A028839 -id:A028839 - cp's OEIS frontend

A053057 Squares whose digit sum is also a square.

0, 1, 4, 9, 36, 81, 100, 121, 144, 169, 196, 225, 324, 400, 441, 484, 529, 900, 961, 1521, 1681, 2025, 2304, 2601, 3364, 3481, 3600, 4489, 4624, 5776, 5929, 7225, 7396, 8100, 8836, 9025, 10000, 10201, 10404, 10609, 10816, 11025, 12100, 12321, 12544, 12769

Offset: 1

Felice Russo, Feb 25 2000

The numbers 81, 100, 121, 144, 169, 196, 225 are seven consecutive squares belonging to this sequence. The next set of seven consecutive squares whose digit sum is also a square is {9999^2, 10000^2, 10001^2, 10002^2, 10003^2, 10004^2, 10005^2}. (See Crux Mathematicorum link.) - Bernard Schott, May 24 2017
The first set of 8 consecutive squares begin at 46045846^2. This was already known in 2016, see MathStackExchange link. - Michel Marcus, May 25 2017
The first run of 9 consecutive squares starts at 302260461719025^2. - Giovanni Resta, Jun 08 2017

			144 is a term: 144 = 12^2 and 1 + 4 + 4 = 9 = 3^2. - _Bernard Schott_, May 24 2017

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press, 2000.

Daniel Mondot, Table of n, a(n) for n = 1..1192
Allan Wm. Johnson Jr., Problem 443, Crux Mathematicorum, Vol. 6, No. 3 (Mar. 1980), page 88.
MathStackExchange, Numbers n such that the digit sum of n^2 is a square, 2015-2016.

Subsequence of A000290.

Cf. A028839, A061910.

[n^2: n in [0..120] | IsSquare(&+Intseq(n^2))];  // Bruno Berselli, May 26 2011

Select[Range[0,115]^2, IntegerQ[Sqrt[DigitSum[#]]]&] (* Stefano Spezia, Mar 07 2024 *)

lista(nn) = for (n=1, nn, if (issquare(sumdigits(n^2)), print1(n^2, ", "));); \\ Michel Marcus, May 25 2017

More terms from James Sellers, Feb 28 2000

A107288 Primes whose digit sum is a square.

Original entry on oeis.org

13, 31, 79, 97, 103, 211, 277, 349, 367, 439, 457, 547, 619, 673, 691, 709, 727, 853, 907, 997, 1021, 1069, 1087, 1201, 1249, 1429, 1447, 1483, 1609, 1627, 1663, 1699, 1753, 1789, 1861, 1879, 1933, 1951, 1987, 2011, 2239, 2293, 2347, 2383, 2437, 2473, 2617, 2671

Offset: 1

Zak Seidov, May 20 2005

Primes in A028839. [K. D. Bajpai, Jul 08 2014]

From Altug Alkan and Waldemar Puszkarz, Apr 10 2016: All terms are congruent to 1 mod 6. Proof: For n > 2, prime(n) is 1 or 5 mod 6. If p is 5 mod 6, then it is of the form 3*k-1. For numbers of this form, the sum of digits is also of this form, as can be seen through the kind of reasoning used in proving that numbers divisible by 3 have the sum of digits divisible by 3. However, 3*k-1 can never be a square, meaning n^2+1 is never divisible by 3: any n is equal to one of 0, 1, 2 mod 3, thus by the rules of modular arithmetic, n^2+1 is 1 or 2 mod 3, never 0. Hence p must be congruent to 1 mod 6.

			79 is in the sequence because it is prime. Also, (7 + 9) = 16 = 4^2.
997 is in the sequence because it is prime. Also, (9 + 9 + 7) = 25 = 5^2.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A007953, A028839, A048519.

Cf. A244863 (Semiprimes whose digit sum is square).

with(numtheory): A107288:= proc() local a; a:=add(i,i = convert((n),base,10))(n); if isprime(n) and root(a,2)=floor(root(a,2)) then RETURN (n); fi; end: seq(A107288 (), n=1..5000); # K. D. Bajpai, Jul 08 2014

bb = {}; Do[If[IntegerQ[Sqrt[Apply[Plus, IntegerDigits[p = Prime[n]]]]], bb = Append[bb, p]], {n, 500}]; bb

lista(nn) = {forprime(p=2, nn, if (issquare(sumdigits(p)), print1(p, ", ")););} \\ Michel Marcus, Apr 09 2016

