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.

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A245440 Primes p == 1 (mod 4) such that p - floor(sqrt(p))^2 and 2p - floor(sqrt(2p))^2 are not squares.

Original entry on oeis.org

353, 373, 449, 461, 521, 541, 593, 653, 673, 757, 769, 797, 821, 829, 941, 953, 1009, 1021, 1061, 1069, 1097, 1193, 1217, 1237, 1249, 1277, 1361, 1381, 1481, 1489, 1549, 1597, 1613, 1621, 1657, 1669, 1693, 1709, 1721, 1733, 1777, 1801, 1877, 1889, 1933, 1949
Offset: 1

Views

Author

Thomas Ordowski, Jul 22 2014

Keywords

Comments

Primes p of the form 4k+1 such that A053610(p) > 2 and A053610(2p) > 2.
Note that p = a^2 + b^2 and 2p = (a+b)^2 + (a-b)^2 is the only way. So according to the definition the greedy algorithm cannot give such the sums of two squares.
Interesting fact: a(n) = A145023(n) for all n < 25. Of course A145023 is a subsequence.
Primes p == 1 (mod 4) such that A245474(p) > 2.

Links

Crossrefs

Cf. A002144, A053610, A145016, A145023, A174806, A245474.

Programs

  • Magma
    [p: p in PrimesUpTo(10000) | p mod 4 eq 1 and not IsSquare(p-Floor(Sqrt(p))^2) and not IsSquare(2*p-Floor(Sqrt(2*p))^2)]; // Vincenzo Librandi, Sep 19 2017
  • Mathematica
    a245440Q[n_Integer] := If[
  And[PrimeQ[n] == True, Mod[n, 4] == 1],
  If[Or[IntegerQ[Sqrt[n - Floor[Sqrt[n]]^2]] == True,
    IntegerQ[Sqrt[2*n - Floor[Sqrt[2*n]]^2]] == True], False, True],
  False]; a245440[n_Integer] :=
Flatten[Position[Thread[a245440Q[Range[n]]],
   True]]; a245440[300000]; (* Michael De Vlieger, Aug 05 2014 *)
  • PARI
    s=[]; forprime(p=2, 3000, if(p%4==1 && !issquare(p-floor(sqrt(p))^2) && !issquare(2*p-floor(sqrt(2*p))^2), s=concat(s, p))); s \\ Colin Barker, Jul 22 2014

Extensions

More terms from Colin Barker, Jul 22 2014

A249298 Smallest positive integer k, such that s-k*n is a square where s is the smallest square >= k*n.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 5, 1, 5, 6, 1, 1, 9, 2, 9, 2, 1, 12, 13, 1, 1, 14, 1, 3, 17, 2, 19, 1, 3, 20, 1, 1, 23, 24, 3, 1, 25, 2, 27, 6, 1, 30, 31, 1, 1, 2, 3, 7, 35, 4, 1, 2, 3, 40, 41, 1, 41, 42, 1, 1, 1, 4, 47, 10, 5, 2, 51, 1, 51, 52, 3, 12, 1, 6, 57, 1, 1, 60, 61, 1, 3, 62, 7, 3, 65, 2
Offset: 1

Views

Author

Valtteri Raiko, Oct 24 2014

Keywords

Comments

For any n>=3, there exists at least one positive integer k, 1 <= k <= n-1 such that the difference between the smallest square >= k*n and k*n is a square. To prove this, consider the multiplier k = n-2. Then (n-2)*n = (n-1)^2-1, thus the difference from the next square is 1, which is a square. If n = 1, k = 1 and if n = 2, k = 2.
Smallest positive integer k such that ceiling(sqrt(k*n))^2-k*n is a square.

Examples

			a(10) = 4, for ceiling(sqrt(10))^2-10 = 6, ceiling(sqrt(2*10))^2-2*10 = 5, ceiling(sqrt(3*10))^2-3*10 = 6 and ceiling(sqrt(4*10))^2-4*10 = 9 = 3^2.

Crossrefs

Cf. A000290, A145236 (equals a(A000040)), A068527 (difference for k=1).
Cf. A145016, A145022, A145023, A145047, A145048, A145049, A145050, A145215.

Programs

  • Mathematica
    dif[n_] := Ceiling[Sqrt[n]]^2 - n;a[k_] := Module[{n = 1}, While[dif[dif[n*k]] != 0, n++]; Return[n]];Table[a[k], {k, 1, 90}]
  • PARI
    a(n) = {k=1; while(!issquare(ceil(sqrt(k*n))^2-k*n), k++); k;} \\ Michel Marcus, Oct 24 2014

A145230 Numbers of different values of the minimal factors s for primes of the form 4k+1 not exceeding 10^n (see A145215).

Original entry on oeis.org

1, 2, 3, 7, 17, 38
Offset: 1

Views

Author

Vladimir Shevelev, Oct 05 2008

Keywords

Crossrefs

A145016 A145022 A145023 A145047 A145048 A145049 A145050 A145215
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