cp's OEIS Frontend

A290727 Analog of A085635, replacing "quadratic residue" (X^2) with "value of X^2+X".

%I A290727 #21 Sep 03 2018 11:07:48
%S A290727 1,2,6,10,14,18,30,42,66,70,90,126,198,210,330,390,450,630,990,1170,
%T A290727 1386,1638,2142,2310,2730,3150,4950,5850,6930,8190,10710,11970,12870,
%U A290727 16830,18018,23562,26334,27846,30030,34650
%N A290727 Analog of A085635, replacing "quadratic residue" (X^2) with "value of X^2+X".
%C A290727 Positions where R(k) = A290731(k)/k achieves a new minimum, i.e., R(k) < R(j), j = 0..k-1, R(0) = 2.
%H A290727 Hugo Pfoertner, <a href="/A290727/b290727.txt">Table of n, a(n) for n = 1..116</a>
%H A290727 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 5.
%t A290727 a290731[n_] := Product[{p, e} = pe; If[p == 2, 2^(e-1), 1+Quotient[p^(e+1), (2p+2)]], {pe, FactorInteger[n]}];
%t A290727 Reap[For[r = 2; k = 1, k <= 35000, k++, t = a290731[k]/k; If[t<r, r = t; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Sep 03 2018, from PARI *)
%o A290727 (PARI) a290731(n)={my(f=factor(n));prod(i=1,#f~,my([p,e]=f[i,]);if(p==2,2^(e-1),1+p^(e+1)\(2*p+2)))} \\ from _Andrew Howroyd_
%o A290727 r=2;for(k=1,40000,t=a290731(k)/k;if(t<r,r=t;print1(k,", "))) \\ _Hugo Pfoertner_, Aug 23 2018
%Y A290727 Cf. A085635, A084848, A290728, A290731.
%K A290727 nonn
%O A290727 1,2
%A A290727 _N. J. A. Sloane_, Aug 10 2017
%E A290727 More terms from _Hugo Pfoertner_, Aug 22 2018
%E A290727 Initial term added by _Hugo Pfoertner_, Aug 23 2018