A098736 -id:A098736 - cp's OEIS frontend

A336826 Bogotá numbers: numbers k such that k = m*p(m) where p(m) is the digital product of m.

0, 1, 4, 9, 11, 16, 24, 25, 36, 39, 42, 49, 56, 64, 75, 81, 88, 93, 96, 111, 119, 138, 144, 164, 171, 192, 224, 242, 250, 255, 297, 312, 336, 339, 366, 378, 393, 408, 422, 448, 456, 488, 497, 516, 520, 522, 525, 564, 575, 648, 696, 704, 738, 744, 755, 777, 792

Offset: 1

Sean A. Irvine, Aug 05 2020

Named Bogotá numbers by Tomás Uribe and Juan Pablo Fernández based on similarity of the construction to the Colombian numbers (A003052).
Some questions about these numbers:
(i) Some Bogotá numbers occur in pairs (such as 24 and 25). Are there infinitely many such pairs?
(ii) More generally, can arbitrarily long sets of consecutive numbers be found all of which are Bogotá numbers?
(iii) Can the gap between two consecutive Bogotá numbers be arbitrarily large? Answer: Yes.
From David A. Corneth, Aug 06 2020: (Start)
The only primes in this sequence are A004022.
To see if a number is a Bogotá number, we only have to look at its 7-smooth divisors. Proof: If a number k is a Bogotá number then k = m*p(m) where p(m) is 7-smooth as it's a product of digits. Furthermore, if k = m*p(m) then p(m) | k. Q.e.d. Below is an example using this idea.
To find Bogotá numbers k up to N we can make a list of 7-smooth numbers up to sqrt(N) and list the factorizations into single-digit numbers of each of these 7-smooth numbers that when concatenated give m such that m * p(m) = k where p(m) is that 7-smooth number.
For example, 10 is a 7-smooth number. Its factorizations into single-digit numbers are 2*5, 5*2, 1*2*5 and so on. This tells us that 10*25 = 250, 10*52 = 520, 10*125 = 1250 all are Bogotá numbers.
Similarily we can find odd Bogotá numbers to then find consecutive Bogotá numbers (See A336864). (End)

			From _David A. Corneth_, Aug 06 2020: (Start)
520 is a term because 52 * p(52) = 52 * 10 = 520.
Example using we only have to look at 7-smooth divisors:
520 is a term as its 7-smooth divisors d are 1, 2, 4, 5, 8, 10, 20, 40. values 520/d are 520, 260, 130, 104, 65, 52, 26, 13 of which 52 * (5*2) = 520 where (5*2) are the products of 52. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)
Math Stackexchange, Gaps between Bogotá numbers, 2020.
Puzzling Stackexchange, Pairs of Bogotá numbers, 2020.

Cf. A003052, A004022, A007954, A098736, A336864, A336876.

f(n) = vecprod(digits(n))*n; \\ A098736
isok(n) = my(k=0); for (k=0, n, if (f(k) == n, return(1))); \\ Michel Marcus, Aug 06 2020

is(n) = { my(f = factor(n), s7 = 1, d, sl = sqrtint(n)); for(i = 1, #f~, if(f[i, 1] > 7, break ); s7 *= f[i, 1]^f[i, 2]; ); d = divisors(s7); for(i = 1, #d, if(d[i] > sl, return(0)); if(n/d[i] * vecprod(digits(n/d[i])) == n, return(1); ) ); 0 } \\ David A. Corneth, Aug 06 2020

A230099 a(n) = n + (product of digits of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 70, 78, 86, 94, 102, 110, 118, 126

Offset: 0

N. J. A. Sloane, Oct 12 2013

A230099, A063114, A098736, A230101 are analogs of A092391 and A062028.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A011540, A092391, A062028, A004207, A230093, A003052, A176995.

Cf. also A063114, A098736, A230101, A230103, A230104, A230105, A062028, A230102, A063108.

a230099 n = a007954 n + n  -- Reinhard Zumkeller, Oct 13 2013

with transforms; [seq(n+digprod(n), n=0..200)];

a(n) = if (n, n + vecprod(digits(n)), 0); \\ Michel Marcus, Dec 18 2018

from math import prod
def a(n): return n + prod(map(int, str(n)))
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

a(n) = n iff n contains a digit 0 (A011540). - Bernard Schott, Jul 31 2023

A336876 a(n) is the least m such that A336826(n) = m*p(m) (where p(m) is the product of decimal digits of m).

Original entry on oeis.org

0, 1, 2, 3, 11, 4, 12, 5, 6, 13, 21, 7, 14, 8, 15, 9, 22, 31, 16, 111, 17, 23, 18, 41, 19, 24, 112, 121, 25, 51, 33, 26, 42, 113, 61, 27, 131, 34, 211, 28, 114, 122, 71, 43, 52, 29, 35, 141, 115, 36, 116, 44, 123, 62, 151, 37, 132, 53, 91, 212, 221, 45, 38

Offset: 1

Rémy Sigrist, Aug 06 2020

Some terms of A336826 have several representations as the product of a number and of its decimal digits; for example 549504 has four such representations: 1696 * 1 * 6 * 9 * 6, 2862 * 2 * 8 * 6 * 2, 3392 * 3 * 3 * 9 * 2 and 3816 * 3 * 8 * 1 * 6.

