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 1-4 of 4 results.

A073412 Lesser of three consecutive nonsquare integers each of which is the sum of two squares.

Original entry on oeis.org

72, 232, 520, 584, 800, 808, 1096, 1152, 1312, 1664, 1744, 1800, 1872, 1960, 2248, 2312, 2384, 2592, 2824, 3328, 3392, 3528, 4112, 4176, 4328, 5120, 5408, 5904, 6056, 6120, 6272, 6352, 6408, 6568, 6920, 8080, 8144, 8296, 8352, 8584, 8712, 9160, 9376
Offset: 1

Views

Author

Jason Earls, Aug 23 2002

Keywords

Comments

a(n) == 0 mod 8. - Zak Seidov, Jan 26 2013
Is this sequence the same as A064715? - Zak Seidov, Jan 26 2013
This sequence is distinct from A064715 since it allows numbers equal to twice a square, like 72, 1152, 2592, 3528, etc. - Giovanni Resta, Jan 29 2013
This sequence lists lesser of three consecutive nonsquare integers each of which is the sum of two squares. So this sequence is a subsequence of A064716. - Altug Alkan, Jul 07 2016

Examples

			232 is here since 232 = 6^2 + 14^2; 233 = 8^2 + 13^2; 234 = 3^2 + 15^2 and 232, 233, 234 are all nonsquares.
288 is not a term because 288 = 12^2 + 12^2, 289 = 8^2 + 15^2, 290 = 1^2 + 17^2 but 289 is also square.

Links

Crossrefs

Cf. A000404,A000415, A001481, A064716.

Programs

  • Maple
    is415:= proc(n) local F;
  if issqr(n) then return false fi;
  F:= select(t -> t[1] mod 4 = 3, ifactors(n)[2]);
  andmap(t -> t[2]::even, F);
end proc:
Q:= select(is415, [seq(seq(8*i+j,j=0..2),i=1..2000)]):
Q[select(t -> Q[t+2]-Q[t]=2, [$1..nops(Q)-2])]; # Robert Israel, Mar 05 2018
  • Mathematica
    nsQ[x_] := !IntegerQ[Sqrt[x]];
prQ[x_] := With[{pr = PowersRepresentations[x, 2, 2]}, pr != {} && AllTrue[pr[[1]], IntegerQ]];
selQ[x_] := nsQ[x] && nsQ[x+1] && nsQ[x+2] && prQ[x] && prQ[x+1] && prQ[x+2];
Select[8 Range[10000], selQ] (* Jean-François Alcover, Jun 11 2020 *)
  • PARI
    isA001481(n) = my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
isok(n) = isA001481(n) && isA001481(n+1) && isA001481(n+2) && !issquare(n) && !issquare(n+1);
lista(nn) = for(n=1, nn, if(isok(8*n), print1(8*n, ", "))); \\ Altug Alkan, Jul 07 2016

Extensions

Edited by Robert Israel, Mar 05 2018

A257118 Smallest of three consecutive prime numbers each of which is the sum of two squares.

Original entry on oeis.org

89, 389, 397, 449, 661, 757, 761, 929, 997, 1193, 1201, 1669, 2213, 2269, 2293, 2593, 2609, 2617, 2741, 3037, 3041, 3209, 3217, 3413, 3433, 3449, 3697, 3877, 4397, 4801, 5189, 5233, 5237, 5569, 5689, 5717, 6101, 6217, 6389, 6469, 6733, 6829, 6833, 6997, 7529
Offset: 1

Views

Author

Abhiram R Devesh, Apr 25 2015

Keywords

Comments

This sequence is a subsequence of A257117.

Examples

			389 = 10^2 + 17^2, 397 = 6^2 + 19^2, and 401 = 1^2 + 20^2, so 389 is a term.
397 = 6^2 + 19^2, 401 = 1^2 + 20^2, and 409 = 3^2 + 20^2, so 397 is a term.

Links

Crossrefs

Cf. A064716 (Smallest member of three consecutive numbers).
Cf. A257117 (Smallest member of two consecutive prime numbers).

Programs

Extensions

Corrected and extended by and prior b-file replaced by Harvey P. Dale, Jul 08 2018

A271536 Integers n such that n^2-1 and n^2 are the sum of two nonzero squares.

Original entry on oeis.org

17, 35, 51, 73, 105, 145, 195, 233, 273, 289, 291, 339, 451, 465, 521, 577, 579, 585, 611, 675, 723, 777, 801, 809, 819, 899, 915, 969, 1043, 1059, 1097, 1153, 1155, 1185, 1225, 1281, 1313, 1347, 1353, 1395, 1545, 1601, 1603, 1665, 1683, 1731, 1745, 1763, 1801, 1873, 1923, 1961, 1971, 2019
Offset: 1

Views

Author

Altug Alkan, Apr 09 2016

Keywords

Comments

Corresponding n^2 values are 289, 1225, 2601, 5329, 11025, 21025, 38025, 54289, 74529, 83521, 84681, ...
If n is in this sequence, so is n^2. Proof:
n is a term of this sequence if and only if n^2 = a^2 + b^2 and n^2-1 = c^2 + d^2 for a,b,c,d are nonzero integers.
If n^2 = a^2 + b^2, then n^4 = (n^2)*(n^2) = (n^2)*(a^2 + b^2) = (n*a)^2 + (n*b)^2.
If n^2-1 = c^2 + d^2, then n^4-1 = (n^2-1)*(n^2+1) = (c^2 + d^2)*(n^2+1) = (c*n + d)^2 + (d*n - c)^2. Note that (d*n - c) > 0 because of definition of n.
Since both (n^2)^2 and (n^2)^2-1 are the sum of two nonzero squares, n^2 must be a term.
With repeating of same procedure, it can be seen that if n is in this sequence, so is n^(2^k), for k >= 0.

Examples

			17 is a term because 17^2 = 8^2 + 15^2 and 17^2 - 1 = 12^2 + 12^2.

Crossrefs

Cf. A000404, A064716.

Programs

A274591 Least number k such that k-1, k, k+1 are the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

73, 22049, 90707201, 4359889
Offset: 1

Views

Author

Altug Alkan, Jun 29 2016

Keywords

Comments

Corresponding triples are (72, 73, 74), (22048, 22049, 22050), (90707200, 90707201, 90707202), (4359888, 4359889, 4359890) for first four terms.
a(6) = 1428907401, a(8) = 2305281745. No more terms < 2*10^11. - Lars Blomberg, Jun 01 2018
a(5) > 10^15, if it exists. - Giovanni Resta, Jun 05 2018

Examples

			a(2) = 22049 because 22048 = 12^2 + 148^2 = 68^2 + 132^2, 22049 = 32^2 + 145^2 = 40^2 + 143^2, 22050 = 21^2 + 147^2 = 105^2 + 105^2.

Crossrefs

Cf. A064716, A274548.
Showing 1-4 of 4 results.

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