A027863 -id:A027863 - cp's OEIS frontend

A027862 Primes of the form j^2 + (j+1)^2.

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681

Offset: 1

Patrick De Geest

Also, primes of the form 4*k+1 which are the hypotenuse of one and only one right triangle with integral legs. - Cino Hilliard, Mar 16 2003
Centered square primes (i.e., prime terms of centered squares A001844). - Lekraj Beedassy, Jan 21 2005
Primes of the form 2*k*(k-1)+1. - Juri-Stepan Gerasimov, Apr 27 2010
Equivalently, primes of the form (m^2+1)/2 (take m=2*j+1). These primes a(n) have nontrivial solutions of x^2 == 1 (Modd a(n)) given by x=x(n)=A002731(n). For Modd n see a comment on A203571. See also A206549 for such solutions for primes of the form 4*k+1, given in A002144.
  E.g., a(3)=41, A002731(3)=9, 9^2=81, floor(81/41)=1 (odd),
  -81 = -2*41 + 1 == 1 (mod 2*41), hence 9^2 == 1 (Modd 41). - Wolfdieter Lang, Feb 24 2012
Also primes of the form 4*k+1 that are the smallest side length of one and only one integer Soddyian triangle (see A230812). - Frank M Jackson, Mar 13 2014
Also, primes of the form (m^2+1)/2. - Zak Seidov, May 01 2014
Note that ((2n+1)^2 + 1)/2 = n^2 + (n+1)^2. - Thomas Ordowski, May 25 2015
Primes p such that 2p-1 is a square. - Thomas Ordowski, Aug 27 2016
Primes in the main diagonal of A000027 when represented as an array read by antidiagonals. - Clark Kimberling, Mar 12 2023
The diophantine equation x^2 + ... + (x + r)^2 = p may be rewritten to A*x^2 + B*x + C = p, where A = (r + 1), B = r*(r + 1), C = r*(r + 1)*(2*r + 1)/6. If gcd(A, B, C) > 1 no solution for a prime p exists. The gcd(A, B, C) = 1 holds only for r = 1, 2, 5 (gcd is the greatest common divisor). For r = 1 we have x^2 + (x + 1)^2 = p, thus for x from A027861 we calculate primes p from A027862. For r = 2 we have x^2 + (x + 1)^2 + (x + 2)^2 = p, thus for x from A027863 we calculate primes p from A027864. For r = 5 we have x^2 + ... + (x + 5)^2 = p, thus for x from A027866 we calculate primes p from A027867. - Ctibor O. Zizka, Oct 04 2023

			13 is in the sequence because it is prime and 13 = 2^2 + 3^2. - _Michael B. Porter_, Aug 27 2016

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 271.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972. pp. 275.

T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers
Daniel Shanks, An analytic criterion for the existence of infinitely many primes of the form 1/2 * (n^2 + 1), Illinois Journal of Mathematics 8:3 (1964), p. 377-379.
Wacław Sierpiński, Sur les nombres triangulaires qui sont sommes de deux nombres triangulaires, Elem. Math., 17 (1962), pp. 63-65.
Panayiotis G. Tsangaris, A sieve for all primes of the form x^2 + (x+1)^2, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25 (1998), pp. 39-53.

Primes p such that A079887(p) = 1.
Cf. A002731 (m values), A027861 (j values), A091277 (prime indices).
Subsequence of A002144 (p=4k+1).
Cf. A001844 (centered squares), A027863, A027864, A027866, A027867, A203571, A206549, A230812.

[ a: n in [0..150] | IsPrime(a) where a is n^2+(n+1)^2 ]; // Vincenzo Librandi, Dec 18 2010

Select[Table[n^2+(n+1)^2,{n,200}],PrimeQ] (* Harvey P. Dale, Aug 22 2012 *)
Select[Total/@Partition[Range[200]^2,2,1],PrimeQ] (* Harvey P. Dale, Apr 20 2016 *)

je=[]; for(n=1,500, if(isprime(n^2+(n+1)^2),je=concat(je,n^2+(n+1)^2))); je

fermat(n) = { for(x=1,n, y=2*x*(x+1)+1; if(isprime(y),print1(y" ")) ) }

a(n) = ((A002731(n)^2 - 1)/2) + 1. - Torlach Rush, Mar 14 2014

a(n) = (A002731(n)^2 + 1)/2. - Zak Seidov, May 01 2014

More terms from Cino Hilliard, Mar 16 2003

A218209 Number of n-digit primes that are of the form (k-2)^2 + (k-1)^2 + k^2.

Original entry on oeis.org

2, 1, 3, 4, 12, 31, 86, 230, 681, 1934, 5634, 15772

Offset: 1

Martin Renner, Oct 23 2012

Cf. A027863, A027864, A218207, A218210, A218211.

n = -1; Table[cnt = 0; While[n++; p = 3*n^2 + 2; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)
Table[Count[Table[Total[Range[n,n+2]^2],{n,-1,600000}],?(PrimeQ[#] && IntegerLength[#]==k&)],{k,12}] (* _Harvey P. Dale, Oct 17 2016 *)

a(n) = A218210(n) - A218210(n-1)

A218210 Number of primes up to 10^n that are of the form (k-2)^2 + (k-1)^2 + k^2.

Original entry on oeis.org

2, 3, 6, 10, 22, 53, 139, 369, 1050, 2984, 8618, 24390

Offset: 1

Martin Renner, Oct 23 2012

There are two primes < 10: 2 and 5.

Cf. A027863, A027864, A218208, A218209, A218212.

n = -1; cnt = 0; Do[While[n++; p = 3*n^2 + 2; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)

a(n) = sum(A218209(k), k=1..n)

A218213 Number of n-digit primes representable as sums of consecutive squares.