			For n = 26:
- A336826(26) = 192,
- the divisors d of 192, alongside d*p(d), are:
  d    d*p(d)
  ---  ------
    1       1
    2       4
    3       9
    4      16
    6      36
    8      64
   12      24
   16      96
   24     192
   32     192
   48    1536
   64    1536
   96    5184
  192    3456
- so a(26) = min(24, 32) = 24.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A336876

Cf. A007954, A098736, A336826, A336879.

C
```
See Links section.
```

A098736(a(n)) = A336826(n).

A336944 Numbers k that have at least two different representations as the product of a number and of its decimal digits.

Original entry on oeis.org

0, 192, 648, 819, 1197, 1536, 4872, 4977, 5976, 7056, 9968, 13608, 20448, 21168, 22176, 22428, 22752, 32040, 33984, 35424, 36864, 37692, 38736, 59778, 64152, 77600, 89928, 96912, 112833, 112896, 113148, 116352, 116736, 120384, 120708, 146412, 154752, 156288, 192888

Offset: 1

Seiichi Manyama, Aug 08 2020

Subsequence of A336826.

			192 = 24 * (2*4) = 32 * (3*2).
549504 = 1696 * (1*6*9*6) = 2862 * (2*8*6*2) = 3392 * (3*3*9*2) = 3816 * (3*8*1*6).
1798848 = 6246 * (6*2*4*6) = 12492 * (1*2*4*9*2) = 33312 * (3*3*3*1*2).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Seiichi Manyama)

Cf. A098736, A336826, A336876.

digprod[n_] := n * Times @@ IntegerDigits[n]; seqQ[0] = True; seqQ[n_] := DivisorSum[n, Boole[digprod[#] == n] &] > 1; Select[Range[0, 2 * 10^5], seqQ] (* Amiram Eldar, Aug 08 2020 *)
Take[Select[Tally[Table[n*Times@@IntegerDigits[n],{n,30000}]],#[[2]]>1&][[;;,1]]//Sort,40] (* Harvey P. Dale, Apr 13 2025 *)

A230101 a(n) = product of n and all its nonzero digits.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 11, 24, 39, 56, 75, 96, 119, 144, 171, 40, 42, 88, 138, 192, 250, 312, 378, 448, 522, 90, 93, 192, 297, 408, 525, 648, 777, 912, 1053, 160, 164, 336, 516, 704, 900, 1104, 1316, 1536, 1764, 250, 255, 520, 795, 1080, 1375, 1680, 1995, 2320, 2655, 360, 366, 744, 1134, 1536, 1950

Offset: 0

N. J. A. Sloane, Oct 12 2013

			a(15) = 15*1*5=75

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A051801, A098736.

with transforms; [seq(n*digprod0(n), n=0..200)];

Table[n*Times@@Select[IntegerDigits[n],#!=0&],{n,0,70}] (* Harvey P. Dale, May 26 2018 *)

a(n) = n*A051801.

A337051 Least positive number that has exactly n different representations as the product of a number and of its decimal digits.

Original entry on oeis.org

2, 1, 192, 1798848, 549504, 20150684596224

Offset: 0

Chai Wah Wu, Aug 12 2020

			1       = 1 * (1).
192     = 24 * (2*4)
        = 32 * (3*2).
1798848 = 6246 * (6*2*4*6)
        = 12492 * (1*2*4*9*2)
        = 33312 * (3*3*3*1*2).
549504  = 1696 * (1*6*9*6)
        = 2862 * (2*8*6*2)
        = 3392 * (3*3*9*2)
        = 3816 * (3*8*1*6).
20150684596224 = 61699872 * (6*1*6*9*9*8*7*2)
               = 123399744 * (1*2*3*3*9*9*7*4*4)
               = 242943246 * (2*4*2*9*4*3*2*4*6)
               = 323924328 * (3*2*3*9*2*4*3*2*8)
               = 416474136 * (4*1*6*4*7*4*1*3*6).

Cf. A098736, A336826, A336876, A336944, A337054.

A337054 Numbers that have at least 3 different representations as the product of a number and of its decimal digits.

Original entry on oeis.org

0, 549504, 1798848, 4193856, 4804128, 5827584, 7426944, 14397696, 34324992, 39401250, 39611040, 42856128, 45312750, 62593440, 81575424, 86171040, 92348928, 140184576, 151600896, 196475328, 221695488, 251584704, 263680704, 271165104, 287945280, 475388928, 499654656

Offset: 1

Chai Wah Wu, Aug 12 2020

Subsequence of A336944. a(2) and a(43) both have 4 representations. The term 1461825635235840 = 696266592*(6*9*6*2*6*6*5*9*2) = 72511192224*(7*2*5*1*1*1*9*2*2*2*4) = 5371199424*(5*3*7*1*1*9*9*4*2*4) = 7161599232*(7*1*6*1*5*9*9*2*3*2) = 1193599872*(1*1*9*3*5*9*9*8*7*2) has 5 representations.

			a(43) = 1578092544 = 342468*(3*4*2*4*6*8) = 913248*(9*1*3*2*4*8) = 97848*(9*7*8*4*8) = 86976*(8*6*9*7*6).

