A053057 -id:A053057 - cp's OEIS frontend

A061910 Positive numbers k such that sum of digits of k^2 is a square.

1, 2, 3, 6, 9, 10, 11, 12, 13, 14, 15, 18, 20, 21, 22, 23, 30, 31, 39, 41, 45, 48, 51, 58, 59, 60, 67, 68, 76, 77, 85, 86, 90, 94, 95, 100, 101, 102, 103, 104, 105, 110, 111, 112, 113, 120, 121, 122, 130, 131, 139, 140, 148, 150, 157, 158, 166, 175, 176, 180, 184, 185

Offset: 1

Asher Auel, May 17 2001

			6^2 = 36 and 3+6 = 9 is a square. 13^2 = 169 and 1+6+9 = 16 is a square.

Bruno Berselli, Table of n, a(n) for n = 1..1000
MathStackExchange, Numbers n such that the digit sum of n2 is a square, 2015-2016.

Cf. A007953, A004159, A053057, A061909, A061911, A061912.

Sequence A293832 gives the start of the first run of n consecutive values.

[ n: n in [1..185] | IsSquare(&+Intseq(n^2)) ];  // Bruno Berselli, Jul 29 2011

readlib(issqr): f := []: for n from 1 to 200 do if issqr(convert(convert(n^2,base,10),`+`)) then f := [op(f), n] fi; od; f;

Select[Range[185], IntegerQ[Sqrt[Total[IntegerDigits[#^2]]]] &] (* Jayanta Basu, May 06 2013 *)

is(n)=n=eval(Vec(Str(n^2)));issquare(sum(i=1,#n,n[i])) \\ Charles R Greathouse IV, Jul 29 2011

select( is_A061910(n)=issquare(sumdigits(n^2)), [0..199]) \\ Includes the initial 0. - M. F. Hasler, Oct 16 2017

from gmpy2 import is_square
A061910 = [n for n in range(1,10**3) if is_square(sum(int(d) for d in str(n*n)))] # Chai Wah Wu, Sep 03 2014

A028839 Sum of digits of n is a square.

Original entry on oeis.org

1, 4, 9, 10, 13, 18, 22, 27, 31, 36, 40, 45, 54, 63, 72, 79, 81, 88, 90, 97, 100, 103, 108, 112, 117, 121, 126, 130, 135, 144, 153, 162, 169, 171, 178, 180, 187, 196, 202, 207, 211, 216, 220, 225, 234, 243, 252, 259, 261, 268, 270, 277, 286, 295, 301, 306, 310

Offset: 1

N. J. A. Sloane

Difference between two consecutive terms is never equal to 8. - Carmine Suriano, Mar 31 2014

In this sequence, there is no number of the form 3*k-1. In other words, if a(n) is not divisible by 9, it must be of the form 3*k+1. - Altug Alkan, Apr 08 2016

			234511 belongs to the sequence as its sum of digits is 16, a square.

Carmine Suriano, Table of n, a(n) for n = 1..1242

Cf. A007953, A028837, A050626.

Cf. A053057 (squares whose digit sum is also a square).

[n: n in [1..400] | IsSquare(&+Intseq(n))];  // Bruno Berselli, May 26 2011

Select[ Range[ 500 ], IntegerQ[ Sqrt[ Apply[ Plus, IntegerDigits[ # ] ] ] ]& ]

isok(n) = issquare(sumdigits(n)); \\ Michel Marcus, Oct 30 2014

More terms from Erich Friedman

A061267 Squares whose sum of digits as well as product of digits is a nonzero square.

Original entry on oeis.org

1, 4, 9, 144, 441, 14884, 44944, 48841, 132496, 214369, 268324, 288369, 294849, 346921, 436921, 511225, 617796, 938961, 1234321, 1336336, 1833316, 2325625, 2356225, 2585664, 2614689, 2778889, 2862864, 3323329, 3767481, 4691556

Offset: 1

Amarnath Murthy, Apr 24 2001

The squares of 969, 9669, 96669, 966669, ... with n 6s belong to this sequence if n = 4*m^2 - 3. The sum of the digits of this number is 36*m^2 and the product of the digits is 108^2 * 20^k, where k = 4xm^2.

			14884 = 122^2 is a member of this sequence as 1+4+8+8+4 = 25 = 5^2 and 1*4*8*8*4 = 1024 = 32^2.

