A337839 - OEIS

%I #29 Oct 07 2020 02:12:12

%S 4,16,81,88,144,250,520,900,1008,1053,1134,2025,2304,2655,3726,4680,

%T 6408,6624,9928,12024,12150,12510,13608,14256,15480,16408,17128,17172,

%U 18304,19152,19288,19602,23310,24336,25110,26550,29358,32896,32968,36864,37485,38592

%N 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.

%C 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.

%C 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?

%C Observation: 7 of the first 13 terms are perfect squares: 4, 16, 81, 144, 900, 2025, 2304 (see examples).

%H Seiichi Manyama, <a href="/A337839/b337839.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Coi#Colombian">Index to sequences related to Colombian numbers</a>.

%e 4 = 2 + 2 = 2 + 2 = 2 * 2 = 2 * 2;

%e 16 = 8 + 8 = 8 + 8 = 4 * 4 = 4 * 4;

%e 81 = 72 + (7+2) = 63 + (6*3) = 9 * 9 = 9 * 9;

%e 144 = 135 + (1+3+5) = 128 + (1*2*8) = 18 * (1*8) = 24 * (2+4).

%t 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 *)

%o (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

%Y Intersection of A176995, A336826, A337718 and A337816.

%Y Intersection of A336983, A337718 and A337816.

%Y Intersection of A337733 and A337816.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Sep 25 2020

%E Terms a(7) and beyond from _Amiram Eldar_, Sep 25 2020