Chai Wah Wu, Table of n, a(n) for n = 1..11157

Cf. A098736, A336826, A336876, A336944, A337051.

A337100 Numbers that have at least 4 different representations as the product of a number and of its decimal digits.

Original entry on oeis.org

0, 549504, 1578092544, 12276847296, 28961412480, 35998381440, 87012926784, 118082893824, 259456659840, 335449175040, 397315715328, 579305502720, 672777778176, 712539265536, 741360356352, 863562591360, 1138944651264, 1264664088576, 1276070713344, 1300488037632

Offset: 1

Chai Wah Wu, Aug 15 2020

Subsequence of A337054. a(61) = 20150684596224 is the smallest positive number with 5 representations. Other terms with 5 representations include 242374224347136, 1461825635235840, 1761950567301120, 3194185120277760, 3415710732779520.

			a(3) = 12276847296 = 676634*(6*7*6*6*3*4) = 773296*(7*7*3*2*9*6) = 2368219*(2*3*6*8*2*1*9) = 12179412*(1*2*1*7*9*4*1*2).

Chai Wah Wu, Table of n, a(n) for n = 1..113

Cf. A098736, A336826, A336876, A336944, A337051, A337054.

A381631 Numbers k such that the product of k and its digits is divisible by the sum of its digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 22, 24, 26, 27, 30, 36, 40, 42, 44, 45, 48, 50, 54, 60, 62, 63, 66, 70, 72, 80, 81, 84, 88, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 116, 117, 120, 123, 126, 130, 132, 133, 134, 135, 138, 140

Offset: 1

Jakub Buczak, Mar 02 2025

Positive integers with the digit 0 (see A011540) are terms, since the product of it and its digits is A098736(k) = 0 which is divisible by any sum of digits.
Terms with a 0 digit form various runs of consecutive terms, such as from 100...00 through to 111...10.
Terms without a 0 digit can form runs of 9 terms: see A381697.
A prime > 7 is never divisible by its sum of digits (because the sum is smaller than the prime) so that primes > 7 occur in this sequence only when their product of digits is divisible by sum of digits (the primes in A038367).

			36 is a term because 36*3*6 is divisible by 3+6.
140 is a term because 140*1*4*0 equals 0, which is trivially divisible by 1+4+0.

Cf. A007953, A098736, A381697.

Cf. A011540, A038367, A005349.

q[k_] := Module[{d = IntegerDigits[k]}, Divisible[k * Times @@ d, Plus @@ d]]; Select[Range[140], q] (* Amiram Eldar, Mar 03 2025 *)

isok(k) = my(d=digits(k)); !((k*vecprod(d)) % vecsum(d)); \\ Michel Marcus, Mar 03 2025

A229544 Numbers k such that k*product_of_digits(k) is a nonzero cube.

Original entry on oeis.org

1, 8, 243, 784, 7776, 9826, 13122, 24389, 26244, 39366, 47628, 55566, 59895, 71442, 82944, 122825, 124416, 226981, 263424, 275625, 316368, 323433, 333396, 588245, 663255, 774144, 843648, 1339893, 1492992, 1613472, 2341344, 3816336, 3981312, 8719893, 8992364, 9393931, 9927988, 11212884, 11239424, 14823774

Offset: 1

Derek Orr, Sep 25 2013

			7776*(7*7*7*6) = 1600030008 = 252^3. Thus, 7776 is a member of this sequence.

Cf. A066565, A098736.

isok(k) = my(p=vecprod(digits(k))); p && ispower(k*p, 3); \\ Michel Marcus, Aug 24 2025

Corrected and extended by Derek Orr, Mar 22 2015

cp's OEIS Frontend

A336826 Bogotá numbers: numbers k such that k = m*p(m) where p(m) is the digital product of m.

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A230099 a(n) = n + (product of digits of n).

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A336876 a(n) is the least m such that A336826(n) = m*p(m) (where p(m) is the product of decimal digits of m).

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A336944 Numbers k that have at least two different representations as the product of a number and of its decimal digits.

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A230101 a(n) = product of n and all its nonzero digits.

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A337051 Least positive number that has exactly n different representations as the product of a number and of its decimal digits.

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A337054 Numbers that have at least 3 different representations as the product of a number and of its decimal digits.

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A337100 Numbers that have at least 4 different representations as the product of a number and of its decimal digits.

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A381631 Numbers k such that the product of k and its digits is divisible by the sum of its digits.