A175396 -id:A175396 - cp's OEIS frontend

A197039 Numbers such that sum of the cube of decimal digits is a perfect square.

1, 4, 9, 10, 12, 21, 22, 40, 48, 84, 88, 90, 100, 102, 120, 123, 126, 132, 162, 168, 186, 201, 202, 210, 213, 216, 220, 231, 261, 312, 321, 333, 400, 408, 480, 612, 618, 621, 681, 804, 808, 816, 840, 861, 880, 900, 1000, 1002, 1020, 1023, 1026, 1032, 1062

Offset: 1

Michel Lagneau, Oct 08 2011

			261 is in the sequence because 2^3+6^3+1^3 = 15^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028839, A175396.

Select[Range[1100],IntegerQ[Sqrt[Apply[Plus,IntegerDigits[#]^3]]]&]

A197125 Numbers such that sum of digits and sum of the square of digits are both a square.

Original entry on oeis.org

1, 4, 9, 10, 40, 90, 100, 400, 900, 1000, 1111, 1177, 1224, 1242, 1339, 1393, 1422, 1717, 1771, 1933, 2124, 2142, 2214, 2241, 2412, 2421, 3139, 3193, 3319, 3391, 3913, 3931, 4000, 4122, 4212, 4221, 4444, 4588, 4669, 4696, 4858, 4885, 4966, 5488, 5848, 5884

Offset: 1

Michel Lagneau, Oct 10 2011

The sequence contains a majority of numbers with two identical digits at least, but there exists a finite subset A = {1, 4, 9, 10, 40, 90, 156789, 156798, ..., 9876510} of 7!+6 = 5046 numbers with distinct decimal digits. The numbers > 90 of A are all permutations of 1567890.

			597618 is in the sequence because :
5+9+7+6+1+8 = 36 = 6^2 ;
5^2+9^2+7^2+6^2+1^2+8^2 = 256 = 16^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028839, A175396.

for n from 1 to 6000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=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: od:if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) then printf(`%d, `, n): else fi:od:

sdQ[n_]:=Module[{idn=IntegerDigits[n]},IntegerQ[Sqrt[Total[idn]]] && IntegerQ[Sqrt[Total[idn^2]]]]; Select[Range[6000],sdQ] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = {A028839} intersection {A175396}.

A223035 Prime numbers whose digits squared sum to a square.

Original entry on oeis.org

2, 3, 5, 7, 43, 263, 269, 1153, 1531, 1933, 2063, 2069, 2287, 2609, 3319, 3391, 3511, 3931, 4003, 4441, 4801, 4889, 5113, 5399, 5939, 6029, 6067, 6203, 6469, 6607, 8849, 9133, 9539, 10111, 10177, 10513, 10531, 10771, 11149, 11213, 11273, 11321, 11491, 11503

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 04 2013

			269 is a prime number, and 2^2+6^2+9^2 = 121 = 11^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Prime numbers from the sequence A175396.

Cf. A000040, A061910, A107288.

Select[Prime[Range[2000]], IntegerQ[Sqrt[Total[IntegerDigits[#]^2]]] &] (* T. D. Noe, Apr 05 2013 *)

ssod<-function(i) sum(as.numeric(strsplit(as.character(i),"")[[1]])^2)
issquare<-function(x) as.integer(sqrt(x))==sqrt(x)
x=as.bigz(c()); i=2
while(length(x)<10000) {if(issquare(ssod(i))) x=c(x,i); i=nextprime(i)}

A307735 Integers k such that if m = k + A003132(k) then k = m - A003132(m).

Original entry on oeis.org

0, 9, 205, 212, 217, 366, 457, 663, 1314, 1315, 1348, 1672, 1742, 1792, 1797, 2005, 2012, 2017, 2129, 2201, 2208, 2213, 2216, 2305, 2404, 2405, 2465, 2564, 2565, 2671, 2741, 2748, 2789, 2829, 3114, 3115, 3205, 3303, 3306, 3394, 3436, 3475, 3696, 3819, 4204, 4205, 4245, 4347, 4475, 4542, 4629, 4647, 4688

Offset: 1

Antonio Roldán, Apr 25 2019

A003132(n) is the sum of the squares of the digits of n.

			205 is in the sequence because 205 + 2^2 + 0^2 + 5^2 = 234 and 234 - 2^2 - 3^2 - 4^2 = 205.

Cf. A003132, A108662, A175396, A209303, A223035, A225049, A258881.

sod2[n_] := Total @ (IntegerDigits[n]^2); aQ[n_] := sod2[n + (s=sod2[n])] == s; Select[Range[0, 4700], aQ] (* Amiram Eldar, Jul 03 2019 *)

for(i = 0 , 5000 , a = i + norml2(digits(i)) ; b = a - norml2(digits(a)) ; if(i == b , print1(i , ", ")))

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

Original entry on oeis.org

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

Michel Lagneau, Oct 10 2011

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.

			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.

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

Cf. A028839, A175396, A197039, A197125.

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:

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 *)

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

A028839 INTERSECT A175396 INTERSECT A197039.

A171976 Numbers n such that the sum of the squares of the digits of n^n is a square.

Original entry on oeis.org

0, 1, 2, 8, 10, 100, 123, 209, 312, 1000, 1668, 2191, 2268, 4767, 9338, 10000, 11004, 12248, 12322, 15926, 17951, 18202, 19764, 21807, 29509, 42647, 43072, 44750, 54237, 56634, 70383, 74032, 85325, 90906, 95261, 100000

Offset: 1

Michel Lagneau, Nov 19 2010

			8 is in the sequence because 8^8 = 16777216 and 1^2+6^2+7^2+7^2+7^2+2^2+1^2+6^2
  = 225 = 15^2.

with(numtheory): digits:=200:nn:=5000:for n from 0 to nn do:l:=length(n^n):n0:=n^n:s:=0:for
  m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s:=s+u^2:od:if sqrt(s)=
  floor(sqrt(s))then printf(`%d, `, n):else fi:od:

Join[{0},Select[Range[100000],IntegerQ[Sqrt[Total[IntegerDigits[ #^#]^2]]]&]] (* Harvey P. Dale, Sep 25 2018 *)

isok(n) = my(d = digits(n^n)); issquare (sum(i=1, #d, d[i]^2)); \\ Michel Marcus, Jan 15 2014

{n: A003132(n^n) in A000290}.

{n: n^n in A175396.}

Edited by D. S. McNeil, Nov 19 2010

Offset corrected and more terms added, Michel Marcus, Jan 15 2014

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A197039 Numbers such that sum of the cube of decimal digits is a perfect square.

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A197125 Numbers such that sum of digits and sum of the square of digits are both a square.

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A223035 Prime numbers whose digits squared sum to a square.

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A307735 Integers k such that if m = k + A003132(k) then k = m - A003132(m).

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

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A171976 Numbers n such that the sum of the squares of the digits of n^n is a square.

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