Original entry on oeis.org

1, 4, 13, 30, 69, 187, 519, 1401, 3889, 10861, 31640, 90735

Offset: 1

Martin Renner, Oct 23 2012

There are no common representations of two, three or six squares for n < 13, so

a(n) = A218207(n) + A218209(n) + A218211(n); n < 13.

Cf. A027861, A027862, A027863, A027864, A027866, A027867, A218207, A218209, A218211.

nn = 8; nMax = 10^nn; t = Table[0, {nn}]; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, If[PrimeQ[s], t[[Ceiling[Log[10, s]]]]++];  k++], {n, Sqrt[nMax]}]; t (* T. D. Noe, Oct 23 2012 *)

a(n) = A218214(n) - A218213(n-1).

A218214 Number of primes up to 10^n representable as sums of consecutive squares.

Original entry on oeis.org

1, 5, 18, 48, 117, 304, 823, 2224, 6113, 16974, 48614, 139349

Offset: 1

Martin Renner, Oct 23 2012

There are no common representations of two, three or six squares for n < 13, so

a(n) = A218208(n) + A218210(n) + A218212(n); n < 13.

			a(1) = 1 because only one prime less than 10 can be represented as a sum of consecutive squares, namely 5 = 1^2 + 2^2.
a(2) = 5 because there are five primes less than 100 representable as a sum of consecutive squares: the aforementioned 5, as well as 13 = 2^2 + 3^2, 29 = 2^2 + 3^2 + 4^2, 41 = 4^2 + 5^2 and 61 = 5^2 + 6^2.

Cf. A027861, A027862, A027863, A027864, A027866, A027867, A163251, A174069, A218208, A218210, A218212, A218213.

nn = 8; nMax = 10^nn; t = Table[0, {nn}]; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, If[PrimeQ[s], t[[Ceiling[Log[10, s]]]]++]; k++], {n, Sqrt[nMax]}]; Accumulate[t] (* T. D. Noe, Oct 23 2012 *)

a(n) = sum(A218213(k),k=1..n)

A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.

Original entry on oeis.org

2, 12, 14, 24, 34, 122, 154, 164, 272, 342, 464, 612, 674, 734, 784, 794, 854, 1174, 1262, 1274, 1364, 1392, 1524, 1554, 1664, 1682, 1844, 1854, 1862, 1892, 1924, 1942, 1994, 2232, 2294, 2354, 2442, 2592, 2802, 2884, 3124, 3164, 3292, 3394, 3544, 3594, 3632, 3724, 3892, 3904, 3922

Offset: 1

Chris Fry, Nov 29 2013

See A027862 for primes of the form x^2+(x+1)^2 = 2x^2+2x+1.
See A027864 for primes of the form x^2+(x+1)^2+(x+2)^2 = 3x^2+6x+5.
It is an open question whether either of these polynomials produces an infinite number of primes.  This sequence lists the values of x that produce a prime in both polynomials. x must be congruent to 0 or 2 (mod 4) and all the generated primes are of the form 4k+1.

			When x=14, 2x^2+2x+1=421 and 3x^2+6x+5=677. 14 is the third value of x for which both these polynomials produce a prime number, so a(3)=14.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 266.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Wikipedia, Hardy and Littlewood's Conjecture F.

Cf. A027862, A027864. Equals n common to A027861 and A027863.

lst = {}; Do[If[And[PrimeQ[n^2 + (n + 1)^2], PrimeQ[n^2 + (n + 1)^2 + (n + 2)^2]], Print[n]; AppendTo[lst, n]], {n, 10000}]
Select[Range[2,4000,2],AllTrue[{(#^2+(#+1)^2),(#^2+(#+1)^2+(#+2)^2)},PrimeQ]&] (* Harvey P. Dale, Jul 30 2023 *)

A242046 Least integer k > n + 1 such that n^2 + (n + 1)^2 + k^2 is prime.

Original entry on oeis.org

2, 6, 4, 6, 24, 14, 8, 12, 16, 24, 24, 14, 14, 24, 16, 20, 42, 20, 26, 54, 30, 26, 30, 28, 26, 54, 42, 38, 42, 34, 40, 48, 38, 36, 36, 44, 48, 102, 42, 46, 54, 44, 50, 48, 60, 54, 66, 50, 54, 54, 54, 54, 54, 56, 64, 84, 58, 62, 84, 64, 66, 78, 64, 66, 84, 74

Offset: 0

Zak Seidov, Aug 24 2014

nonn

If n is in A027863 then a(n) = n + 2, otherwise a(n) > n + 2. All terms are even. Corresponding primes are 5, 41, 29, 61, 617, ...

Jens Kruse Andersen, Table of n, a(n) for n = 0..10000

Cf. A027863.

lk[n_]:=Module[{k=n+2,c=n^2+(n+1)^2},While[!PrimeQ[c+k^2],k++];k]; Array[ lk,70,0] (* Harvey P. Dale, Aug 06 2015 *)

a(n)=k=n+2; while(!isprime(n^2+(n+1)^2+k^2), k++); k
vector(100,n,a(n-1)) \\ Jens Kruse Andersen, Aug 26 2014

Corrected and extended by Jens Kruse Andersen, Aug 26 2014

cp's OEIS Frontend

A027862 Primes of the form j^2 + (j+1)^2.

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A218209 Number of n-digit primes that are of the form (k-2)^2 + (k-1)^2 + k^2.

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A218210 Number of primes up to 10^n that are of the form (k-2)^2 + (k-1)^2 + k^2.

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A218213 Number of n-digit primes representable as sums of consecutive squares.

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A218214 Number of primes up to 10^n representable as sums of consecutive squares.

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A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.

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A242046 Least integer k > n + 1 such that n^2 + (n + 1)^2 + k^2 is prime.

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