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A367140
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a(n) = Sum_{prime p|n} p^A001222(n).
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2
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0, 2, 3, 4, 5, 13, 7, 8, 9, 29, 11, 35, 13, 53, 34, 16, 17, 35, 19, 133, 58, 125, 23, 97, 25, 173, 27, 351, 29, 160, 31, 32, 130, 293, 74, 97, 37, 365, 178, 641, 41, 378, 43, 1339, 152, 533, 47, 275, 49, 133, 298, 2205, 53, 97, 146, 2417, 370, 845, 59, 722, 61
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OFFSET
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1,2
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COMMENTS
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The definition implies a(n) >= n, with equality only when n is a term in A000961.
This sequence contains sums of distinct prime powers, but not all such sums are terms (6 is not a term since it cannot be expressed as the sum of powers of distinct primes). If m (a non prime power) is a term it must occur as a(n) = m for some n < m, for if not there is no way it can occur later (if so we would have n > m and a(n) = m, but then a(n) < n; contradiction); see Example.
Some primes occur twice; once as fixed points a(p) = p, and once as a(m) = p for some m < p (e.g. 13 = a(6) = a(13) and 29 = a(10) = a(29)).
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LINKS
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Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
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FORMULA
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a(n) = n for n a term >1 in A000961.
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EXAMPLE
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a(1) = 0, the empty sum.
a(6) = a(2*3) = 2^2 + 3^2 = 13.
a(12) = a(2^2*3) = 2^3 + 3^3 = 8 + 27 = 35.
a(18) = a(2^1*3^2) = 2^3 + 3^3 = 35.
15 is expressible as the sum of prime powers (2^2 + 11^1) but it is not a term since it has not occurred prior to a(15), likewise 18 (5 + 13)) is not a term since it has not occurred prior to a(18).
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MATHEMATICA
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Table[Function[k, DivisorSum[n, #^k &, PrimeQ]][PrimeOmega[n]], {n, 61}] (* Michael De Vlieger, Nov 06 2023 *)
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PROG
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(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]^bigomega(f)); \\ Michel Marcus, Nov 06 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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