

A337839


Numbers that can be written as (k + sum of digits of k) for some k, then as (m + product of digits of m) for some m, also as (q * product of digits of q) for some q, and finally as (t * sum of digits of t) for some t.


1



4, 16, 81, 88, 144, 250, 520, 900, 1008, 1053, 1134, 2025, 2304, 2655, 3726, 4680, 6408, 6624, 9928, 12024, 12150, 12510, 13608, 14256, 15480, 16408, 17128, 17172, 18304, 19152, 19288, 19602, 23310, 24336, 25110, 26550, 29358, 32896, 32968, 36864, 37485, 38592
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OFFSET

1,1


COMMENTS

Equivalently, Bogotá numbers that are not Colombian and that can be written as (m + product of digits of m) for some m and also as (t * sum of digits of t) for some t.
The only primes that can belong to this sequence are repunits > 11 whose indices are in A004023. It is known that these primes belong to A336983 but do they belong also to A337718 and A337816?
Observation: 7 of the first 13 terms are perfect squares: 4, 16, 81, 144, 900, 2025, 2304 (see examples).


LINKS



EXAMPLE

4 = 2 + 2 = 2 + 2 = 2 * 2 = 2 * 2;
16 = 8 + 8 = 8 + 8 = 4 * 4 = 4 * 4;
81 = 72 + (7+2) = 63 + (6*3) = 9 * 9 = 9 * 9;
144 = 135 + (1+3+5) = 128 + (1*2*8) = 18 * (1*8) = 24 * (2+4).


MATHEMATICA

m = 40000; Select[Intersection @@ Union /@ Transpose[Table[{n + (s = Plus @@ (d = IntegerDigits[n])), n + (p = Times @@ d), n*s, n*p}, {n, 1, m}]], # <= m &] (* Amiram Eldar, Sep 25 2020 *)


PROG

(PARI) lista(nn) = {my(vd = vector(nn, k, digits(k)), vs = vector(nn, k, vecsum(vd[k])), vp = vector(nn, k, vecprod(vd[k])), vsp = Set(vector(nn, k, k+vp[k])), vss = Set(vector(nn, k, k+vs[k])), vps = Set(vector(nn, k, k*vs[k])), vpp = Set(vector(nn, k, k*vp[k])), vk = vector(nn, k, k)); Vec(setintersect(vk, setintersect(vsp, setintersect(vss, setintersect(vps, vpp))))); } \\ Michel Marcus, Oct 01 2020


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



