A336826 -id:A336826 - cp's OEIS frontend

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).

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
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See Links section.
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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 *)

A337718 Numbers that can be written as (m + product of digits of m) for some m.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 54, 55, 56, 58, 60, 62, 65, 66, 67, 68, 70, 74, 75, 78, 80, 81, 85, 86, 88, 89, 90, 92, 94, 95, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset: 1

Bernard Schott, Sep 16 2020

Every integer that contains a digit 0 is a term (A011540).
When R_m with m >= 1 is in A002275, then R_m + 1 is a term (A047855 \ {1}).
Near similar:
-> Not-Colombian (A176995) are numbers that can be written as (m + sum of digits of m) for some m.
-> Bogotá numbers (A336826) are numbers that can be written as (m * product of digits of m) for some m.

			10 = 5 + 5 = 10 + (1*0) and 22 = 16 + (1*6) are terms.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequences: A011540, A047855 \ {1}.
Range of A230099.
Cf. A176995 (not Colombian), A336826 (Bogotá numbers).

m = 100; Select[Union[Table[n + Times @@ IntegerDigits[n], {n, 0, m}]], # <= m &] (* Amiram Eldar, Sep 16 2020 *)

isok(m) = {if (m==0, return (1)); for (k=1, m,  if (k+vecprod(digits(k)) == m, return (1)););} \\ Michel Marcus, Sep 17 2020

from math import prod
def b(n): return n + prod(map(int, str(n)))
def aupto(n): return sorted(set(b(m) for m in range(n+1) if b(m) <= n))
print(aupto(109)) # Michael S. Branicky, Jan 09 2023

A336983 Bogota numbers that are not Colombian numbers.

Original entry on oeis.org

4, 11, 16, 24, 25, 36, 39, 49, 56, 81, 88, 93, 96, 111, 119, 138, 144, 164, 171, 192, 224, 242, 250, 297, 336, 339, 366, 393, 408, 422, 448, 456, 488, 497, 516, 520, 522, 564, 575, 696, 704, 744, 755, 777, 792, 795, 819, 848, 884, 900, 912, 933, 944, 966, 992

Offset: 1

Bernard Schott, Aug 10 2020

Equivalently, numbers m that are of the form k + sum of digits of k for some k (A176995), and also of the form q * product of digits of q for some q (A336826).
Repunits are trivially Bogota numbers but the indices m of the repunits R_m that are not Colombian numbers are in A337139; also, all known repunit primes are terms (A004023) [see examples for primes R_2, R_19 and R_23].
35424 is the smallest term that belongs to both A230094 and A336944 (see last example).

			R_2 = 11 = 10 + (1+0) = 11 * (1*1) is a term;
24 = 21 + (2+1) = 12 * (1*2) is a term;
39 = 33 + (3+3) = 13 * (1*3) is a term;
R_19 = 1111111111111111079 + (16*1+7+9) = 1111111111111111111 * (1^19) hence R_19 is a term;
R_23 = 11111111111111111111077 + (20*1+7+7) = 11111111111111111111111 * (1^23) hence R_23 is a term;
42 = 21 * (2*1) is a Bogota number but there does not exist m < 42 such that 42 = m + sum of digits of m, hence 42 that is also a Colombian number is not a term.
35424 = 35406 + (3+5+4+0+6) = 35397 + (3+5+3+9+7) = 2214 * (2*2*1*4) = 492 * (4*9*2).

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Puzzling Stackexchange, Pairs of Bogotá Numbers.
Index to sequences related to Colombian numbers.

Intersection of A176995 and A336826.
Cf. A003052 (Colombian), A336984 (Bogota and Colombian), A336985 (Colombian not Bogota), A336986 (not Colombian and not Bogota).
Cf. A230093, A230094, A336944, A337139.

m = 1000; Intersection[Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Aug 10 2020 *)

lista(nn) = Vec(setintersect(Set(vector(nn, k, k+sumdigits(k))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ Michel Marcus, Aug 20 2020

A358350 Numbers that can be written as (m + sum of digits of m + product of digits of m) for some m.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 23, 24, 26, 27, 29, 30, 32, 33, 34, 35, 38, 42, 43, 44, 46, 48, 50, 53, 54, 55, 56, 58, 62, 63, 66, 68, 69, 73, 74, 76, 77, 78, 80, 82, 83, 86, 88, 90, 92, 95, 97, 98, 99, 101, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 1

Bernard Schott, Nov 11 2022

Integers that are in A161351.
(i) Can arbitrarily long sets of consecutive integers be found in this sequence?
(ii) Is the gap between two consecutive terms bounded?
A000533 \ {1} is a subsequence.
This has the same asymptotic density, approximately 0.9022222, as A176995, since the asymptotic density of non-pandigital numbers is 0. - Charles R Greathouse IV, Nov 16 2022

			A161351(23) = 23 + (2+3) + (2*3) = 34 so 34 is a term.
There is no integer du_10 such that du + (d+u) + (d*u) = 31, so 31 is not a term.

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

Range of A161351.
Similar: A176995 (m+digitsum), A336826 (m*digitprod), A337718 (m+digitprod).
Cf. A000533.

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; With[{m = 110}, Select[Union[Table[f[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Nov 11 2022 *)

f(n) = my(d=digits(n)); n + vecsum(d) + vecprod(d); \\ A161351
lista(nn) = select(x->(x<=nn), Set(vector(nn, k, f(k)))); \\ Michel Marcus, Nov 12 2022

from math import prod
def sp(n): d = list(map(int, str(n))); return sum(d) + prod(d)
def ok(n): return any(m + sp(m) == n for m in range(n))
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Dec 19 2022

a(n) ~ kn with k approximately 1.108374, see comments. - Charles R Greathouse IV, Nov 16 2022

A336984 Colombian numbers that are also Bogotá numbers.

