A336876 -id:A336876 - cp's OEIS frontend

A336879 a(n) is the product of the decimal digits of A336876(n).

0, 1, 2, 3, 1, 4, 2, 5, 6, 3, 2, 7, 4, 8, 5, 9, 4, 3, 6, 1, 7, 6, 8, 4, 9, 8, 2, 2, 10, 5, 9, 12, 8, 3, 6, 14, 3, 12, 2, 16, 4, 4, 7, 12, 10, 18, 15, 4, 5, 18, 6, 16, 6, 12, 5, 21, 6, 15, 9, 4, 4, 20, 24, 3, 8, 6, 8, 14, 27, 9, 20, 24, 1, 18, 8, 9, 10, 6, 16

Offset: 1

Rémy Sigrist, Aug 06 2020

			For n = 26:
- A336876(26) = 24,
- so a(26) = 2*4 = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A336879

Cf. A007954, A336826, A336876.

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a(n) = A007954(A336876(n)).

a(n) * A336876(n) = A336826(n).

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

Original entry on oeis.org

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

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 *)

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

A336879 a(n) is the product of the decimal digits of A336876(n).

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

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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