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A163671 Expansion of Sum_( x^k / (1 - x^(k^2)) ).

%I A163671 #23 Feb 16 2025 08:33:11
%S A163671 1,2,2,2,2,3,2,2,2,3,2,3,2,3,2,2,2,3,2,3,3,3,2,2,2,3,2,2,2,5,2,2,2,3,
%T A163671 2,3,2,3,3,2,2,4,2,2,2,3,2,3,2,3,2,3,2,3,3,3,3,3,2,2,2,3,2,2,2,4,2,3,
%U A163671 2,3,2,3,2,3,3,2,2,4,2,3,2,3,2,4,2,3,2,2,2,4,2,2,3,3,2,2,2,3,2,3,2,4,2,2,4
%N A163671 Expansion of Sum_( x^k / (1 - x^(k^2)) ).
%C A163671 Number of divisors d of n such that n/d == 1 (mod d). Such divisors must be unitary divisors.
%C A163671 Up to n = 10000, the largest value in this sequence is 7; the first occurrence of each integer up to 7 in the sequence is 1, 2, 6, 42, 30, 210, 2310; which except for 42 is the primorial numbers. However, a(30030) = 7.
%C A163671 If n is the product of two distinct primes then a(n) = 2 if and only if there are no nontrivial groups of order n. This relation does not hold if n is the product of 3 or more distinct primes or is not squarefree.
%C A163671 First occurrences: a(1) = 1, a(2) = 2, a(6) = 3, a(42) = 4, a(30) = 5, a(210) = 6, a(2310) = 7, a(87780) = 8, a(53130) = 9, a(7375830) = 10, a(172320330) = 11. - _Charles R Greathouse IV_, Jun 01 2016
%H A163671 Franklin T. Adams-Watters, <a href="/A163671/b163671.txt">Table of n, a(n) for n = 1..10000</a>
%H A163671 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor.</a>
%F A163671 a(n) = Sum_{k=1..n} (floor((k^2-k)/n) - floor((k^2-k-1)/n))*(floor(n/k) - floor((n-1)/k)). - _Anthony Browne_, Jun 01 2016
%F A163671 2 <= a(n) <= 2*omega(n) for n > 1. In particular a(p^e) = 2 for each prime p and each e > 0. - _Charles R Greathouse IV_, Jun 01 2016
%t A163671 Table[Sum[(Floor[(k^2 - k)/n] - Floor[(k^2 - k - 1)/n])*(Floor[n/k] - Floor[(n - 1)/k]), {k, 1, n}], {n, 1, 50}] (* _G. C. Greubel_, Aug 02 2017 *)
%o A163671 (PARI) al(n)=local(m);m=sqrtint(n);Vec(sum(k=1,m,(x^k+x*O(x^n))/(1-x^(k^2))+x^(m+1)/(1-x)))
%o A163671 (PARI) a(n)=local(d,r);r=0;d=divisors(n);for(k=1,#d,if((n-d[k])%d[k]^2==0,r++));r
%o A163671 (PARI) a(n)=sumdiv(n,d,(n-d)%d^2==0) \\ _Charles R Greathouse IV_, Jun 01 2016
%Y A163671 Cf. A000005, A001227, A002110, A003277, A050384, A006881, A034444.
%K A163671 nonn
%O A163671 1,2
%A A163671 _Franklin T. Adams-Watters_, Aug 02 2009