Hani Samamah - cp's OEIS frontend

A346563 a(n) = n + A007978(n).

3, 5, 5, 7, 7, 10, 9, 11, 11, 13, 13, 17, 15, 17, 17, 19, 19, 22, 21, 23, 23, 25, 25, 29, 27, 29, 29, 31, 31, 34, 33, 35, 35, 37, 37, 41, 39, 41, 41, 43, 43, 46, 45, 47, 47, 49, 49, 53, 51, 53, 53, 55, 55, 58, 57, 59, 59, 61, 61, 67, 63, 65, 65, 67, 67

Offset: 1

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

Beginning at n=3, a(n) represents the maximum length of consecutive numbers that are divisible by the product of their nonzero digits in base n. In particular, if n=10, the sequence of numbers that are divisible by the product of their nonzero digits is given by A055471.

			For n=6, the least non-divisor of 6 is 4, so a(6) = 6+4 = 10. As seen in the Comments section, 55980, 55981, ..., 55989 form a sequence of length 10, where every number is divisible by the product of its nonzero digits in base n=6. Work has been done to show that 10 is the maximum length for such sequences.

Cf. A000027, A007978, A055471.

a[n_] := Module[{k = 1}, While[Divisible[n, k], k++]; n + k]; Array[a, 100] (* Amiram Eldar, Jul 23 2021 *)

a(n) = my(k=2); while(!(n % k), k++); n+k; \\ Michel Marcus, Jul 23 2021

goal = 100
these = []
n = 1
while n <= goal:
    k = 1
    while n % k == 0:
        k = k + 1
    these.append(n + k)
    n += 1
print(these)

a(2k+1) = 2k+3.

a(2k) >= 2k+3.

A346537 Squares that are divisible by the product of their nonzero digits.

Original entry on oeis.org

1, 4, 9, 36, 100, 144, 400, 900, 1024, 1296, 2304, 2500, 2916, 3600, 10000, 11664, 12100, 14400, 22500, 32400, 40000, 41616, 78400, 82944, 90000, 102400, 110224, 121104, 122500, 129600, 152100, 176400, 186624, 200704, 202500, 219024, 230400, 250000, 260100, 291600

Offset: 1

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

			For the perfect square 1024 = 32^2 the product of its nonzero digits is 8 which divides 1024.

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 and A055471.

Select[Range[500]^2, Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* Amiram Eldar, Jul 23 2021 *)

isok(m) = issquare(m) && !(m % vecprod(select(x->(x>0), digits(m))));
lista(nn) = for (m=1, nn, if (isok(m^2), print1(m^2, ", "))); \\ Michel Marcus, Jul 23 2021

from math import prod
def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
def ok(sqr): return sqr > 0 and sqr%nzpd(sqr) == 0
print(list(filter(ok, (i*i for i in range(541))))) # Michael S. Branicky, Jul 23 2021

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

A346657 Numbers that are not divisible by the product of their nonzero digits.

Original entry on oeis.org

13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 27 2021

			The product of the nonzero digits of 42 is 4*2 = 8, which does not divide 42.

Complement of A055471.

Select[Range[100], !Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* Amiram Eldar, Jul 27 2021 *)

isok(k) = k % vecprod(select(x->(x>0), digits(k))); \\ Michel Marcus, Jul 28 2021

from math import prod
def ok(n): return n > 0 and n%prod([int(d) for d in str(n) if d != '0'])
print(list(filter(ok, range(88)))) # Michael S. Branicky, Jul 27 2021

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User: Hani Samamah

A346563 a(n) = n + A007978(n).

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A346537 Squares that are divisible by the product of their nonzero digits.

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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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A346657 Numbers that are not divisible by the product of their nonzero digits.

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Mathematica

PARI

Python