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A283848 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

%I A283848 #29 Mar 23 2019 12:38:15
%S A283848 8,23,32,86,128,339,512,1332,2048,5298,8192,21066,32768,83987,131072,
%T A283848 334966,524288,1336988,2097152,5338206,8388608,21321234,33554432,
%U A283848 85176636,134217728,340338398,536870912,1360073016,2147483648,5435820051,8589934592,21727481616,34359738368,86853790498,137438953472
%N A283848 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.
%H A283848 Robert Israel, <a href="/A283848/b283848.txt">Table of n, a(n) for n = 3..3318</a>
%H A283848 Shinsaku Fujita, <a href="https://doi.org/10.1246/bcsj.20160369">alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method</a>, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
%F A283848 If n is even, a(n) = (2*n)^(-1)*Sum_{d|n, d even} phi(d)*4^(n/d) + 5*2^(n-2). - _Robert Israel_, Aug 23 2018 after Fujita (2017), Eq. (101) (set n=2, m=n).
%F A283848 If n is odd, a(n) = 2^n. For the even bisection see A284711.
%p A283848 f:= proc(n) uses numtheory;
%p A283848   if n::even then (2*n)^(-1)*add(phi(d)*4^(n/d),d=select(type,divisors(n),even))+5*2^(n-2)
%p A283848   else 2^n
%p A283848   fi
%p A283848 end proc:
%p A283848 map(f, [$1..40]); # _Robert Israel_, Aug 23 2018
%t A283848 a[n_] := If[EvenQ[n], (2n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Select[ Divisors[n], EvenQ]}] + 5*2^(n-2), 2^n];
%t A283848 Table[a[n], {n, 3, 40}] (* _Jean-François Alcover_, Mar 23 2019, after _Robert Israel_ *)
%o A283848 (PARI) a(n) = if (n%2, 2^n, (2*n)^(-1)*sumdiv(n, d, if (!(d%2), eulerphi(d)*4^(n/d))) + 5*2^(n-2)); \\ _Michel Marcus_, Mar 23 2019
%Y A283848 The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K A283848 nonn
%O A283848 3,1
%A A283848 _N. J. A. Sloane_, Apr 01 2017
%E A283848 Edited and more terms by _Robert Israel_, Aug 23 2018