A056899 -id:A056899 - cp's OEIS frontend

A243449 Primes of the form n^2 + 14.

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 *)

A302434 Number of primes of the form b^2 + 2 for b <= 10^n.

Original entry on oeis.org

4, 12, 69, 447, 3423, 27869, 236985, 2054022, 18127693, 162237123

Offset: 1

Seiichi Manyama, Apr 07 2018

From Jacques Tramu, Sep 13 2018: (Start)
Table C(i) = a(i)/(n*log(n)), with n = 10^i:
a(1)  =         4     C(1)  = 0.92103404
a(2)  =        12     C(2)  = 0.55262042
a(3)  =        69     C(3)  = 0.47663511
a(4)  =       447     C(4)  = 0.41170221
a(5)  =      3423     C(5)  = 0.39408744
a(6)  =     27869     C(6)  = 0.38502446
a(7)  =    236985     C(7)  = 0.38197469
a(8)  =   2054022     C(8)  = 0.37836484
a(9)  =  18127693     C(9)  = 0.37566500
a(10) = 162237123     C(10) = 0.37356478
(End)

			a(1) = 4 because there are 4 primes of the form b^2+2 for b <= 10: 2, 3, 11 and 83.

Number of primes of the form b^2+m for b <= 10^n: A302443 (m=-3), A302442 (m=-2), A206709 (m=1), this sequence (m=2), A302435 (m=3).

Cf. A056899.

PARI
```
{a(n) = sum(k=0, 10^n, isprime(k^2+2))}
```

a(10) from Jacques Tramu, Sep 13 2018

A386245 Composite numbers k such that A075255(k) is a square.

Original entry on oeis.org

4, 6, 22, 135, 166, 444, 454, 636, 650, 854, 886, 1086, 1122, 1196, 1431, 1928, 2182, 2244, 2316, 2702, 3046, 3464, 3510, 3770, 4004, 4054, 4125, 4476, 4671, 5052, 5106, 5394, 5450, 6435, 6502, 6750, 8076, 8264, 8500, 9170, 9471, 9726, 10035, 10386, 10648, 10659, 11228, 11495, 11515, 11935, 12732

Offset: 1

Will Gosnell and Robert Israel, Jul 16 2025

nonn

Composite numbers k such that k - sopfr(k) is a square, where sopfr(k) is the sum of prime factors of k with multiplicity.
Includes 2*p for p in A056899, but no odd semiprimes.
Is this sequence disjoint from A386246?

			a(3) = 22 is a term because 22 = 2 * 11 is composite and 22 - (2 + 11) = 9 is a square.

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

Cf. A056899, A075255, A386246.

filter:= proc(n) local t;
 if isprime(n) then return false fi;
 issqr(n - add(t[1]*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$4..20000]);

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

A164520 Primes p such that p-2 is the product of exactly 2 distinct cubes of primes.

Original entry on oeis.org

274627, 328511, 1860869, 2146691, 2924209, 9129331, 9938377, 10503461, 15438251, 24642173, 26730901, 28372627, 39651823, 61629877, 105823819, 125751503, 136590877, 151419439, 194104541, 426957779, 573856193

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 14 2009

nonn

			274627 - 2 = 5^3*13^3, 328511 - 2 = 3^3*23^3,..

Chai Wah Wu, Table of n, a(n) for n = 1..10124

Cf. A056899, A144953, A164517, A164518, A164519

f3[n_]:=FactorInteger[n][[1,2]]==3&&Length[FactorInteger[n]]==2&&FactorInteger[n][[2,2]]==3; lst={};Do[p=Prime[n];If[f3[p-2],AppendTo[lst,p]],{n,4,4*9!}];lst

forprime(p=3,1e9,if(ispower(p-2,3,&n)&&!issquare(n)&&bigomega(n)==2,print1(p",")))

Program by Charles R Greathouse IV, Oct 12 2009

A232011 Numbers n such that (3n)^2 + 2 is prime.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 19, 27, 33, 35, 37, 39, 41, 49, 57, 73, 75, 79, 81, 83, 85, 91, 99, 101, 103, 107, 115, 117, 121, 123, 125, 129, 139, 141, 143, 147, 149, 151, 159, 167, 171, 183, 185, 187, 191, 201, 203, 217, 225, 227, 233, 237, 251, 259, 269, 273, 279

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Corresponding values of such primes are in A056899(n) for n>2.

Supersequence of A126960 (primes p such that (3p)^2 + 2 is prime).

			7 is in sequence because (3*7)^2 + 2 = 443 (prime).

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000040, A056899, A067201, A126960.

Select[Range[279], PrimeQ[9 #^2 + 2] &] (* T. D. Noe, Nov 27 2013 *)

is(n)=isprime(9*n^2+2) \\ Charles R Greathouse IV, Nov 25 2013

a(n) = sqrt(A056899(n+2) - 2)/3 = A067201(n+2)/3.

A260930 Differences between the numbers n such that n^2 + 2 is prime.

