A290729 - OEIS

%I #24 Nov 03 2018 19:44:14

%S 1,5,7,11,13,17,19,23,25,35,55,65,77,91,119,133,143,175,275,325,385,

%T 455,595,665,715,935,1001,1309,1463,1547,1729,1925,2275,2975,3325,

%U 3575,4675,5005,6545,7315,7735,8645

%N Analog of A085635, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2".

%C Positions k where R(k) = A290732(k)/k, achieves a new minimum.

%H Hugo Pfoertner, <a href="/A290729/b290729.txt">Table of n, a(n) for n = 1..182</a>

%H Andreas Enge, William Hart, Fredrik Johansson, <a href="http://arxiv.org/abs/1608.06810">Short addition sequences for theta functions</a>, arXiv:1608.06810 [math.NT], 2016-2018. See Table 6.

%t a[n_] := Product[{p, e} = pe; If[p <= 3, p^e, (p^e - p^(e-1))/2 + (p^(e-1) - p^(Mod[e+1, 2]))/(2*(p+1)) + 1], {pe, FactorInteger[n]}];

%t r = 2; Reap[For[j=1, j <= 10^4, j = j+1, t = a[j]/j; If[t<r, r = t; Sow[j] ]]][[2, 1]] (* _Jean-François Alcover_, Oct 02 2018, after _Hugo Pfoertner_ *)

%o (PARI) a290732(n)={my(f=factor(n));prod(k=1,#f~,my([p,e]=f[k,]);if(p<=3,p^e,(p^e-p^(e-1))/2+(p^(e-1)-p^((e+1)%2))/(2*(p+1))+1))}

%o my(r=2);for(j=1,10001,my(t=a290732(j)/j);if(t<r,r=t;print1(j,", "))) \\ _Hugo Pfoertner_, Aug 26 2018

%Y Cf. A000326, A085635, A084848, A290727, A290728, A290730, A290732.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Aug 10 2017

%E a(1) corrected by _Hugo Pfoertner_, Aug 26 2018