A118547 -id:A118547 - cp's OEIS frontend

A118720 Cubes which are divisible by the sum of their digits.

1, 8, 27, 216, 512, 1000, 1728, 4913, 5832, 8000, 13824, 17576, 19683, 27000, 35937, 46656, 59319, 64000, 74088, 110592, 125000, 157464, 185193, 216000, 287496, 314432, 328509, 343000, 373248, 421875, 474552, 512000, 592704, 658503, 729000

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

Intersection of A000578 and A005349. - Robert Israel, Jun 27 2017

			4913 is in the sequence because it is a cube, the sum of its digits is 4+9+1+3=17 and 4913 is divisible by 17.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A005349, A118547.

select(t -> t mod convert(convert(t,base,10),`+`) = 0, map(t -> t^3, [$1..100])); # Robert Israel, Jun 28 2017

Select[Range[100]^3,Divisible[#,Total[IntegerDigits[#]]]&]  (* Harvey P. Dale, Apr 25 2011 *)

isok(n) = ispower(n,3) && ((n % sumdigits(n)) == 0); \\ Michel Marcus, Jun 28 2017

A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 30, 36, 39, 42, 45, 48, 49, 51, 52, 54, 60, 63, 65, 66, 68, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 111, 112, 117, 120, 126, 132, 138, 140, 144, 148, 150, 156, 162, 168, 174, 180, 182, 190, 198, 200, 201, 204, 207

Offset: 1

Vighnesh Patil, Jul 06 2025

			15 is a term 15^2 = 225; digit sum of 225 = 2 + 2 + 5 = 9; 225 mod 9 = 0, so 15 is included.
18 is a term 18^2 = 324; digit sum of 324 = 3 + 2 + 4 = 9; 324 mod 9 = 0, so 16 is included.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Cf. A005349, A007953, A034706, A118547, A385640.

digitSum := n -> add(i,i=convert(n, base, 10)):
isok := n -> modp(n^2, digitSum(n^2)) = 0:
select(isok, [$1..400])[];

DigitSum[n_] := Total[IntegerDigits[n]];
Select[Range[400], Mod[#^2, DigitSum[#^2]] == 0 &]

isok(k) = (k^2 % sumdigits(k^2)) == 0; \\ Michel Marcus, Jul 06 2025

def digit_sum(n): return sum(int(d) for d in str(n))
def ok(n): return (n**2) % digit_sum(n**2) == 0
print([n for n in range(1, 1000) if ok(n)])

a(n) = sqrt(A118547(n)). - Michel Marcus, Jul 07 2025

A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.

Original entry on oeis.org

1, 4, 9, 36, 100, 144, 400, 900, 1296, 2304, 2916, 3600, 10000, 11664, 12100, 14400, 22500, 32400, 40000, 41616, 82944, 90000, 121104, 122500, 129600, 152100, 176400, 186624, 202500, 219024, 230400, 260100, 291600, 360000, 419904, 435600, 504100

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 23 2021

			For the perfect square 144 = 12^2, the sum of its digits is 9, which divides 144, and the product of its nonzero digits is 16, which also divides 144 so 144 is a term of the sequence.

H. G. Grundman, Sequences of consecutive n-Niven numbers, Fibonacci Quarterly (1994) 32 (2): 174-175.
Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below.)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.

Intersection of A000290, A005349 and A055471.

Cf. A118547, A342262, A342650.

Select[Range[720]^2, And @@ Divisible[#, {Plus @@ (d = IntegerDigits[#]), Times @@ Select[d, #1 > 0 &]}] &] (* Amiram Eldar, Jul 23 2021 *)

from math import prod
def sumd(n): return sum(map(int, str(n)))
def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
def ok(sqr): return sqr > 0 and sqr%sumd(sqr) == 0 and sqr%nzpd(sqr) == 0
print(list(filter(ok, (i*i for i in range(1001)))))
# Michael S. Branicky, Jul 23 2021

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A118720 Cubes which are divisible by the sum of their digits.

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A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

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A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.

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