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

1,2

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.

Links

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

Formula

A028839 INTERSECT A175396 INTERSECT A197039.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A028839, A175396, A197039, A197125.

Sequence in context: A112401 A178360 A197125 * A115688 A115710 A082749

Adjacent sequences: A197126 A197127 A197128 * A197130 A197131 A197132

Keyword

nonn,base

Author

Michel Lagneau, Oct 10 2011

Status

approved