Original entry on oeis.org

1, 2, 6, 6, 6, 12, 6, 6, 12, 24, 18, 6, 6, 6, 6, 24, 24, 48, 6, 12, 6, 6, 6, 18, 24, 6, 6, 12, 24, 6, 12, 6, 6, 12, 30, 6, 6, 12, 6, 6, 24, 24, 12, 36, 6, 6, 12, 30, 6, 42, 24, 6, 18, 12, 42, 24, 30, 12, 18, 30, 18, 12, 6, 6, 24, 24, 12, 12, 30, 24, 36, 42, 18

Offset: 1

Michel Lagneau, Aug 04 2015

nonn

Sequence A067201 has the values of n. This sequence is the first differences of A067201.

a(n) is divisible by 6 for n>2.

			a(6)=12 because A067201(7) - A067201(6) = 33 - 21 = 12.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A056899 (primes of the form n^2+2), A067201 (values of n).

i0:=0:
for k from 1 to 1500 do:
   p:=k^2+2:
   if isprime(p) then printf(`%d, `,k-i0):i0:=k:
   else
   fi:
od:

Differences[Select[Range[1500], PrimeQ[2 + #^2] &, 100]]

first(m)=my(u=vector(m+1),v=vector(m),r=0);for(i=1,m+1,while(!isprime(r^2 + 2),r++);u[i]=r;r++);for(i=1,m,v[i]=u[i+1]-u[i]);v; \\ Anders Hellström, Aug 14 2015

A164521 Primes of the form A162142(k) - 2.

Original entry on oeis.org

3373, 753569, 2146687, 3048623, 6539201, 8120599, 10218311, 17373977, 18609623, 19034161, 32461757, 44738873, 59776469, 69426529, 72511711, 77854481, 88121123, 116930167, 133432829, 299418307, 338608871, 413493623, 458314009, 679151437

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 14 2009

nonn

Primes p such that p+2 is the cube of a squarefree semiprime, i.e., such that p+2 = q^3*r^3 where q and r are two distinct primes.

			3373 + 2 = 3375 = 3^3*5^3. 753569 + 1 = 753571 = 7^3*13^3.

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

Cf. A006881, A056899, A144953, A162142, A164517, A164518, A164519, A164520.

N:= 10^10: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor((N+2)^(1/3)/3))]):
R:= NULL:
for i from 1 to nops(P) do
    for j from 1 to i-1 do
      p:= (P[i]*P[j])^3-2;
      if p > N then break fi;
      if isprime(p) then R:= R, p fi
od od:
sort([R]); # Robert Israel, Jun 05 2018

f3[n_]:=FactorInteger[n][[1,2]]==3&&Length[FactorInteger[n]]==2&&FactorInteger[n][[2, 2]]==3; lst={};Do[p=Prime[n];If[f3[p+2],AppendTo[lst,p]],{n,4,4*9!}];  lst
csfsQ[n_]:=Module[{c=Surd[n+2,3]},SquareFreeQ[c]&&PrimeOmega[c]==2]; Select[Prime[Range[353*10^5]],csfsQ] (* Harvey P. Dale, Jan 07 2018 *)

Edited and examples corrected by R. J. Mathar, Aug 21 2009

A199854 Primes of the form 1 + m^2 + n^2 with gcd(m,n)=1.

Original entry on oeis.org

3, 11, 59, 83, 107, 131, 179, 227, 251, 347, 443, 467, 563, 587, 971, 1019, 1091, 1187, 1259, 1283, 1307, 1451, 1523, 1571, 1619, 1811, 1907, 1931, 2027, 2099, 2411, 2459, 2579, 2819, 2939, 2963, 3203, 3251, 3299, 3371, 3467, 3491, 3539, 3779, 3803, 3923, 3947

Offset: 1

Michel Marcus, Dec 22 2012

nonn

			First such decompositions are 3 = 1 + 1^2 + 1^2, 11 = 1 + 1^2 + 3^2, 59 = 1 + 3^2 + 7^2.
First instance of several decompositions for the same prime: 131 = 1 + 3^2 + 11^2 = 1 + 7^2 + 9^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.
J. Wu, Primes of the form 1 + m^2 + n^2 in short intervals, Proc. Amer. Math. Soc. 126 (1998), 1-8.

Cf. A056899 (when the decomposition has m=1).

filter:= proc(n) local S,x,y;
 if not isprime(n) then return false fi;
 S:= map(t -> subs(t,[x,y]),[isolve](x^2 + y^2 = n-1));
 ormap(t -> t[1] > 0 and t[2] >= t[1] and igcd(t[1],t[2])=1, S)
end proc:
select(filter, [seq(i,i=3..5000,2)]); # Robert Israel, Sep 30 2024

hasform(p) = {q = p - 1; for (k = 1, q/2, if (issquare(k) && issquare(q-k) && (gcd(k, q-k)==1), return(1));); return(0);}

cp's OEIS Frontend

A243449 Primes of the form n^2 + 14.

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Magma

Mathematica

A302434 Number of primes of the form b^2 + 2 for b <= 10^n.

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Comments

Examples

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PARI

Extensions

A386245 Composite numbers k such that A075255(k) is a square.

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Maple

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

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Maple

Mathematica

Formula

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

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Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

A164520 Primes p such that p-2 is the product of exactly 2 distinct cubes of primes.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A232011 Numbers n such that (3n)^2 + 2 is prime.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A260930 Differences between the numbers n such that n^2 + 2 is prime.

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