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A118314
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Divisor-indexed recurrence.
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2
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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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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.
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FORMULA
| 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).
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EXAMPLE
| a(5) = a(1) = 1 because 1 is the only proper divisor of 5; indeed, this applies to any prime, a(p) = 1.
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}.
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.
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CROSSREFS
| Cf. A000040, A000961, A007304.
Sequence in context: A079617 A079616 A097283 * A002033 A074206 A173801
Adjacent sequences: A118311 A118312 A118313 * A118315 A118316 A118317
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), May 14 2006
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EXTENSIONS
| Offset corrected by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 19 2011
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