A238903 - OEIS

%I #16 Dec 09 2024 23:22:32

%S 0,1,3,6,11,18,36,43,56,61,106,136,168,181,206,411,431,511,518,536,

%T 606,613,1056,1068,1388,1631,1636,1668,1686,1693,1806,1813,1956,1981,

%U 2068,2081,3363,3411,3418,3631,3693,3763,4106,4331,5136,5318,5411,5606,5868,6011,6036,6236,6238,6256,6431,6456,6581,10568,10668,10813,11581,11588,11806,11888

%N Integers k such that (k^2 + (k+1)^2) has no square proper substring.

%C Inspired by (and program used from) A238334.

%C Note that (m^2+(m+1)^2), for m>0, always ends with 5. Any other patterns?

%C From _Robert Israel_, Dec 09 2024: (Start)

%C The last two digits of k^2 + (k+1)^2 (if more than 2 digits) are 01, 05, 13, 21, 25, 41, 45, 61, 65, 81, or 85.  The only ones of these that don't contain the squares 0, 1, 4, or 25 are 65 and 85, so all terms k > 3 of this sequence have k^2 + (k+1)^2 ending in 65 or 85. (End)

%H Robert Israel, <a href="/A238903/b238903.txt">Table of n, a(n) for n = 1..1000</a>

%e 1^2 + 2^2 = 5, 3^2 + 4^2 = 25, 6^2 + 7^2 = 85.

%p filter:= proc(m) local n,i,j,S;

%p   n:= m^2 + (m+1)^2;

%p S:= {seq(seq(floor((n mod 10^i)/10^j),j=0..i-1),i=1 .. ilog10(n)+1)} minus {n};

%p   not ormap(issqr,S);

%p end proc:

%p select(filter, [$0..20000]); # _Robert Israel_, Dec 09 2024

%Y Cf. A001844, A130448, A238334, A238335.

%K nonn,base

%O 1,3

%A _Zak Seidov_, Mar 07 2014