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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A225892 Numbers of the form p^2 * (p^2 + 1) where p is in A225856.

Original entry on oeis.org

20, 90, 650, 14762, 28730, 83810, 130682, 280370, 708122, 924482, 1875530, 4881890, 7893290, 12120842, 13849562, 20155610, 25416722, 28403570, 38956322, 47465210, 62750162, 88538690, 104070602, 112561490, 141170042, 163060130, 260160770, 294517082, 352294130
Offset: 1

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Author

Rafael Parra Machio, May 20 2013

Keywords

Examples

			a(2) = 3^2(3^2+1) = 3^4+3^2 = 90.
a(5) = 13^2(13^2+1) = 13^4+13^2 = 28730.

Links

Crossrefs

Cf. A071253, A225856 (primes p such that p^2+1 is squarefree).

Programs

  • Mathematica
    p = Select[Prime[Range[60]], SquareFreeQ[#^2+1]&]; p^2 * (p^2+1)
#^2 (#^2+1)&/@Select[Prime[Range[50]],SquareFreeQ[#^2+1]&] (* Harvey P. Dale, Jun 17 2022 *)

Formula

a(n) = A071253(A225856(n)). - Amiram Eldar, Feb 23 2021

A308935 a(n) is the smallest m > n such that n^2*(n^2 + 1) divides m^2*(m^2 + 1).

Original entry on oeis.org

2, 8, 12, 64, 18, 216, 35, 112, 360, 818, 660, 348, 208, 2744, 693, 4096, 493, 450, 3420, 4832, 1071, 2112, 1242, 13824, 7800, 17576, 1998, 4368, 10133, 1560, 1178, 1280, 3597, 3060, 8582, 46656, 5032, 1292, 29640, 12768, 1189, 14868, 3182, 13112, 36468, 6670
Offset: 1

Views

Author

Rémy Sigrist, Jul 01 2019

Keywords

Comments

For any n > 0, a(n) exists as n^2*(n^2+1) divides (n^3)^2*((n^3)^2+1).
Tsz Ho Chan proved that a(n) >> n*log(n)^(1/8)/log(log(n))^12.

Examples

			For n = 2:
- A071253(3) mod A071253(2) = 10,
- A071253(4) mod A071253(2) = 12,
- A071253(5) mod A071253(2) = 10,
- A071253(6) mod A071253(2) = 12,
- A071253(7) mod A071253(2) = 10,
- A071253(8) mod A071253(2) = 0,
- hence a(2) = 8.

Links

Crossrefs

Cf. A065876, A071253.

Programs

  • Magma
    a:=[]; for n in [1..50] do m:=n+1; while not IsIntegral( (m^2*(m^2 + 1))/(n^2*(n^2 + 1) ))  do m:=m+1; end while; Append(~a,m); end for; a; // Marius A. Burtea, Dec 20 2019
  • Mathematica
    a[n_] := With[{n2 = n^2(n^2+1)}, For[m = n+1, True, m++, If[Divisible[ m^2(m^2+1), n2], Print[n, " ", m]; Return[m]]]];
a /@ Range[100] (* Jean-François Alcover, Dec 20 2019 *)
  • PARI
    a(n, f = x->x^2*(x^2+1)) = my (fn=f(n)); for (m=n+1, oo, if (f(m)%fn==0, return (m)))
  • Python
    def A308935(n):
    n2, m, m2 = n**2*(n**2+1), n+1, ((n+1)**2*((n+1)**2+1)) % (n**2*(n**2+1))
    while m2:
        m2, m = (m2 + 2*(2*m+1)*(m**2+m+1)) % n2, (m+1) % n2
    return m # Chai Wah Wu, Jul 01 2019

Formula

a(n) <= n^3.

A338485 Primitive numbers that are the sum of the squares of two of their distinct divisors.

Original entry on oeis.org

20, 90, 272, 468, 650, 1332, 2450, 2900, 3600, 4160, 6642, 7650, 10100, 10388, 14762, 16400, 20880, 25578, 27540, 28730, 38612, 42048, 50850, 50960, 54900, 65792, 83810, 90650, 98100, 116948, 125712, 130682, 141570, 142400, 149940, 160400, 194922, 206100, 214650
Offset: 1

Views

Author

Bernard Schott, Oct 30 2020

Keywords

Comments

If m is a term of A337988 then k^2*m is another term for any k in N*; so, there exist primitive terms m as 20, 90, 272,... in the sense that m' is not a term for any m' = m/k^2, k>1.

