A197129 Numbers such that the sum, sum of the squares, and sum of the cubes of the decimal digits are each a perfect square.

1, 4, 9, 10, 40, 90, 100, 400, 900, 1000, 1111, 1224, 1242, 1339, 1393, 1422, 1933, 2124, 2142, 2214, 2241, 2412, 2421, 3139, 3193, 3319, 3391, 3913, 3931, 4000, 4122, 4212, 4221, 4444, 4669, 4696, 4966, 6469, 6496, 6649, 6694, 6946, 6964, 9000, 9133, 9313

Offset: 1

Author

Michel Lagneau, Oct 10 2011

Keywords

• nonn
• base

Comments

Each number > 90 contains at least two identical digits because the sequence A197125 contains a subset of numbers all of whose digits are distinct and are all the permutations of 1567890. But 1^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 1926 is not square. Consequently, it is impossible to find numbers > 90 with distinct digits in this sequence.

Examples

			4669 is in the sequence because:
4   + 6   + 6   + 9   = 25   = 5^2;
4^2 + 6^2 + 6^2 + 9^2 = 169  = 13^2;
4^3 + 6^3 + 6^3 + 9^3 = 1225 = 35^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A028839, A175396, A197039, A197125.

Programs

Maple

for n from 1 to 10000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:for m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10): n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:od:if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) and sqrt(s3)=floor(sqrt(s3))then printf(`%d, `, n): else fi:od:

Mathematica

sdQ[n_]:=Module[{idn=IntegerDigits[n]},IntegerQ[Sqrt[Total[idn]]] && IntegerQ[Sqrt[Total[idn^2]]]&&IntegerQ[Sqrt[Total[idn^3]]]]; Select[ Range[ 10000],sdQ] (* Harvey P. Dale, Oct 25 2011 *)
psQ[n_]:=With[{idn=IntegerDigits[n]},AllTrue[{Sqrt[Total[idn]],Sqrt[Total[idn^2]],Sqrt[Total[idn^3]]},IntegerQ]]; Select[Range[10000],psQ] (* Harvey P. Dale, Nov 17 2024 *)

PARI

is(n)=my(v=eval(Vec(Str(n))));issquare(sum(i=1,#v,v[i]))&&issquare(sum(i=1,#v,v[i]^2))&&issquare(sum(i=1,#v,v[i]^3)) \\ Charles R Greathouse IV, Oct 10 2011

Formula

A028839 INTERSECT A175396 INTERSECT A197039.