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.

A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

Original entry on oeis.org

9, 16, 128, 243, 576, 1024, 6561, 8192, 9604, 11664, 28224, 36864, 51984, 65536, 97344, 140625, 177147, 250000, 275625, 345744, 419904, 450241, 524288, 614656, 717409, 746496, 1028196, 1058841, 1399489, 1500625, 1590121, 1750329, 1806336, 1882384, 2359296, 3326976, 4194304
Offset: 1

Views

Author

Altug Alkan, Jan 10 2016

Keywords

Comments

Intersection of A001597 and A003325.
Motivation for this sequence is the equation m^k = x^3 + y^3 where x,y,m > 0 and k >= 2.
Obviously, because of Fermat's Last Theorem, a(n) cannot be a cube.
A050802 is a subsequence.
Obviously, this sequence contains all numbers of the form 2^(3*n+1) and 3^(3*n-1), for n > 0.

Examples

			9 is a term because 9 = 3^2 = 1^3 + 2^3.
16 is a term because 16 = 2^4 = 2^3 + 2^3.
243 is a term because 243 = 3^5 = 3^3 + 6^3.

Links

Crossrefs

Cf. A001597, A003325, A050802, A085332, A013730, A013733, A266927.

Programs

  • PARI
    T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=2, 1e7, if(ispower(n) && is(n), print1(n, ", ")))

A273279 Least perfect power that is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

8, 125, 3125, 4225, 1953125, 48828125, 105625, 274625, 762939453125, 2640625, 476837158203125, 17850625, 1221025, 34328125, 186264514923095703125, 1650390625, 446265625, 1160290625, 41259765625, 4291015625, 45474735088646411895751953125, 30525625
Offset: 1

Views

Author

Altug Alkan, May 19 2016

Keywords

Comments

Least m^k that is the sum of two nonzero squares in exactly n ways where m > 0 and k >= 2.
Terms of this sequence are 2^3, 5^3, 5^5, 65^2, 5^10, 5^11, 325^2, 65^3, ...
Prime powers that are listed in this sequence are 2^3, 5^3, 5^5, 5^10, 5^11, ...

Examples

			8 is a term because 8 = 2^3 = 2^2 + 2^2.
125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.

Links

Crossrefs

Cf. A001597, A006339, A016032, A025426, A266927, A273238.

Programs

  • Mathematica
    p = Select[Prime@ Range@ 90, Mod[#, 4] == 1 &]; f[w_] := Times @@ (Take[p, Length@w]^Reverse[w]); c[w_] := Floor[(1/2) Times @@ (w+1)];r[w_] := Block[{v, k = If[Length@w == 1, 1,2]}, While[(v = cn[k w]) < trg, k++]; If[v == trg, b = Min[b, f[k*w]]]; If[cn[w] < trg, r[Append[w, 1]]; v=w; v[[-1]]++; r[v]]]; a[1]=8; a[n_] := (b=Infinity; trg = n; r[{2}]; r[{1, 1}]; b); Array[a, 50] (* Giovanni Resta, May 19 2016 *)

Extensions

a(9)-a(22) from Giovanni Resta, May 19 2016

A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.

Original entry on oeis.org

49, 81, 121, 125, 169, 196, 216, 225, 243, 289, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2401, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125, 3136
Offset: 1

Views

Author

Altug Alkan, Jan 23 2016

Keywords

Comments

Intersection of A001597 and A004432.
Note that this sequence is not the complement of A267321. This sequence is a subsequence for complement of A267321.
Sequence focuses on the equation m^k = x^2 + y^2 + z^2 where x > y > z > 0 and m > 0, k >= 2.
Corresponding exponents are 2, 4, 2, 3, 2, 2, 3, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 4, 2, 2, ...

Examples

			49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.
81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.
121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.

Crossrefs

Cf. A001597, A004432, A266927, A267321.

Programs

  • Mathematica
    fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1800, fQ@ # && Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Jan 24 2016, after Ant King at A001597 *)
  • PARI
    isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));

A270739 Prime powers (p^k, p prime, k > 1) of the form x^2 + y^2 where x and y are nonzero integers.

Original entry on oeis.org

8, 25, 32, 125, 128, 169, 289, 512, 625, 841, 1369, 1681, 2048, 2197, 2809, 3125, 3721, 4913, 5329, 7921, 8192, 9409, 10201, 11881, 12769, 15625, 18769, 22201, 24389, 24649, 28561, 29929, 32761, 32768, 37249, 38809, 50653, 52441, 54289, 58081, 66049, 68921, 72361, 76729, 78125, 78961, 83521
Offset: 1

Views

Author

Altug Alkan, Mar 22 2016

Keywords

Comments

Subsequence of A266927.
Among the Gaussian integers, these numbers have two distinct prime factors, and four or more prime factors when counted with multiplicity. - Alonso del Arte, Mar 22 2016

Examples

			125 is a term because 125 = 5^3 = 5^2 + 10^2.
169 is a term because 169 = 13^2 = 5^2 + 12^2.
512 is a term because 512 = 2^9 = 16^2 + 16^2.

Crossrefs

Cf. A000404, A000961, A266927.

Programs

  • PARI
    isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
forcomposite(n=4, 1e5, if(isprimepower(n) && isA000404(n), print1(n, ", ")));
Showing 1-4 of 4 results.

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