A115689 -id:A115689 - cp's OEIS frontend

A115690 Squares whose digit reversal is a powerful(1) number (A001694).

1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 576, 676, 900, 961, 1089, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 25281, 27225, 40000, 40401, 40804, 44100, 44521, 44944, 48400, 48841, 57600

Offset: 1

Giovanni Resta, Jan 31 2006

If x is a member, then so is 100*x. - Robert Israel, Mar 16 2020

			25281=159^2 and 18252=2^2*3^3*13^2 is powerful.

Robert Israel, Table of n, a(n) for n = 1..1500

Subsequence of A115656.
A033294 is a subsequence, and the main entry for this sequence.
Cf. A001694, A115689.

filter:= proc(n) local L,i,x;
  L:= convert(n,base,10);
  x:=add(L[-i]*10^(i-1),i=1..nops(L));
  andmap(t -> t[2]>=2, ifactors(x)[2]):
end proc:select(filter, [seq(i^2,i=1..10^4)]); # Robert Israel, Mar 16 2020

is(k) = ispowerful(fromdigits(Vecrev(digits(k))));
select(is, vector(300, n, n^2)) \\ Michel Marcus, Nov 01 2022

Trivially, n^2 <= a(n) <= 100^(n-1). - Charles R Greathouse IV, Nov 01 2022

A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

Original entry on oeis.org

21, 960540, 16418498901144294337512360, 436066841882071117095002459324085167366543342937477344818646196279385305441506861017701946929489111120

Offset: 1

Michel Lagneau, Feb 20 2010

nonn

A. Nitaj proved Erdős's conjecture (1975) and claimed that there exist infinitely many triples of 3-powerful numbers a,b,c with (a,b) = 1, such that a+b=c, because the equation x^3 + y^3 = 6z^3 admits an infinite number of solutions, and given by the recurrence equations (see formula). It is proved that a=x(k)^3, b=y(k)^3, and c=6c^3, and are 3-powerful numbers for each k >= 1.

			37^3 + 17^3 = 6*21^3.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 348.
Mordell, L. J. (1969). Diophantine equations. Academic Press. ISBN 0-12-506250-8

P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]
A. Nitaj, On a conjecture of Erdős on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
Wikipedia, Diophantine equation

Cf. A050240, A050241, A057521, A060859, A113839, A115645, A115651, A115676, A115686, A115687, A115689, A115691, A115693, A115695, A115697, A116064, A140172.

x0:=37:y0:=17:z0:=21: for p from 1 to 5 do: x1:=x0*(x0^3+ 2*y0^3):y1:=-y0*(2*x0^3+ y0^3):z1:=z0*(x0^3- y0^3): print(z1) : x0 :=x1 :y0 :=y1 :z0 :=z1 :od :

We generate the solutions (x(k),y(k),z(k)) from the initial solution x(0) = 37, y(0)=17, z(0)=21 x(k+1) = x(k)*(x(k)^3 + 2*y(k)^3) y(k+1) = -y(k)*(2*x(k)^3 + y(k)^3) z(k+1) = z(k)*(x(k)^3 - y(k)^3).

cp's OEIS Frontend

A115690 Squares whose digit reversal is a powerful(1) number (A001694).

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