

			For n = 17, let P = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,...} be the set of primes, then P - 17 = {-15,...,-4,0,2,6,12,14,20,24,26,30,36,...}. The first positive square in P - 17 is 36 with p = 53, so a(17) = 53. The square arising here is usually 1.

Zak Seidov, Table of n, a(n) for n = 1..10000

These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.
Cf. A002496, A056899, A049423, A005473, A056905, A056909.
Cf. A056896 (where p-n can be 0).

SearchLimit := 100;
for n from 1 to 400 do
k := 0: c := true:
while(c and k < SearchLimit) do
    k := k + 1:
    c := not isprime(k^2+n):
end do:
if k = SearchLimit then error("Search limit reached!") fi;
a[n] := k^2 + n end do: seq(a[j], j=1..400);
# Edited and SearchLimit introduced by Peter Luschny, Feb 05 2019

spsq[n_]:=Module[{p=NextPrime[n]},While[!IntegerQ[Sqrt[p-n]],p= NextPrime[ p]];p]; Array[spsq,70] (* Harvey P. Dale, Nov 10 2017 *)

for(n=1, 100, for(k=1, 100, if(isprime(k^2+n), print1(k^2+n, ", "); break()))) \\ Jianing Song, Feb 04 2019

a(n) = forprime(p=n,, if ((p-n) && issquare(p-n), return (p))); \\ Michel Marcus, Feb 05 2019

a(n) = min{p : p - n = x^2 for some x > 0, p is prime}.

Does a(n) exist for all n? - Jianing Song, Feb 04 2019

A098062 Primes of the form n^2 + 4n + 8.

Original entry on oeis.org

13, 29, 53, 173, 229, 293, 733, 1093, 1229, 1373, 2029, 2213, 3253, 4229, 4493, 5333, 7229, 7573, 9029, 9413, 10613, 13229, 13693, 15629, 18229, 18773, 21613, 24029, 26573, 27893, 31333, 33493, 37253, 41213, 42853, 46229, 47093, 54293, 55229

Offset: 1

Giovanni Teofilatto, Sep 12 2004

Or, primes that are equal to the mean of 7 consecutive squares. - Zak Seidov, Apr 14 2007
Sum of 7 consecutive squares starting with m^2 is equal to 7*(13 + 6*m + m^2) and mean is (13 + 6*m + m^2)=(m+3)^2+4. Hence a(n)=A005473(n+1). Note that only nonnegative m's are considered. - Zak Seidov, Apr 14 2007
a(n)==1 (mod 4).
a(n)= A005473(n+1). - Zak Seidov, Apr 12 2007

			13 = (0^2 + ... + 6^2)/7, 29 = (2^2 + ... + 8^2)/7 = 29, 53 = (4^2 + ... + 10^2)/7 = 53.

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

Cf. A005473, A056899, A067201, A007591, A129389, A129412.

[a: n in [0..250] | IsPrime(a) where a is n^2 + 4*n + 8]; // Vincenzo Librandi, Jul 17 2012

Select[ Table[ n^2 + 4n + 8, {n, 240}], PrimeQ[ # ] &] (* Robert G. Wilson v, Sep 14 2004 *)

for(n=0,240,if(isprime(p=n^2+4*n+8),print1(p,","))) \\ Klaus Brockhaus

Edited, corrected and extended by Robert G. Wilson v and Klaus Brockhaus, Sep 14 2004

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A243449 Primes of the form n^2 + 14.

Original entry on oeis.org

23, 239, 743, 1103, 2039, 5639, 7583, 8663, 27239, 33503, 38039, 42863, 59063, 81239, 88223, 91823, 119039, 131783, 140639, 164039, 189239, 205223, 245039, 263183, 288383, 328343, 342239, 378239, 393143, 400703, 431663, 439583, 514103, 660983, 710663, 950639

Offset: 1

Vincenzo Librandi, Jun 05 2014

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

Cf. A121250 (associated n).

Cf. primes of the form n^2+k: A144255 (k=1), A056899 (k=2), A049423 (k=3), A005473 (k=4), A056905 (k=5), A056909 (k=6), A079138 (k=7), A138338 (k=8), A138353 (k=9), A138355 (k=10), A138362 (k=11), A138368 (k=12), A138375 (k=13), this sequence (k=14), A243450 (k=15), A243451 (k=16), A228244 (k=17), A174812 (k=42).

[a: n in [0..1000] | IsPrime(a) where a is n^2+14];

Select[Table[n^2 + 14, {n, 0, 2000}], PrimeQ]
Select[Range[1,1001,2]^2+14,PrimeQ] (* Harvey P. Dale, May 30 2023 *)

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, 1, 2, -3, -6, 6, -8, -11, 11, 12, 14, -16, 16, 17, 19, -21, -23, -26, 27, -28, 32, -34, -36, 36, -39, 39, -41, 42, 44, -46, 46, -48, -49, 51, 52, -53, -58, 62, 64, 67, -68, -71, 71, -76, 77, 79, 81, -84, -89, 91, 96, -99, -101, 101, 102, -104, -111, 111, -113

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 5 = (6*k+1)^2 + 4, which is four more than a square. Except for a(0), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 12*2 + 5 = 173 which is prime (as well as being four more than a square).

This sequence and formula, together with A056908 and its formula, generate all primes of the form k^2+4, i.e., A005473. Except for the first term, this sequence is a subsequence of A047201. Cf. A056900, A056902, A056904, A056906.

A059844 a(n) = smallest nonzero square x^2 such that n+x^2 is prime.