Terms a(47) and a(48) added by K. D. Bajpai, Jul 08 2014

A001102 Numbers k such that k / (sum of digits of k) is a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 81, 100, 144, 150, 192, 200, 225, 288, 300, 320, 324, 375, 400, 441, 500, 512, 600, 640, 648, 700, 704, 735, 800, 832, 882, 900, 960, 1014, 1088, 1200, 1452, 1458, 1521, 1815, 2023

Offset: 1

N. J. A. Sloane, Bill Moran (moran1(AT)llnl.gov)

The sequence is infinite since if m = 10^(2*j) then m / digitsum(m) = m. - Marius A. Burtea, Dec 21 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of Niven numbers (A005349); cf. A028839.

a001102 n = a001102_list !! (n-1)
a001102_list =
   filter (\x -> a010052 (x `div` (a007953 x)) == 1) a005349_list
-- Reinhard Zumkeller, Aug 17 2011

[n: n in [1..1000] | IsIntegral(n/(&+Intseq(n))) and IsSquare(n/(&+Intseq(n)))]; // Marius A. Burtea, Dec 21 2018

Select[Range[2200],IntegerQ[Sqrt[#/Total[IntegerDigits[#]]]]&] (* Harvey P. Dale, Feb 25 2012 *)

a(n) mod A007953(a(n)) = 0 and A010052(a(n) / A007953(a(n))) = 1. - Reinhard Zumkeller, Aug 17 2011

A050626 Product of digits of n is a nonzero square.

Original entry on oeis.org

1, 4, 9, 11, 14, 19, 22, 28, 33, 41, 44, 49, 55, 66, 77, 82, 88, 91, 94, 99, 111, 114, 119, 122, 128, 133, 141, 144, 149, 155, 166, 177, 182, 188, 191, 194, 199, 212, 218, 221, 224, 229, 236, 242, 248, 263, 281, 284, 289, 292, 298, 313, 326, 331, 334, 339, 343

Offset: 1

Patrick De Geest, Jun 15 1999

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

Cf. A007954, A028845, A028839.

[ n: n in [1..350] | s gt 0 and IsSquare(s) where s is &*Intseq(n,10) ];  // Bruno Berselli, May 26 2011

Select[Range[500],DigitCount[#,10,0]==0&&IntegerQ[Sqrt[Times@@ IntegerDigits[ #]]]&] (* Harvey P. Dale, Jun 09 2020 *)

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

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

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

Original entry on oeis.org

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]]]&]

A236748 Positive integers k such that k^2 divided by the digital sum of k is a square.

Original entry on oeis.org

1, 4, 9, 10, 18, 22, 27, 36, 40, 45, 54, 63, 72, 81, 88, 90, 100, 108, 112, 117, 126, 130, 135, 144, 153, 162, 171, 180, 196, 202, 207, 216, 220, 225, 234, 243, 252, 261, 268, 270, 306, 310, 315, 324, 333, 342, 351, 360, 376, 400, 405, 414, 423, 432, 441

Offset: 1

Colin Barker, Jan 30 2014

Subsequence of A028839 (sum of digits of n is a square). - Jon Perry and Michel Marcus, Oct 30 2014
A028839 is the sequence of positive integers such that n^2 divided by the sum of the digits is a rational square. For this sequence, it is required to be an integer square. - Franklin T. Adams-Watters, Oct 30 2014
The sequence is infinite since if m = 10^j then m^2 / digitsum(m) = m^2. - Marius A. Burtea, Dec 21 2018

			153 is in the sequence because the digital sum of 153 is 9, and 153^2/9 = 2601 = 51^2.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A001102, A007953, A236749, A236750, A236751.

Cf. A028839.

[n: n in [1..1500] | IsIntegral((n^2)/(&+Intseq(n))) and IsSquare((n^2)/(&+Intseq(n)))]; // Marius A. Burtea, Dec 21 2018

filter:= n -> issqr(n^2/convert(convert(n,base,10),`+`)):
select(filter, [$1..10000]); # Robert Israel, Oct 30 2014

Select[Range[500],IntegerQ[Sqrt[#^2/Total[IntegerDigits[#]]]]&] (* Harvey P. Dale, Nov 19 2014 *)

s=[]; for(n=1, 600, d=sumdigits(n); if(n^2%d==0 && issquare(n^2\d), s=concat(s, n))); s

A028837 Iterated sum of digits of n is a square.

