cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A001481 Numbers that are the sum of 2 squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.
Closed under multiplication. - David W. Wilson, Dec 20 2004
Also, numbers whose cubes are the sum of 2 squares. - Artur Jasinski, Nov 21 2006 (Cf. A125110.)
Terms are the squares of smallest radii of circles covering (on a square grid) a number of points equal to the terms of A057961. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 16 2007. [Comment corrected by T. D. Noe, Mar 28 2008]
Numbers with more 4k+1 divisors than 4k+3 divisors. If a(n) is a member of this sequence, then so too is any power of a(n). - Ant King, Oct 05 2010
A000161(a(n)) > 0; A070176(a(n)) = 0. - Reinhard Zumkeller, Feb 04 2012, Aug 16 2011
Numbers that are the norms of Gaussian integers. This sequence has unique factorization; the primitive elements are A055025. - Franklin T. Adams-Watters, Nov 25 2011
These are numbers n such that all of n's odd prime factors congruent to 3 modulo 4 occur to an even exponent (Fermat's two-squares theorem). - Jean-Christophe Hervé, May 01 2013
Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. The present sequence without 0 is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. Then A055025 (norms of Gaussian primes) gives the "prime divisors" of the square lattice. - Jean-Christophe Hervé, May 01 2013
For any i,j > 0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
The sequence is closed under multiplication. Primitive elements are in A055025. The sequence can be split into 3 multiplicatively closed subsequences: {0}, A004431 and A125853. - Jean-Christophe Hervé, Nov 17 2013
Generalizing Jasinski's comment, same as numbers whose odd powers are the sum of 2 squares, by Fermat's two-squares theorem. - Jonathan Sondow, Jan 24 2014
By the 4 squares theorem, every nonnegative integer can be expressed as the sum of two elements of this sequence. - Franklin T. Adams-Watters, Mar 28 2015
There are never more than 3 consecutive terms. Runs of 3 terms start at 0, 8, 16, 72, ... (A082982). - Ivan Neretin, Nov 09 2015
Conjecture: barring the 0+2, 0+4, 0+8, 0+16, ... sequence, the sum of 2 distinct terms in this sequence is never a power of 2. - J. Lowell, Jan 14 2022
All the areas of squares whose vertices have integer coordinates. - Neeme Vaino, Jun 14 2023
Numbers represented by the definite binary quadratic forms x^2 + 2nxy + (n^2+1)y^2 for any integer n. This sequence contains the even powers of any integer. An odd power of a number appears only if the number itself belongs to the sequence. The equation given in the comment by Boris Putievskiy 2013 is Brahmagupta's identity with n = 1. It proves that any set of numbers of the form a^2 + nb^2 is closed under multiplication. - Klaus Purath, Sep 06 2023

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
  • David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
  • L. Euler, (E388) Vollständige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 417.
  • S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
  • G. H. Hardy, Ramanujan, pp. 60-63.
  • P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Expos. Math. 17 (1999), pp. 289-312.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Disjoint union of A000290 and A000415.
Complement of A022544.
Cf. A004018, A000161, A002654, A064533, A055025, A002828, A000378, A025284-A025320, A125110, A118882, A125022.
A000404 gives another version. Subsequence of A091072, supersequence of A046711.
Cf. A057961, A232499, A077773, A363763.
Column k=2 of A336820.

Programs

  • Haskell
    a001481 n = a001481_list !! (n-1)
a001481_list = [x | x <- [0..], a000161 x > 0]
-- Reinhard Zumkeller, Feb 14 2012, Aug 16 2011
  • Magma
    [n: n in [0..160] | NormEquation(1, n) eq true]; // Arkadiusz Wesolowski, May 11 2016
  • Maple
    readlib(issqr): for n from 0 to 160 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi: od: od:
  • Mathematica
    upTo = 160; With[{max = Ceiling[Sqrt[upTo]]}, Select[Union[Total /@ (Tuples[Range[0, max], {2}]^2)], # <= upTo &]]  (* Harvey P. Dale, Apr 22 2011 *)
Select[Range[0, 160], SquaresR[2, #] != 0 &] (* Jean-François Alcover, Jan 04 2013 *)
  • PARI
    isA001481(n)=local(x,r);x=0;r=0;while(x<=sqrt(n) && r==0,if(issquare(n-x^2),r=1);x++);r \\ Michael B. Porter, Oct 31 2009
  • PARI
    is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,2]%2 && f[i,1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Aug 24 2012
  • PARI
    B=bnfinit('z^2+1,1);
is(n)=#bnfisintnorm(B,n) \\ Ralf Stephan, Oct 18 2013, edited by M. F. Hasler, Nov 21 2017
  • PARI
    list(lim)=my(v=List(),t); for(m=0,sqrtint(lim\=1), t=m^2; for(n=0, min(sqrtint(lim-t),m), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016
  • PARI
    is_A001481(n)=!for(i=2-bittest(n,0),#n=factor(n)~, bittest(n[1,i],1)&&bittest(n[2,i],0)&&return) \\ M. F. Hasler, Nov 20 2017
  • Python
    from itertools import count, islice
from sympy import factorint
def A001481_gen(): # generator of terms
    return filter(lambda n:(lambda m:all(d & 3 != 3 or m[d] & 1 == 0 for d in m))(factorint(n)),count(0))
A001481_list = list(islice(A001481_gen(),30)) # Chai Wah Wu, Jun 27 2022

