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Numbers of the form (7^i)*(13^j).
9

%I #16 Oct 22 2024 15:12:21

%S 1,7,13,49,91,169,343,637,1183,2197,2401,4459,8281,15379,16807,28561,

%T 31213,57967,107653,117649,199927,218491,371293,405769,753571,823543,

%U 1399489,1529437,2599051,2840383,4826809,5274997,5764801,9796423,10706059,18193357,19882681

%N Numbers of the form (7^i)*(13^j).

%H Amiram Eldar, <a href="/A108056/b108056.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = (7*13)/((7-1)*(13-1)) = 91/72. - _Amiram Eldar_, Sep 23 2020

%F a(n) ~ exp(sqrt(2*log(7)*log(13)*n)) / sqrt(91). - _Vaclav Kotesovec_, Sep 23 2020

%t n = 10^7; Flatten[Table[7^i*13^j, {i, 0, Log[7, n]}, {j, 0, Log[13, n/7^i]}]] // Sort (* _Amiram Eldar_, Sep 23 2020 *)

%o (PARI) list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=7));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jun 28 2011

%o (Python)

%o from sympy import integer_log

%o def A108056(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o def f(x): return n+x-sum(integer_log(x//13**i,7)[0]+1 for i in range(integer_log(x,13)[0]+1))

%o return bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

%Y Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466.

%K nonn,easy

%O 1,2

%A Douglas Winston (douglas.winston(AT)srupc.com), Jun 02 2005

%E More terms from _Amiram Eldar_, Sep 23 2020