A336944 -id:A336944 - cp's OEIS frontend

A336983 Bogota numbers that are not Colombian numbers.

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

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.

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.

cp's OEIS Frontend

A336983 Bogota numbers that are not Colombian numbers.

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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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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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