

A336826


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


16



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
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OFFSET

1,3


COMMENTS

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 7smooth divisors. Proof: If a number k is a Bogotá number then k = m*p(m) where p(m) is 7smooth 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 7smooth numbers up to sqrt(N) and list the factorizations into singledigit numbers of each of these 7smooth numbers that when concatenated give m such that m * p(m) = k where p(m) is that 7smooth number.
For example, 10 is a 7smooth number. Its factorizations into singledigit 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)


LINKS

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.


EXAMPLE

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 7smooth divisors:
520 is a term as its 7smooth 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)


PROG

(PARI) 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
(PARI) 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


CROSSREFS

Cf. A003052, A004022, A007954, A098736, A336864, A336876.
Sequence in context: A239435 A190559 A316621 * A212017 A236171 A312838
Adjacent sequences: A336823 A336824 A336825 * A336827 A336828 A336829


KEYWORD

nonn,base


AUTHOR

Sean A. Irvine, Aug 05 2020


STATUS

approved