Original entry on oeis.org

1, 9, 42, 64, 75, 255, 312, 378, 525, 648, 738, 1111, 1278, 2224, 2448, 2784, 2817, 3504, 3864, 3875, 4977, 5238, 5495, 5888, 8992, 9712, 10368, 11358, 11817, 12348, 12875, 13136, 13584, 13775, 13832, 13944, 15351, 15384, 15744, 15900, 16912, 17768, 18095, 19344, 20448

Offset: 1

Bernard Schott, Aug 22 2020

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), but are of the form q * product of digits of q for some q (A336826).
Repunits are trivially Bogotá numbers but the indices m of the repunits R_m that are Colombian numbers are in A337208. No known prime belongs to this sequence (see A004023).
A336983, A336985, A336986 and this sequence form a partition of the set of positive integers N*.

			42 = 21 * (2*1) is a Bogotá number and there does not exist m < 42 such that 42 = m + sum of digits of m, hence 42 is a Colombian number and 42 is a term.
56 = 14 * (1*4) is a Bogotá number but as 56 = 46 + (4+6), 56 is not a Colombian number, hence 56 is not a term.
648 = 36 * (3*6) = 81 * (8*1) but there does not exist m < 648 such that 648 = m + sum of digits of m, hence 648 is a Colombian number, so 648 is a term that has two different representations as the product of a number and of its decimal digits.

David A. Corneth, Table of n, a(n) for n = 1..10000
Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Puzzling Stackexchange, Pairs of Bogotá Numbers.
Index to sequences related to Colombian numbers.

Intersection of A003052 and A336826.
Cf. A336983 (Bogotá and not Colombian), A336985 (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).
Cf. A336944, A336983, A337208.

m = 21000; Intersection[Complement[Range[m], Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &]], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Aug 22 2020 *)

lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ Michel Marcus, Aug 23 2020

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.

A336985 Colombian numbers that are not Bogotá numbers.

Original entry on oeis.org

3, 5, 7, 20, 31, 53, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 266, 277, 288, 299, 310, 323, 334, 345, 356, 367, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 536, 547, 558, 569, 580, 591, 602, 613

Offset: 1

Bernard Schott, Aug 26 2020

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), and that are not of the form q * product of digits of q for any q (complement of A336826).
As repunits are trivially Bogotá numbers, there are not repunits in the data.
A336983, A336984, A336986 and this sequence form a partition of the set of positive integers N*

			7 is a term because there are not k < 7  such that 7 = k + sum of digits of k, and that are not q such that 7 = q * product of digits of q.
13 is not of the form q * product of digits of q for any q <= 13, so 13 is not a Bogotá number, but 13 = 11 + (1+1) is not Colombian, hence 13 is not a term.
42 is Colombian because there does not exist m < 42 such that 42 = m + sum of digits of m, but as 42 = 21 * (2*1) is a Bogota number, 42 is not a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Index to sequences related to Colombian or self numbers.

Cf. A003052 (Colombian), A176995 (not Colombian), A336826 (Bogotá numbers), A336983 (Bogotá not Colombian), A336984 (Bogotá and Colombian), this sequence (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).

m = 600; Intersection[Complement[Range[m], Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &]], Complement[Range[m], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]]] (* Amiram Eldar, Aug 26 2020 *)

lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), setminus([1..nn], Set(vector(nn, k, k*vecprod(digits(k))))))); \\ Michel Marcus, Aug 26 2020

A336986 Numbers that are not Colombian and not Bogotá.

Original entry on oeis.org

2, 6, 8, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 87, 89, 90, 91

Offset: 1

Bernard Schott, Aug 22 2020

Equivalently, numbers m that are of the form k + sum of digits of k for some k (A176995), but are not of the form q * product of digits of q for any q.
As repunits are trivially Bogotá numbers, there are not repunits in the data.
A336983, A336984, A336985 and this sequence form a partition of the set of positive integers N*.

			13 = 11 + (1+1) is not Colombian and 13 is not of the form q * product of digits of q for any q <= 13, so 13 is not a Bogotá number, hence 13 is a term.
39 = 33 + (3+3) is not Colombian but 39 = 13 * (1*3) is a Bogotá number, hence 39 is not a term.
42 = 21 * (2*1) is a Bogotá number but there does not exist k < 42 such that 42 = k + sum of digits of k, hence 42 is a Colombian number and 42 is not a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Index to sequences related to Colombian numbers.

Cf. A003052 (Colombian), A176995 (not Colombian), A336826 (Bogotá), A336983 (Bogotá and not Colombian), A336984 (Bogotá and Colombian), A336985 (Colombian not Bogotá), this sequence (not Colombian and not Bogotá).

m = 100; Intersection[Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &], Complement[Range[m], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]]] (* Amiram Eldar, Aug 22 2020 *)

lista(nn) = Vec(setintersect(Set(vector(nn, k, k+sumdigits(k))), setminus([1..nn], Set(vector(nn, k, k*vecprod(digits(k))))))); \\ Michel Marcus, Aug 23 2020

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.

cp's OEIS Frontend

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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A337718 Numbers that can be written as (m + product of digits of m) for some m.

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A336983 Bogota numbers that are not Colombian numbers.

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A358350 Numbers that can be written as (m + sum of digits of m + product of digits of m) for some m.

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A336984 Colombian numbers that are also Bogotá numbers.

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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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A336985 Colombian numbers that are not Bogotá numbers.

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