A000113 - OEIS

%I #26 Oct 23 2022 02:54:44

%S 1,3,4,3,6,12,8,6,4,18,12,12,14,24,24,6,18,12,20,18,32,36,24,24,30,42,

%T 12,24,30,72,32,12,48,54,48,12,38,60,56,36,42,96,44,36,24,72,48,24,56,

%U 90,72,42,54,36,72,48,80,90,60,72,62,96,32,12,84,144,68

%N Number of transformation groups of order n.

%C A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Jan 03 2017

%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 139.

%H T. D. Noe, <a href="/A000113/b000113.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F Let psi(m) = A001615(m) (Dedekind's psi function). Write n = 2^i*3^j*k, where (6,k) = 1 and let i' = floor(i/2) for i < 6, i' = 3 for i >= 6; let j' = 0 for j = 0 or 1, j' = 1 for j >= 2. Then a(n) = psi(n/(2^i'*3^j')) = psi(n)/(2^i'*3^j').

%F Multiplicative with a(2^e)=3*2^Floor[(e-1)/2] for 0<e<7, a(2^e)=3*2^(e-4) for e>=7, a(3^e)=4 for 0<e<3, a(3^e)=4*3^(e-2) for e>=3 and a(p^e)=(p+1)*p^(e-1) for p>3. - _T. D. Noe_, Nov 14 2006

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 43501/(7680*Pi^2) = 0.573902... . - _Amiram Eldar_, Oct 23 2022

%e G.f. = x + 3*x^2 + 4*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + ...

%t psi[n_] := n*DivisorSum[n, MoebiusMu[#]^2/#&]; a[n_] := (i=IntegerExponent[ n, 2]; j=IntegerExponent[n, 3]; ip = If[i<6, Floor[i/2], 3]; jp = If[j<2, 0, 1]; psi[n]/(2^ip*3^jp)); Array[a, 60] (* _Jean-François Alcover_, Feb 04 2016 *)

%t a[ n_] := If[ n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors @ n}] / (2^Min[3, Quotient[IntegerExponent[n, 2], 2]] 3^Boole[1 < IntegerExponent[n, 3]]) ]; (* _Michael Somos_, Jan 03 2017 *)

%o (PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, moebius(d)^2 / d) / (2^min(3, valuation(n, 2)\2) * 3^(1 < valuation(n, 3))))}; /* _Michael Somos_, Jan 03 2017 */

%Y Cf. A001615.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_