Examples

			20 = 2^2 + 4^2 and there is no k>1 such that 20/k^2 is another term, so 20 is in the sequence.
90 = 3^2 + 9^2 and there is no k>1 such that 90/k^2 is another term, so 90 is in the sequence.
468 = 12^2 + 18^2 and there is no k>1 such that 468/k^2 is another term, so 468 is in the sequence.

Links

Crossrefs

Subsequence of A337988.
A071253 is a subsequence.

Programs

  • Mathematica
    sumdivQ[n_] := AnyTrue[Most @ Divisors[n], (s = n - #^2) > 0 && s != n/2 && IntegerQ@Sqrt[s] && Divisible[n, Sqrt[s]] &]; s = Select[Range[200000], sumdivQ]; seq = {s[[1]]}; Do[If[! AnyTrue[s[[1 ;; k - 1]], IntegerQ@Sqrt[s[[k]]/#] &], AppendTo[seq, s[[k]]]], {k, 2, Length[s]}]; seq (* Amiram Eldar, Oct 31 2020 *)
  • PARI
    isok(m) = {my(d=divisors(m)); for (i=2, #d, for (j=1, i-1, if (d[i]^2+d[j]^2 == m, return (1)); ); ); } \\ A337988
isprim(x, vp) = {for (i=1, #vp, my(y = x/vp[i]); if ((denominator(y)==1) && issquare(y), return (0));); return(1);}
lista(nn) = {my(vp = []); for (n=1, nn, if (isok(n) && isprim(n, vp), vp = concat(vp, n));); vp;} \\ Michel Marcus, Oct 30 2020

Extensions

More terms from Michel Marcus, Oct 30 2020

A358317 Ordered squares of the chord lengths of the parabola y=x^2, where the chord ends are all possible points of the parabola with integer coordinates.

Original entry on oeis.org

0, 2, 4, 10, 16, 18, 20, 26, 36, 50, 64, 68, 80, 82, 90, 98, 100, 122, 144, 148, 162, 170, 180, 196, 226, 234, 242, 250, 256, 260, 272, 290, 320, 324, 338, 362, 400, 404, 442, 450, 484, 490, 500, 530, 576, 578, 580, 592, 612, 626, 650, 676, 720, 722, 730, 738, 784, 788, 810, 842, 882, 900, 962, 980
Offset: 1

Views

Author

Nicolay Avilov, Nov 09 2022

Keywords

Comments

Numbers of the form (x^2 - z^2)^2 + (x-z)^2 for integers x and z, so that terms are sums of 2 squares (subset of A001481).
Numbers of the form m^2*(k^2 + 1) for integers m and k of the same parity.
Chords starting at the origin (z=0, or m=k) are terms A071253(x).

Examples

			0 is a term since it is the square of the chord length from (0,0) to (0,0).
10 = 1^2 + 3^2 is a term since it is the square of the chord length from (1,1) to (2,4).

Links

Crossrefs

Cf. A001481, A071253.

Programs

  • Python
    # Program from Oleg Sorokin
from math import isqrt
limit = 2000
s = set()
end = isqrt(limit)
for m in range(0, end+1):
    for k in range(m%2, end+1, 2):
        c = m**2*(k**2+1)
        if c > limit:
            break
        s.add(c)
print(sorted(s))
  • Python
    from itertools import count, islice
from sympy import divisors, integer_nthroot
def A358317_gen(startvalue=0): # generator of terms >= startvalue
    for n in count(max(startvalue,0)):
        if n == 0:
            yield 0
        else:
            for d in divisors(n,generator=True):
                a, b = integer_nthroot(d,2)
                if b:
                    c, e = integer_nthroot(n//d-1,2)
                    if e and not (c^a)&1:
                        yield n
                        break
A358317_list = list(islice(A358317_gen(),30)) # Chai Wah Wu, Nov 24 2022
Previous Showing 21-24 of 24 results.

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