Original entry on oeis.org

1, 1, 4, 1, 36, 1, 4, 9, 4, 1, 36, 1, 4, 9, 4, 1, 36, 1, 4, 9, 16, 1, 36, 49, 4, 81, 4, 1, 144, 1, 16, 9, 4, 9, 36, 1, 4, 9, 4, 1, 576, 1, 4, 9, 16, 1, 36, 25, 4, 9, 16, 1, 36, 25, 4, 81, 4, 1, 324, 1, 36, 9, 4, 9, 36, 1, 4, 81, 4, 1, 36, 1, 16, 9, 4, 25, 36, 1, 4, 9, 16, 1, 144, 25, 4, 81

Offset: 1

Labos Elemer, Feb 26 2001

nonn

a(n) = 1 for n in A006093. - Robert Israel, Dec 31 2023

			a(24) = 49 because 49 + 24 = 73 is prime and 1 + 24 = 25, 4 + 24 = 28, 9 + 24 = 33, 16 + 24 = 40, 25 + 24 = 49, and 36 + 24 = 60 are composite.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002496, A006093, A056899, A049423, A005473, A056905, A056909.

f:= proc(n) local x;
 for x from 1 + (n mod 2) by 2  do
  if isprime(n+x^2) then return x^2 fi;
 od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 31 2023

sqs[n_]:=Module[{q=1},While[!PrimeQ[n+q],q=(Sqrt[q]+1)^2];q]; Array[ sqs,90] (* Harvey P. Dale, Aug 11 2017 *)

a(n) + n is the smallest prime of the form x^2 + n.

A129412 Numbers k such that mean of 7 consecutive squares starting with k^2 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 14, 24, 30, 32, 34, 42, 44, 54, 62, 64, 70, 82, 84, 92, 94, 100, 112, 114, 122, 132, 134, 144, 152, 160, 164, 174, 180, 190, 200, 204, 212, 214, 230, 232, 240, 242, 250, 252, 262, 264, 272, 274, 284, 290, 300, 304, 310, 314, 344, 354, 370, 372

Offset: 1

Zak Seidov, Apr 14 2007

Sum of 7 consecutive squares starting with k^2 is equal to 7*(13 + 6*k + k^2) and mean is (13 + 6*k + k^2) = (k+3)^2+4. Hence a(n) = A007591(n+1)-3.

			(0^2+...+6^2)/7=13 prime, (2^2+...+8^2)/7=29 prime, (4^2+...+10^2)/7=53 prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005473, A056899, A067201, A007591, A129389, A098062.

Select[Range[0,400],PrimeQ[Mean[Range[#,#+6]^2]]&]  (* Harvey P. Dale, Apr 23 2011 *)

is(n)=isprime(n^2+6*n+13) \\ Charles R Greathouse IV, Jun 13 2017

A089747 Numbers n such that n^2 - 2n + 5 is prime.

Original entry on oeis.org

0, 2, 4, 6, 8, 14, 16, 18, 28, 34, 36, 38, 46, 48, 58, 66, 68, 74, 86, 88, 96, 98, 104, 116, 118, 126, 136, 138, 148, 156, 164, 168, 178, 184, 194, 204, 208, 216, 218, 234, 236, 244, 246, 254, 256, 266, 268, 276, 278, 288, 294, 304, 308, 314, 318, 348, 358, 374

Offset: 1

Giovanni Teofilatto, Jan 08 2004

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna, 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano, 1997.

Cf. A005473 gives primes, A007591.

[n: n in [0..100]|IsPrime(n^2-2*n+5)] // Vincenzo Librandi, Nov 16 2010

Select[Range[0,400,2],PrimeQ[#^2-2#+5]&] (* Harvey P. Dale, Jan 04 2015 *)

is(n)=isprime(n^2-2*n+5) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A007591(n-1) + 1 for n > 1. [corrected by Georg Fischer, Jun 20 2020]

A346145 Primes of the form k^2 + 25.

Original entry on oeis.org

29, 41, 61, 89, 281, 349, 509, 601, 701, 809, 1049, 1181, 1321, 1789, 2141, 2729, 3389, 4649, 5209, 5501, 5801, 8861, 9241, 9629, 10429, 11261, 11689, 12569, 15401, 15901, 17449, 17981, 18521, 19069, 21341, 21929, 23741, 24989, 26921, 27581, 33149, 39229, 40829, 41641, 42461, 45821, 46681, 52009

Offset: 1

Todor Szimeonov, Jul 06 2021

nonn

k^2 + 25 = (k+5i)*(k-5i), where i is the imaginary unit.

Todor Szimeonov, A dramatic encounter

Cf. A002144, A002496, A005473, A138353, A243451.

Select[Range[230]^2 + 25, PrimeQ] (* Amiram Eldar, Jul 06 2021 *)

list(lim)=my(v=List(),p); forstep(k=2,sqrtint(lim\1-25),2, if(isprime(p = k^2+25), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 06 2021

a(n) >> n log^2 n (Brun sieve). - Charles R Greathouse IV, Jul 06 2021

cp's OEIS Frontend

A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

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A098062 Primes of the form n^2 + 4n + 8.

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A243449 Primes of the form n^2 + 14.

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A056907 Numbers k such that 36*k^2 + 12*k + 5 is prime (sorted by absolute values with negatives before positives).

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A059844 a(n) = smallest nonzero square x^2 such that n+x^2 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A129412 Numbers k such that mean of 7 consecutive squares starting with k^2 is prime.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A089747 Numbers n such that n^2 - 2n + 5 is prime.

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Author

Keywords

References

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A346145 Primes of the form k^2 + 25.

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Author

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).