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
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
Programs
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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:
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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
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