Amarnath Murthy, Infinitely many common members of Smarandache Additive as well as Multiplicative Square sequence, (to be published in the Smarandache Notions Journal)
Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

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

Intersection of A050626, A028839, and A000290.
A061869 allows values with zero product.
Cf. A053057, A053059.

d[n_]:=IntegerDigits[n]; iQ[n_]:=IntegerQ[Sqrt[n]]; Select[Range[2500]^2,iQ[Plus@@(x=d[#])] && iQ[Times@@x] && FreeQ[x,0] &] (* Jayanta Basu, May 19 2013 *)

is(n)=my(v=digits(n),pr=prod(i=1,#v,v[i])); pr && issquare(pr) && issquare(n) && issquare(sumdigits(n)) \\ Charles R Greathouse IV, May 19 2013

More terms from Larry Reeves (larryr(AT)acm.org), May 11 2001

A061268 Numbers k such that k^2 has property that the sum of its digits and the product of its digits are nonzero squares.

Original entry on oeis.org

1, 2, 3, 12, 21, 122, 212, 221, 364, 463, 518, 537, 543, 589, 661, 715, 786, 969, 1111, 1156, 1354, 1525, 1535, 1608, 1617, 1667, 1692, 1823, 1941, 2166, 2235, 2337, 2379, 2515, 2943, 2963, 3371, 3438, 3631, 3828, 4018, 4077, 4119, 4271, 4338, 4341, 4471

Offset: 1

Amarnath Murthy, Apr 24 2001

See A061267 for the corresponding squares (the so-called ultrasquares). - M. F. Hasler, Oct 25 2022

			212^2 = 44944, 4+4+9+4+4 = 25 = 5^2 and 4*4*9*4*4 = 2304 = 48^2.

Amarnath Murthy, Infinitely many common members of the Smarandache Additive as well as multiplicative square sequence, (To be published in Smarandache Notions Journal).
Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Cf. A061267 (the corresponding squares), A053057 (squares with square digit sum), A053059 (squares with square product of digits).

Sequence A061868 allows digit products = 0.

select( {is_A061268(n)=vecmin(n=digits(n^2))&&issquare(vecprod(n))&&issquare(vecsum(n))}, [1..4567]) \\ M. F. Hasler, Oct 25 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 11 2001

A061269 Squares with nonzero digits such that (1) each digit is a square and (2) the sum of the digits is a square.

Original entry on oeis.org

1, 4, 9, 144, 441, 44944

Offset: 1

Amarnath Murthy, Apr 24 2001

Note that (1) implies that the product of the digits is a square.

Next term, if it exists, is > 90000000000. - Larry Reeves (larryr(AT)acm.org), May 11 2001

			For example, 44944 = 212^2, each digit is a square, sum of digits = 4+4+9+4+4 = 25 = 5^2.

Amarnath Murthy, The Smarandache multiplicative square sequence is infinite, (to be published in Smarandache Notions Journal).
Amarnath Murthy, Infinitely many common members of the Smarandache additive as well as multiplicative square sequence, (to be published in Smarandache Notions Journal).

Felice Russo, A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory, Lupton, AZ: American Research Press, 2000.

If zeros are allowed as digits, the result is A061270.
A subsequence of A006716.
Cf. A053057, A053059, A061267, A061268, A061269, A061270.

For[n = 1, n < 100000, n++, a := DigitCount[n^2]; If[a[[2]] == 0, If[a[[3]] == 0, If[a[[5]] == 0, If[a[[6]] == 0, If[a[[7]] == 0, If[a[[8]] == 0, If[a[[10]] == 0, If[Sqrt[Sum[a[[i]]*i, {i, 1, 10}]] == Floor[Sqrt[Sum[a[[i]]*i, {i, 1, 10}]]], Print[n^2]]]]]]]]]] (* Stefan Steinerberger, Mar 15 2006 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

A061090 Squares the sum of the squares of whose digits are squares.

Original entry on oeis.org

1, 4, 9, 100, 400, 676, 841, 900, 1444, 4225, 10000, 24025, 40000, 42025, 42436, 43264, 66049, 67600, 84100, 90000, 109561, 119716, 144400, 155236, 239121, 244036, 248004, 252004, 335241, 355216, 362404, 373321, 422500, 643204, 664225

Offset: 1

Amarnath Murthy, Apr 19 2001

Contains 10^(2k) for all k.

More generally, if k is in this sequence so is 100k. - Charles R Greathouse IV, Sep 20 2012

			676 = 26^2, 6^2 + 7^2 + 6^2 = 121 = 11^2;
1444 = 38^2, 1^2 + 4^2 + 4^2 + 4^2 = 49 = 7^2.

A. Murthy, Smarandache Pythagoras additive square sequence (to be published in Smarandache Notions Journal).

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

Cf. A053057.

readlib(issqr): for n from 1 to 2000 do L1 := convert(n^2, base, 10): if issqr(sum(L1[i]^2, i=1..nops(L1))) then printf(`%d,`,n^2) fi: od:

Select[Range[1000]^2,IntegerQ[Sqrt[Total[IntegerDigits[#]^2]]]&] (* Harvey P. Dale, Apr 26 2025 *)

ssd(n)=n=digits(n);sum(i=1,#n,n[i]^2)
v=List();for(n=1,1e4,if(issquare(ssd(n^2)),listput(v,n^2))); Vec(v) \\ Charles R Greathouse IV, Sep 20 2012

Corrected and extended by James Sellers, Apr 20 2001

A061270 Squares such that each digit is a square and the sum of the digits is a square.