Formula

n = square * 2^{0 or 1} * {product of distinct primes == 1 (mod 4)}.
The number of integers less than N that are sums of two squares is asymptotic to constant*N/sqrt(log(N)), hence lim_{n->infinity} a(n)/n = infinity.
Nonzero terms in expansion of Dirichlet series Product_p (1 - (Kronecker(m, p) + 1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -1.
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533. - Charles R Greathouse IV, Apr 16 2012
There are B(x) = x/sqrt(log x) * (K + B2/log x + O(1/log^2 x)) terms of this sequence up to x, where K = A064533 and B2 = A227158. - Charles R Greathouse IV, Nov 18 2022

Extensions

Deleted an incorrect comment. - N. J. A. Sloane, Oct 03 2023

A000161 Number of partitions of n into 2 squares.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.
Number of similar sublattices of square lattice with index n.
Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013
a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

Examples

			25 = 3^2+4^2 = 5^2, so a(25) = 2.

References

  • J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

Links

Crossrefs

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.
Cf. A000925, A247367.
Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

Programs

  • Haskell
    a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011
  • Maple
    A000161 := proc(n) local i,j,ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007
  • Mathematica
    Length[PowersRepresentations[ #,2,2]] &/@Range[0,150] (* Ant King, Oct 05 2010 *)
  • PARI
    a(n)=sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))) \\ for illustrative purpose
  • PARI
    A000161(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n),issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007
  • PARI
    A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018
  • Python
    from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

Formula

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007
Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

A025284 Numbers that are the sum of 2 nonzero squares in exactly 1 way.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 72, 73, 74, 80, 82, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 128, 136, 137, 146, 148, 149, 153, 157, 160, 162, 164, 169, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

A025426(a(n)) = 1. - Reinhard Zumkeller, Aug 16 2011

Links

Crossrefs

Cf. A000404, A025426, A081324, A125022, A018825, A001481, A007692, A243148.

Programs

  • Haskell
    import Data.List (elemIndices)
a025284 n = a025284_list !! (n-1)
a025284_list = elemIndices 1 a025426_list
-- Reinhard Zumkeller, Aug 16 2011
  • Mathematica
    selQ[n_] := Length[ Select[ PowersRepresentations[n, 2, 2], Times @@ # != 0 &]] == 1; Select[Range[300], selQ] (* Jean-François Alcover, Oct 03 2013 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 2], {n, 0, 300}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

Formula

A243148(a(n),2) = 1. - Alois P. Heinz, Feb 25 2019

A081324 Twice a square but not the sum of 2 distinct squares.

Original entry on oeis.org

0, 2, 8, 18, 32, 72, 98, 128, 162, 242, 288, 392, 512, 648, 722, 882, 968, 1058, 1152, 1458, 1568, 1922, 2048, 2178, 2592, 2888, 3528, 3698, 3872, 4232, 4418, 4608, 4802, 5832, 6272, 6498, 6962, 7688, 7938, 8192, 8712, 8978, 9522, 10082, 10368, 11552
Offset: 1

