A053059 -id:A053059 - cp's OEIS frontend

A006716 Squares with digits 1, 4, 9.

1, 4, 9, 49, 144, 441, 1444, 11449, 44944, 991494144, 4914991449, 149991994944, 9141411499911441, 199499144494999441, 9914419419914449449, 444411911999914911441, 419994999149149944149149944191494441

Offset: 1

N. J. A. Sloane, revised Jul 10 2015

This is probably a finite sequence, but that is only a conjecture.

Since 1, 4 and 9 are squares, all terms are in A053059. - Rabii Younès, Mar 17 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 234.

Patrick De Geest, Squares containing at most three distinct digits, Index entries for related sequences
A. Ottens, The arithmetic-digits-squares-three.digits problem
Eric Weisstein's World of Mathematics, Square Number

Subsequence of A019544 and A053059.
Cf. A027675 (square roots), A061269.
For other digit groups {0,1,2} through {7,8,9}, see also: A058411, ..., A058472, A058473, A058474.

a(n) = A027675(n)^2. - M. F. Hasler, Nov 15 2017

a(13) corrected by Neven Juric (neven.juric(AT)apis-it.hr), May 14 2003

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

A061867 Squares whose product of digits is also a square (allowing zeros).

Original entry on oeis.org

0, 1, 4, 9, 49, 100, 144, 289, 400, 441, 900, 1024, 1089, 1444, 1600, 2025, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 3025, 3600, 4096, 4900, 5041, 6084, 6400, 7056, 7744, 8100, 9025, 9409, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236

Offset: 1

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

Felice Russo, A set of new Smarandache Functions, Sequences and conjectures in number theory, pp-64, American Research Press, Lupton USA.
Amarnath Murthy, Exploring some new ideas on Smarandache Type Sets, Functions and sequences, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy, On the infinitude of Smarandache multiplicative square sequence, (to be published in Smarandache Notions Journal).

Sequence A053059 removes the numbers with zeros.

Select[Range[0,200]^2,IntegerQ[Sqrt[Times@@IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 03 2014 *)

isok(n) = {if (! issquare(n), return (0)); digs = digits(n, 10); issquare(prod(i=1, #digs, digs[i]));}  \\ Michel Marcus, Aug 02 2013

a(n) ~ n^2. - Charles R Greathouse IV, Sep 19 2012

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

A285079 Oblong numbers the product of whose digits are positive oblong numbers.

Original entry on oeis.org

2, 6, 12, 56, 132, 156, 756, 2756, 4556, 6162, 6972, 7656, 13572, 21756, 31152, 33672, 45156, 61752, 84972, 153272, 166872, 279312, 467172, 626472, 661782, 1273512, 1412532, 1541322, 1568756, 1596432, 1786232, 1867322, 2678132, 2817362, 3416952, 3521252

Offset: 1

Melvin Peralta, Apr 09 2017

Oblong numbers are numbers of the form k*(k+1) (A002378).

Robert Israel and Giovanni Resta, Table of n, a(n) for n = 1..391 (terms < 10^25, first 221 terms from Robert Israel)

Cf. A002378, A053059, A069077.

filter:= proc(x) local t; t:= convert(convert(x,base,10),`*`);
t > 0 and issqr(1+4*t); end proc:
select(filter, [seq(x*(1+x),x=1..10^4)]); # Robert Israel, Apr 14 2017

f[x_] := Sqrt[1 + 4 (Times @@ IntegerDigits[x])]; Select[Table[n (n + 1), {n, 1, 10000}], f[#] > 1 && Mod[f[#], 2] == 1 &]

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

A006716 Squares with digits 1, 4, 9.

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

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

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

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A061867 Squares whose product of digits is also a square (allowing zeros).

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

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A285079 Oblong numbers the product of whose digits are positive oblong numbers.

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Maple

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

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