A050626 -id:A050626 - cp's OEIS frontend

A028839 Sum of digits of n is a square.

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

A028845 Iterated product of digits of n is a nonzero square.

Original entry on oeis.org

1, 4, 9, 11, 14, 19, 22, 27, 33, 39, 41, 72, 89, 91, 93, 98, 111, 114, 119, 122, 127, 133, 139, 141, 172, 189, 191, 193, 198, 212, 217, 221, 249, 266, 271, 277, 294, 313, 319, 331, 333, 338, 346, 364, 379, 383, 391, 397, 411, 429, 436, 463, 492, 626, 634, 643

Offset: 1

N. J. A. Sloane

			E.g. 27 -> 2*7 = 14 -> 1*4 = 4 is a square.

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

Cf. A031347, A050626, A028837.

ipdQ[n_]:=MemberQ[{9,4,1},NestWhile[Times@@IntegerDigits[#]&,n,#>9&]]; Select[Range[700],ipdQ] (* Harvey P. Dale, Apr 15 2018 *)

Extended (and corrected) by Patrick De Geest, Jun 15 1999

A237767 Integers whose product of digits is a nonzero cube.

Original entry on oeis.org

1, 8, 11, 18, 24, 39, 42, 81, 88, 93, 111, 118, 124, 139, 142, 181, 188, 193, 214, 222, 241, 248, 284, 319, 333, 389, 391, 398, 412, 421, 428, 444, 469, 482, 496, 555, 649, 666, 694, 777, 811, 818, 824, 839, 842, 881, 888, 893, 913, 931

Offset: 1

Derek Orr, Feb 12 2014

No number with a 0 in it (A011540) can be in this sequence. If a number is in this sequence, then so is its reversal of digits (A004086) and other permutations of its digits. - Alonso del Arte, Feb 20 2014

			3*9*1 = 27 = 3^3, thus 391 is a member of this sequence.
3*9*8 = 216 = 6^3, thus 398 is a member of this sequence.
4*2*8 = 64 = 4^3, thus 428 is a member of this sequence.

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

Cf. A007954, A050626.

filter:= proc(n) local T;
  T:= Statistics:-Tally(convert(n,base,10),output=table);
  if assigned(T[0]) then return false fi;
  eval(T[2] + 2*T[4] + T[6] mod 3, T = [0$6]) = 0
  and eval(T[3] + T[6] + 2*T[9] mod 3, T = [0$9]) = 0
  and member(T[5] mod 3, [0,'T[5]'])
  and member(T[7] mod 3, [0,'T[7]'])
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 16 2025

pdcQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&IntegerQ[ Surd[ Times@@idn,3]]]; Select[Range[1000],pdcQ] (* Harvey P. Dale, Aug 25 2017 *)

s=[]; for(n=1, 1000, t=eval(Vec(Str(n))); d=prod(i=1, #t, t[i]); if(d>0 && ispower(d, 3), s=concat(s, n))); s \\ Colin Barker, Feb 17 2014

def DigitProd(x):
  total = 1
  for i in str(x):
    total *= int(i)
  return total
def Cube(x):
  for n in range(1,10**3):
    if DigitProd(x) == n**3:
      return True
    if DigitProd(x) < n**3:
      return False
  return False
x = 1
while x < 1000:
  if Cube(x):
    print(x)
  x += 1

from math import prod
from sympy import integer_nthroot
def ok(n): return (p:=prod(map(int, str(n)))) > 0 and integer_nthroot(p, 3)[1]
print([k for k in range(10**3) if ok(k)]) # Michael S. Branicky, Jun 16 2025

There are between 9^(k-6) and 9^k k-digit members of this sequence, so a(n) >> n^1.04 and in particular this sequence has density 0. - Charles R Greathouse IV, Feb 21 2014

Name edited by Michel Marcus, Jun 16 2025

A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

Original entry on oeis.org

11, 14, 41, 49, 94, 111, 119, 122, 128, 133, 155, 166, 182, 188, 191, 199, 212, 218, 221, 229, 236, 263, 281, 289, 292, 298, 313, 326, 331, 362, 368, 386, 449, 494, 515, 551, 559, 595, 616, 623, 632, 638, 661, 683, 779, 797, 812, 818, 821, 829, 836, 863, 881

Offset: 1

Ryan Bresler, May 29 2021

If k is in the sequence then all anagrams of k are in the sequence. - David A. Corneth, May 29 2021

Trivially, this sequence has infinite elements. A031974 is an infinite sequence that is found in this sequence - Ryan Bresler, May 30 2021

			11 is a term because its digit sum is 2 (prime) and its digit product is 1 (perfect square > 0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A028834 and A050626.
Subsequence of A052382.
A031974 is a subsequence of this sequence.

q:= n-> (l-> not 0 in l and isprime(add(i, i=l)) and
         issqr(mul(i, i=l)))(convert(n, base, 10)):
select(q, [$0..999])[];  # Alois P. Heinz, May 29 2021

from math import prod
from sympy import isprime, integer_nthroot
def ok(n):
  d = list(map(int, str(n)))
  return 0 not in d and isprime(sum(d)) and integer_nthroot(prod(d), 2)[1]
print(list(filter(ok, range(1000)))) # Michael S. Branicky, May 29 2021

cp's OEIS Frontend

A028839 Sum of digits of n is a square.

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

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A028845 Iterated product of digits of n is a nonzero square.

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A237767 Integers whose product of digits is a nonzero cube.

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A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

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Maple

Python