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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A304436 Numbers n such that n^6 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 33, 34, 35, 36, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 56, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 81, 82, 85, 87, 89, 90, 91, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 122, 123, 125, 126, 130, 135, 136, 137, 140, 143, 145, 146, 148, 149, 150
Offset: 1

Views

Author

M. F. Hasler, May 25 2018

Keywords

Comments

Motivated by the search for solutions to a^n + b^(2n+2)/4 = (perfect square), which arises when searching for solutions to x^n + y^(n+1) = z^(n+2) of the form x = a*z, y = b*z. It turns out that many solutions are of the form a^n = d*(b^(n+1) + d), where d is a perfect power.

Examples

			5^6 = 35^2 + 120^2, 10^6 = 280^2 + 960^2, ...

Crossrefs

Cf. A304433, A304434, A304435, A001597 (perfect powers).

Programs

  • Maple
    LIM:=200^6: P:={seq(seq(x^k, k=3..floor(log[x](LIM))), x=2..floor(LIM^(1/3)))}:
is_A304436:= proc(n) local N, S;  N:= n^6;  if remove(t -> subs(t, x)<=1 or subs(t, y)<=1 or subs(t, x-y)=0, [isolve(x^2+y^2=N)]) <> [] then return true fi;  S:= map(t ->N-t, P minus {N, N/2});  (S intersect P <> {}) or (select(issqr, S) <> {})
end proc: # adapted from code by Robert Israel for A304434
  • Mathematica
    LIM = 200^6;
P = Union@ Flatten@ Table[Table[x^k, {k, 3, Floor[Log[x, LIM]]}], {x, 2, Floor[LIM^(1/3)]}];
filterQ[n_] := Module[{M = n^6, S}, If[Solve[x > 1 && y > 1 && x^2 + y^2 == M, {x, y}, Integers] != {}, Return [True]]; S = M - (P ~Complement~ {M, M/2}); S ~Intersection~ P != {} || Select[S, IntegerQ[Sqrt[#]]&] != {}];
Reap[For[n = 1, n <= 150, n++, If[filterQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Aug 12 2020, after Maple *)
  • PARI
    L=200^6;P=List(); for(x=2,sqrtnint(L,3),for(k=3,logint(L,x),listput(P,x^k)));#P=Set(P) \\ This P = A076467 \ {1} = A111231 \ {0} up to limit L.
is(n,e=6)={for(i=1,#s=sum2sqr(n=n^e),vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=1,#P, n>P[i]||return; ispower(n-P[i])&& P[i]*2 != n && return(1))} \\ Needs the above P computed up to L >= n^6. For sum2sqr() see A133388.

A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

Original entry on oeis.org

2, 3, 5, 15, 26, 46, 82, 89, 90, 129, 323, 362, 401, 420, 610, 624, 840, 2024, 2703, 2808, 6888, 12099, 15963, 19320, 24650, 29930, 33490, 36482, 39203, 45795, 47523, 52440, 66050, 69168, 83408, 94248, 94863, 103683, 114284, 164399, 185364, 206442, 222785, 227530, 229180
Offset: 1

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Comments

Within the range of the data, a(n)^2 = A340642(n), i.e., no 3 immediately consecutive perfect powers x^p1, y^p2, z^p3 with min (p1, p2, p3) > 2 are seen. Is there a counterexample?

Links

Crossrefs

Cf. A000290, A001597, A025479, A076467, A097054, A111245, A153158, A340642, A340700, A340701.

Programs

  • PARI
    a340643(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(issquare(n1)&p2>2&p0>2, print1(sqrtint(n1),", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340643(10^8)
  • PARI
    upto(n) = {n *= n; my(v = List(), res = List([2])); for(i = 2, sqrtnint(n, 3), for(e = 3, logint(n, i), listput(v, i^e) ); ); listsort(v, 1); for(i = 1, #v - 1, if(sqrtint(v[i]) + 1 == sqrtint(v[i+1]) - issquare(v[i+1]), listput(res, sqrtint(v[i+1]-issquare(v[i+1]))); ) ); res }

Extensions

More terms from David A. Corneth, Jan 14 2021

A139308 Lucky numbers that are pure (perfect) powers with exponent greater than 2.

Original entry on oeis.org

729, 2187, 29791, 50625, 50653, 83521, 130321, 132651, 166375, 185193, 226981, 300763, 707281, 804357, 823543, 1295029, 1860867, 1874161, 2685619, 3176523, 4782969, 5545233, 5764801, 7189057, 9393931, 12977875, 13845841, 16194277
Offset: 1

Views

Author

Walter Kehowski, Jun 07 2008

Keywords

Comments

Intersection of A000959 and A076467.

Examples

			a(1) = 729 = 3^6 and 729 is lucky, i.e., A000959(118) = 729.

Links

Crossrefs

Cf. A000959, A076467, A031162.

Programs

  • Mathematica
    t = 2 Range@ 10000000 - 1; f[n_] := Block[{k = t[[n]]}, t = Delete[t, Table[{k}, {k, k, Length@t, k}]]]; Do[f@n, {n, 2, 1000000}]; fQ[n_] := ! IntegerQ@ Log[2, n] && GCD @@ Last /@ FactorInteger@n > 2; Select[t, fQ] (* Robert G. Wilson v, Oct 16 2010 *)

Extensions

a(16)-a(28) from Robert G. Wilson v, Oct 16 2010

A298591 Numbers which are the sum of two distinct perfect powers x^k + y^m with x, y, k, m >= 2.

Original entry on oeis.org

12, 13, 17, 20, 24, 25, 29, 31, 33, 34, 35, 36, 40, 41, 43, 44, 45, 48, 52, 53, 57, 58, 59, 61, 63, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 100, 104, 106, 108, 109, 113, 116, 117, 125, 127, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 149, 150
Offset: 1

Views

Author

M. F. Hasler, May 26 2018

Keywords

Comments

The number of terms between 2^(n-1) and 2^n-1 is, for n = 1, 2, 3, ...: 0, 0, 0, 2, 6, 17, 24, 69, 129*, 215, 425, 891, 1571, 2994, 5655*, 10535, 20132, 38840, 73510, 140730, 268438*, 514262, ... (For terms with * the next larger power of 2 is in the sequence, so it would be, e.g., ..., 130, 214, ... if we count from 2^n+1 to 2^(n+1).) At 2^22 this corresponds to a density of about 25%, decreasing by about 1% at each power of 2.

Examples

			12 = 2^2 + 2^3, 13 = 2^2 + 3^2, 17 = 2^3 + 3^2, ...

Links

Crossrefs

Cf. A076467, A111231, A304433, A304434, A304435, A304436.

Programs

  • Maple
    N:= 1000: # for all terms <= N
PP:= {seq(seq(x^k,k=2..floor(log[x](N))),x=2..floor(sqrt(N)))}:
sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],i=1..j-1),j=2..nops(PP))},N),list)); # Robert Israel, May 27 2018
  • Mathematica
    max = 150; Table[If[x^k == y^m, Nothing, x^k + y^m], {x, 2, Sqrt[max-4]}, {y, x, Sqrt[max-4]}, {k, 2, Log[2, max-4]}, {m, 2, Log[2, max-4]}] // Flatten // Select[#, # <= max &]& // Union (* Jean-François Alcover, Sep 18 2018 *)
  • PARI
    is(n,A=A076467,s=sum2sqr(n))={for(i=1,#s, vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=2,#A, n>A[i]||return; ispower(n-A[i]) && A[i]*2!=n && return(1))} \\ A076467 must be computed up to limit n. See A133388 for sum2sqr.
Previous Showing 31-34 of 34 results.

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