A336826 - OEIS

%I #45 Jan 14 2021 16:29:51

%S 0,1,4,9,11,16,24,25,36,39,42,49,56,64,75,81,88,93,96,111,119,138,144,

%T 164,171,192,224,242,250,255,297,312,336,339,366,378,393,408,422,448,

%U 456,488,497,516,520,522,525,564,575,648,696,704,738,744,755,777,792

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

%C Named Bogotá numbers by Tomás Uribe and Juan Pablo Fernández based on similarity of the construction to the Colombian numbers (A003052).

%C Some questions about these numbers:

%C (i) Some Bogotá numbers occur in pairs (such as 24 and 25). Are there infinitely many such pairs?

%C (ii) More generally, can arbitrarily long sets of consecutive numbers be found all of which are Bogotá numbers?

%C (iii) Can the gap between two consecutive Bogotá numbers be arbitrarily large? Answer: Yes.

%C From _David A. Corneth_, Aug 06 2020: (Start)

%C The only primes in this sequence are A004022.

%C 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.

%C 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.

%C 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.

%C Similarily we can find odd Bogotá numbers to then find consecutive Bogotá numbers (See A336864). (End)

%H Seiichi Manyama, <a href="/A336826/b336826.txt">Table of n, a(n) for n = 1..10000</a>

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a336/A336826.java">Java program</a> (github)

%H Math Stackexchange, <a href="https://math.stackexchange.com/questions/3718212/gaps-between-bogot%c3%a1-numbers/3718477#3718477">Gaps between Bogotá numbers</a>, 2020.

%H Puzzling Stackexchange, <a href="https://puzzling.stackexchange.com/questions/98998/pairs-of-bogot%c3%a1-numbers?noredirect=1#comment281441_98998">Pairs of Bogotá numbers</a>, 2020.

%e From _David A. Corneth_, Aug 06 2020: (Start)

%e 520 is a term because 52 * p(52) = 52 * 10 = 520.

%e Example using we only have to look at 7-smooth divisors:

%e 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)

%o (PARI) f(n) = vecprod(digits(n))*n; \\ A098736

%o isok(n) = my(k=0); for (k=0, n, if (f(k) == n, return(1))); \\ _Michel Marcus_, Aug 06 2020

%o (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

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

%K nonn,base

%O 1,3

%A _Sean A. Irvine_, Aug 05 2020