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 A118314 Divisor-indexed recurrence. 2

%I

%S 1,1,1,2,1,3,1,4,2,3,1,8,1,3,3,8,1,6,1,8,3,3,1,20,2,3,4,8,1,13,1,16,3,

%T 3,3,24,1,3,3,20,1,13,1,8,8,3,1,40,2,8,3,8,1,20,3,8,3,3,1,44

%N Divisor-indexed recurrence.

%C a(n) depends only on a(1) and a(2) and the exponents of the prime factorization, with multiplicity, also called the prime signature of n (see A025487), rather than the specific distinct prime factors. For a(1) = a(2) = 1 and for distinct primes p, q, r: a(p) = 1, a(p^n) = 2^(n-1), a(pq) = 3, a(pqr) = 13, a(p^2 q) = 8, a(p^3 q) = 20, a(p^4 q) = 40, a(p^2 q^2) = 24, a(p^2 qr) = 44.

%F a(1) = a(2) = 1; a(n) = SUM[d proper divisor of n] a(d). a(n) = SUM[d|n and d<n] a(d).

%e a(5) = a(1) = 1 because 1 is the only proper divisor of 5; indeed, this applies to any prime, a(p) = 1.

%e a(6) = a(1) + a(2) + a(3) = 1 + 1 + 1 = 3, since the proper divisors of n are {1,2,3}; this applies to any nonsquare semiprime a(pq) = a(1) + a(p) + a(q) = 1 + 1 + 1 = 3, since the proper divisors of pq in A006881 are {1,p,q}.

%e a(30) = a(1) + a(2) + a(3) + a(5) + a(6) + a(10) + a(15) = 1 + 1 + 1 + 1 + 3 + 3 + 3 = 13; this applies to any sphenic number A007304.

%Y Cf. A000040, A000961, A007304.

%K easy,nonn

%O 1,4

%A _Jonathan Vos Post_, May 14 2006

%E Offset corrected by _Reinhard Zumkeller_, May 19 2011

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