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Numbers k such that k+digitsum(k^2) is a square.
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%I #37 May 28 2023 08:45:40

%S 0,17,62,71,117,125,197,206,296,297,305,413,414,557,558,692,702,711,

%T 863,864,872,873,1062,1070,1071,1268,1493,1502,1727,1736,1737,1745,

%U 1998,2006,2267,2276,2285,2564,2565,2573,2879,2888,2889,3221,3222

%N Numbers k such that k+digitsum(k^2) is a square.

%C Conjecture: there are infinitely many pairs of consecutive terms. Example: (296,297); (413,414); (863,864).

%e k=17 is a term because k^2=289 and 17+2+8+9=36=6^2.

%t Select[Range[0, 3300], IntegerQ[Sqrt[# + Plus @@ IntegerDigits[#^2]]] &] (* _Amiram Eldar_, May 18 2023 *)

%o (PARI) isok(k)=issquare(k+sumdigits(k^2))

%Y Cf. A004159, A066564.

%K nonn,easy,base

%O 1,2

%A _Alexandru Petrescu_, May 17 2023