Views

Author

Benoit Cloitre, Apr 20 2003

Keywords

Comments

Conjecture: for n>1 this is A050804.
From Altug Alkan, Apr 12 2016: (Start)
Conjecture is true. Proof :
If n = a^2 + b^2, where a and b are nonzero integers, then n^3 = (a^2 + b^2)^3 = A^2 + B^2 = C^2 + D^2 where;
A = 2*a^2*b + (a^2-b^2)*b = 3*a^2*b - b^3,
B = 2*a*b^2 - (a^2-b^2)*a = 3*a*b^2 - a^3,
C = 2*a*b^2 + (a^2-b^2)*a = 1*a*b^2 + a^3,
D = 2*a^2*b - (a^2-b^2)*b = 1*a^2*b + b^3.
Obviously, A, B, C, D are always nonzero because a and b are nonzero integers. Additionally, if a^2 is not equal to b^2, then (A, B) and (C, D) are distinct pairs, that is, n^3 can be expressible as a sum of two nonzero squares more than one way. Since we know that n is a sum of two nonzero squares if and only if n^3 is a sum of two nonzero squares (see comment section of A000404); if n^3 is the sum of two nonzero squares in exactly one way, n must be a^2 + b^2 with a^2 = b^2 and n is the sum of two nonzero squares in exactly one way. That is the definition of this sequence, so this sequence is exactly A050804 except "0" that is the first term of this sequence. (End) [Edited by Altug Alkan, May 14 2016]
Conjecture: sequence consists of numbers of form 2*k^2 such that sigma(2*k^2)==3 (mod 4) and k is not divisible by 5.
The reason of related observation is that 5 is the least prime of the form 4*m+1. However, counterexamples can be produced. For example 57122 = 2*169^2 and sigma(57122) == 3 (mod 4) and it is not divisible by 5. - Altug Alkan, Jun 10 2016
For n > 0, this sequence lists numbers n such that n is the sum of two nonzero squares while n^2 is not. - Altug Alkan, Apr 11 2016
2*k^2 where k has no prime factor == 1 (mod 4). - Robert Israel, Jun 10 2016

Links

Crossrefs

Cf. A050804, A025284, A063725, A125022, A018825, A000404, A004431.

Programs

  • Haskell
    import Data.List (elemIndices)
a081324 n = a081324_list !! (n-1)
a081324_list = 0 : elemIndices 1 a063725_list
-- Reinhard Zumkeller, Aug 17 2011
  • Maple
    map(k -> 2*k^2, select(k -> andmap(t -> t[1] mod 4 <> 1, ifactors(k)[2]), [$0..100])); # Robert Israel, Jun 10 2016
  • Mathematica
    Select[ Range[0, 12000], MatchQ[ PowersRepresentations[#, 2, 2], {{n_, n_}}] &] (* Jean-François Alcover, Jun 18 2013 *)
  • PARI
    concat([0,2],apply(n->2*n^2, select(n->vecmin(factor(n)[, 1]%4)>1, vector(100,n,n+1)))) \\ Charles R Greathouse IV, Jun 18 2013

Formula

A063725(a(n)) = 1. [Reinhard Zumkeller, Aug 17 2011]
a(n) = 2*A004144(n-1)^2 for n > 1. - Charles R Greathouse IV, Jun 18 2013

Extensions

a(19)-a(45) from Donovan Johnson, Nov 15 2009
Offset corrected by Reinhard Zumkeller, Aug 17 2011

A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

Original entry on oeis.org

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269, 272
Offset: 1

Views

Author

Jean-Christophe Hervé, Nov 16 2013

Keywords

Comments

Numbers with exactly one prime factor of form 4*k+1, that must have multiplicity one, and no prime factor of the form 4*k+3 with odd multiplicity. There is thus no square in the sequence.
These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.
Numbers such that A004018(a(n)) = 8.
The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.
Also a subsequence of A191217: the two sequences are equal up to a(76) = 320, then A191217(77) = 325, the value which is missing from this sequence, as a(77) = 328 = A191217(78). (3125 is also missing from this sequence, although present in A191217, and it is the 31st such number). - Corrected by Antti Karttunen, May 14 2022.
Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - Altug Alkan, Jul 01 2016
Sequence A125022 (numbers with a unique partition as the sum of 2 squares x^2 + y^2), but without any terms of A028982 (squares and twice squares) that might occur there. - Antti Karttunen, May 14 2022

Examples

			a(1) = 5 = 4+1, a(2) = 10 = 9+1, a(3) =  13 = 9+4. However 2 = 1+1, 4 = 4+0, 8 = 4+4 are excluded because the unique decomposition of these numbers in two squares is not with two distinct nonzero squares; 25, 50, 100 are also excluded because there are two decompositions of these numbers in two squares (including one with equal or zero squares).

Links

Crossrefs

Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).
Subsequence of A004431, of A084645, of A125022, and of A191217.
Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).

Programs

  • PARI
    isok(n) = {f = factor(n); nb1 = 0; for (i=1, #f~, p = f[i, 1]; ep = f[i, 2]; if (p % 4 == 1, nb1 ++; if (ep != 1, return (0))); if (p % 4 == 3, if (ep % 2, return (0)));); return (nb1 == 1);} \\ Michel Marcus, Nov 17 2013

Formula

Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.
{k | A004018(k) = 8}.

A125021 Even numbers with a unique partition as the sum of 2 squares x^2 + y^2.

