A003591 Numbers of form 2^i*7^j, with i, j >= 0.
1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 196, 224, 256, 343, 392, 448, 512, 686, 784, 896, 1024, 1372, 1568, 1792, 2048, 2401, 2744, 3136, 3584, 4096, 4802, 5488, 6272, 7168, 8192, 9604, 10976, 12544, 14336, 16384, 16807, 19208, 21952, 25088
Offset: 1
Keywords
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 100 terms from Vincenzo Librandi)
- Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
Programs
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GAP
Filtered([1..30000],n->PowerMod(14,n,n)=0); # Muniru A Asiru, Mar 19 2019
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Haskell
import Data.Set (singleton, deleteFindMin, insert) a003591 n = a003591_list !! (n-1) a003591_list = f $ singleton 1 where f s = y : f (insert (2 * y) $ insert (7 * y) s') where (y, s') = deleteFindMin s -- Reinhard Zumkeller, May 16 2015 -
Magma
[n: n in [1..26000] | PrimeDivisors(n) subset [2,7]]; // Bruno Berselli, Sep 24 2012
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Mathematica
fQ[n_] := PowerMod[14,n,n]==0; Select[Range[30000], fQ] (* Vincenzo Librandi, Feb 04 2012 *)
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PARI
list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
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PARI
isA003591(n)=n>>=valuation(n,2);ispower(n,,&n);n==1||n==7 \\ Charles R Greathouse IV, Jun 28 2011
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Python
from sympy import integer_log def A003591(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return n+x-sum((x//7**i).bit_length() for i in range(integer_log(x,7)[0]+1)) return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024
Formula
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(14*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
Sum_{n>=1} 1/a(n) = (2*7)/((2-1)*(7-1)) = 7/3. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(2)*log(7)*n)) / sqrt(14). - Vaclav Kotesovec, Sep 22 2020
Comments