Original entry on oeis.org

1, 4, 9, 10, 13, 18, 19, 22, 27, 28, 31, 36, 37, 40, 45, 46, 49, 54, 55, 58, 63, 64, 67, 72, 73, 76, 81, 82, 85, 90, 91, 94, 99, 100, 103, 108, 109, 112, 117, 118, 121, 126, 127, 130, 135, 136, 139, 144, 145, 148, 153, 154, 157, 162, 163, 166, 171, 172, 175, 180

Offset: 1

N. J. A. Sloane

			E.g. 58 -> 5+8 = 13 -> 1+3 = 4 is a square.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A010888, A028839, A028845.

LinearRecurrence[{1,0,1,-1},{1,4,9,10},60] (* Harvey P. Dale, Jan 26 2015 *)

a(n) = a(n-3)+9. If n is a multiple of 3 then a(n) = 3n, otherwise a(n) = 3n-2. Numbers of form {0, 1, 4} modulo 9 - Henry Bottomley, Jun 30 2000
a(1)=1, a(2)=4, a(3)=9, a(4)=10, a(n)=a(n-1)+a(n-3)-a(n-4). - Harvey P. Dale, Jan 26 2015
G.f.: x*(1+3*x+5*x^2) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Sep 22 2016
E.g.f.: (exp(x)*(9*x - 4) + 4*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Mar 07 2024

More terms from Patrick De Geest, Jun 15 1999

A225535 Numbers whose cubed digits sum to a cube, and have more than one nonzero digit.

Original entry on oeis.org

168, 186, 345, 354, 435, 453, 534, 543, 618, 681, 816, 861, 1068, 1086, 1156, 1165, 1516, 1561, 1608, 1615, 1651, 1680, 1806, 1860, 3045, 3054, 3405, 3450, 3504, 3540, 4035, 4053, 4305, 4350, 4503, 4530, 5034, 5043, 5116, 5161, 5304, 5340, 5403, 5430, 5611

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

			5^3 + 6^3 + 1^3 + 1^3 = 343, which is 7^3.

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

Cf. A225534 (cubed digits sum to a prime), A197039 (square), A046459. A055012.
Cf. A165330 (cube cycle), A046197 (cubic fixed points), A000578 (cubes).
Cf. A052034 (squared digits sum to a prime), A028839, A117685.
Cf. A164882 (n such that sum of the cubes of the digits of n^3 is perfect cube). - Zak Seidov, May 21 2013

fQ[n_] := Module[{d = IntegerDigits[n]}, Count[d, 0] + 1 < Length[d] && IntegerQ[Total[d^3]^(1/3)]]; Select[Range[5611], fQ] (* T. D. Noe, May 19 2013 *)

y=rep(0,10000); len=0; x=0; library(gmp);
digcubesum<-function(x) sum(as.numeric(unlist(strsplit(as.character(as.bigz(x)),split="")))^3);
iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0));
nonzerodig<-function(x) sum(strsplit(as.character(x),split="")[[1]]!="0");
which(sapply(1:6000,function(x) nonzerodig(x)>1 & iscube(digcubesum(x))))

A244863 Semiprimes whose digit sum is a perfect square.

Original entry on oeis.org

4, 9, 10, 22, 121, 169, 178, 187, 202, 259, 295, 301, 358, 394, 466, 493, 529, 538, 565, 583, 655, 718, 745, 763, 781, 799, 817, 835, 862, 871, 889, 898, 934, 943, 961, 979, 1003, 1111, 1159, 1177, 1186, 1195, 1267, 1285, 1294, 1339, 1357, 1366, 1393, 1438, 1465

Offset: 1

K. D. Bajpai, Jul 07 2014

Subsequence of A028839.

			178 is in the sequence because 178 = 2*89 (semiprime) and 1+7+8 = 16 (square).
187 is in the sequence because 187 = 11*17 (semiprime) and 1+8+7 = 16 (square).

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A007953, A028839, A048519.

Cf. A107288 (Primes whose digit sum is square).

select(n -> numtheory:-bigomega(n)=2 and issqr(convert(convert(n,base,10),`+`)),
[$1..3000]); # Robert Israel, Jul 09 2014

Select[Range[3000], PrimeOmega[#] == 2 && IntegerQ[Sqrt[Apply [Plus, IntegerDigits[#]]]] &]

cp's OEIS Frontend

A053057 Squares whose digit sum is also a square.

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A107288 Primes whose digit sum is a square.

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A001102 Numbers k such that k / (sum of digits of k) is a square.

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A050626 Product of digits of n is a nonzero square.

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Magma

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

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Mathematica

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

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Mathematica

A236748 Positive integers k such that k^2 divided by the digital sum of k is a square.

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