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 18, 20, 26, 32, 34, 36, 40, 52, 58, 64, 68, 72, 74, 80, 82, 90, 98, 104, 106, 116, 122, 128, 136, 144, 146, 148, 160, 162, 164, 178, 180, 194, 196, 202, 208, 212, 218, 226, 232, 234, 242, 244, 256, 272, 274, 288, 292, 296, 298, 306, 314
Offset: 1

Views

Author

Artur Jasinski, Nov 16 2006, corrected Nov 18 2006

Keywords

Links

Crossrefs

Cf. A002145, A124982, A125022.

Programs

  • Mathematica
    Select[2 * Range[0, 200], Length @ PowersRepresentations[#, 2, 2] == 1 &] (* Amiram Eldar, Mar 12 2020 *)

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Original entry on oeis.org

0, 1, 8, 10, 16, 41, 45, 53, 65, 81, 128, 130, 136, 146, 160, 178, 200, 226, 256, 313, 317, 325, 337, 353, 373, 397, 425, 457, 493, 533, 577, 625, 648, 650, 656, 666, 680, 698, 720, 746, 776, 810, 848, 890, 936, 986, 1040, 1098, 1160, 1201, 1205, 1213, 1225, 1226
Offset: 1

Views

Author

Felix Huber, Jan 25 2025

Keywords

Comments

Numbers k for which exists at least one solution to k = x^2 + (z^2 - x)^2 in integers x and z with x >= 0 and z >= sqrt(2*x).
Subsequence of A001481.

Examples

			10 is in the sequence because 10 = 1^2 + 3^2 and 1 + 3 = 2^2.
81 is in the sequence because 81 = 0^2 + 9^2 and 0 + 9 = 3^2.

Links

Crossrefs

Cf. A000161, A000290, A001481, A004018, A025284-A025320, A091072, A125110, A118882, A125022, A380072, A380073, A380074.

Programs

  • Maple
    # Calculates the first 10005 terms.
A379925:=proc(K)
    local i,j,L;
    L:={};
    for i from 0 to floor(sqrt((K+1)^2)/2) do
        for j from 0 to floor(sqrt((K+1)^2/2-i^2)) do
            if issqr(i+j) then
                L:=L union {i^2+j^2}
            fi
        od
    od;
    return op(L)
end proc;
A379925(1737);
  • PARI
    isok(n)=my(x=0, r=0); while(x<=sqrt(n) && r==0, if(issquare(n-x^2) && issquare(x+sqrtint(n-x^2)), r=1); x++); r; \\ Michel Marcus, Feb 10 2025

Formula

k = m^(4*j) is in the sequence for nonnegative integers m and j (not both 0) because x = 0 and z = m^j is a solution to m^(4*j) = x^2 + (z^2 - x)^2.

A277857 Numbers that are the sum of 2 squares with a unique partition and also the sum of 3 nonnegative cubes with a unique partition.

Original entry on oeis.org

1, 2, 8, 9, 10, 16, 17, 29, 36, 64, 72, 73, 80, 81, 128, 136, 153, 160, 197, 218, 232, 244, 277, 281, 288, 314, 349, 397, 405, 433, 466, 468, 512, 514, 521, 557, 576, 577, 584, 586, 593, 637, 640, 648, 701, 738, 757, 794, 801, 853, 857, 881, 882, 953, 980, 1024, 1028, 1088, 1152, 1217, 1224, 1249, 1268, 1280, 1332, 1341, 1396
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 02 2016

Keywords

Comments

Primes in this sequence are 2, 17, 29, 73, 197, 277, 281, 349, 397, 433, 521, 557, 577, 593, 701, 757, 853, 857, 881, 953, ... (subsequence of A002313).

Examples

			a(1) = 1 because 1 = 0^2 + 1^2 and 1 = 0^3 + 0^3 + 1^3;
a(2) = 2 because 2 = 1^2 + 1^2 and 2 = 0^3 + 1^3 + 1^3;
a(3) = 8 because 8 = 2^2 + 2^2 and 8 = 0^3 + 0^3 + 2^3;
a(4) = 9 because 9 = 0^2 + 3^2 and 9 = 0^3 + 1^3 + 2^3;
a(5) = 10 because 10 = 1^2 + 3^2 and 10 = 1^3 + 1^3 + 2^3, etc.

Links

Crossrefs

Cf. A001481, A002313, A003072, A004825, A125022.

Programs

  • Mathematica
    Select[Range[1400], Length[PowersRepresentations[#1, 2, 2]] == 1 && Length[PowersRepresentations[#1, 3, 3]] == 1 & ]
Showing 1-8 of 8 results.

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