Original entry on oeis.org

0, 1, 4, 9, 100, 144, 400, 441, 900, 10000, 10404, 14400, 40000, 40401, 44100, 44944, 90000, 1000000, 1004004, 1040400, 1440000, 4000000, 4004001, 4040100, 4410000, 4494400, 9000000, 9941409, 11909401, 100000000, 100040004, 100400400

Offset: 1

Amarnath Murthy, Apr 24 2001

			44944 = 212^2, each digit is a square, sum of digits = 4 + 4 + 9 + 4 + 4 = 25 = 5^2.

Amarnath Murthy, Smarandache Additive square sequence is infinite. (To be published in Smarandache Notions Journal.)
Amarnath Murthy, Infinitely many common members of the Smarandache Additive as well as multiplicative square sequence. (To be published in Smarandache Notions Journal.)
Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000.

Cf. A053057, A053059, A061267, A061268, A061269.

More terms from Larry Reeves (larryr(AT)acm.org), May 11 2001

a(1)=0 inserted by Sean A. Irvine, Jan 29 2023

A180737 a(n) is the largest n-digit square whose sum of digits is also a square.

Original entry on oeis.org

9, 81, 961, 9025, 96721, 990025, 9966649, 99980001, 999761161, 9999400009, 99998250625, 999946000729, 9999989500176, 99999960000004, 999999455981824, 9999998600000049, 99999998724439696, 999999998000000001

Offset: 1

Daniel Mondot, Oct 08 2010

Daniel Mondot, Table of n, a(n) for n = 1..258

Cf. A007953, A053057, A180738.

lnds[n_]:=Module[{s=Floor[Sqrt[10^n-1]]},While[!IntegerQ[Sqrt[Total[ IntegerDigits[ s^2]]]],s--];s^2]; Array[lnds,20] (* Harvey P. Dale, Mar 28 2013 *)

A371004 Fourth powers whose digital sum is also a fourth power.

Original entry on oeis.org

0, 1, 10000, 14641, 100000000, 104060401, 146410000, 1000000000000, 1004006004001, 1040604010000, 1464100000000, 4228599998736, 8670998958336, 9748688599521, 9948826238976, 12598637895936, 19226786746896, 19896452775936, 20699669996721, 23768199069696, 26599197668481

Offset: 1

Stefano Spezia, Mar 08 2024

Among the terms of this sequence, there are:
  the numbers of the form 10^(4*k) with k >= 0;
  the numbers of the form (10^i + 10^j)^4 with i > j >= 0.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000583, A007953, A038444, A053057, A053058, A371047.

Select[Range[0,2500]^4, IntegerQ[DigitSum[#]^(1/4)]&]

a(n) = A371047(n)^4.

A061272 Squares such that (1) each digit is a square, (2) the sum of squares of the digits is a square.

Original entry on oeis.org

0, 1, 4, 9, 100, 400, 900, 1444, 10000, 40000, 90000, 144400, 1000000, 4000000, 9000000, 14440000, 94109401, 100000000, 400000000, 900000000, 1444000000, 9410940100, 10000000000, 10100049001, 40000000000, 90000000000, 144400000000, 414441100441, 941094010000

Offset: 1

Amarnath Murthy, Apr 24 2001

			1444 = 38^2, each digit is a square, Sum of the squares of digits = 1+16+16+16 = 49 = 7^2.

Amarnath Murthy, Smarandache Pythagoras Additive Square Sequence. (To be published in Smarandache Notions Journal).

Cf. A053057, A053059, A061267, A061268, A061269, A061270, A061090.

okQ[n_]:=Module[{fd=FromDigits[n]},IntegerQ[Sqrt[fd]]&&IntegerQ[ Sqrt[ Total[n^2]]]]; FromDigits/@Select[Tuples[{0,1,4,9},8],okQ] (* Harvey P. Dale, May 12 2011 *)

Corrected and extended by Harvey P. Dale, May 12 2011

More terms from Jason Yuen, Aug 27 2025

cp's OEIS Frontend

A061910 Positive numbers k such that sum of digits of k^2 is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

A028839 Sum of digits of n is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A061267 Squares whose sum of digits as well as product of digits is a nonzero square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A061268 Numbers k such that k^2 has property that the sum of its digits and the product of its digits are nonzero squares.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A061269 Squares with nonzero digits such that (1) each digit is a square and (2) the sum of the digits is a square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A061090 Squares the sum of the squares of whose digits are squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A061270 Squares such that each digit is a square and the sum of the digits is a square.

Views