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A161510
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Number of primes formed as the sum of distinct divisors of n, counted with repetition.
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3
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0, 2, 1, 4, 1, 6, 1, 6, 2, 7, 1, 20, 1, 5, 4, 11, 1, 16, 1, 19, 5, 5, 1, 66, 2, 5, 4, 17, 1, 64, 1, 18, 4, 6, 6, 120, 1, 5, 5, 63, 1, 62, 1, 18, 11, 5, 1, 237, 1, 15, 3, 18, 1, 47, 6, 60, 5, 7, 1, 863, 1, 3, 20, 31, 6, 58, 1, 16, 3, 62, 1, 808, 1, 4, 13, 16, 4, 56, 1, 216, 5, 5, 1, 839, 5, 5
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OFFSET
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1,2
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COMMENTS
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That is, if a number has d divisors, then we compute all 2^d sums of distinct divisors and count how many primes are formed. Sequence A093893 lists the n that produce no primes except for the primes that divide n. The Mathematica code works well for numbers up to about 221760, which has 168 divisors and creates a polynomial of degree 950976. The coefficients of the prime powers of that polynomial sum to 28719307224839120896278355000770621322645671888269, the number of primes formed by the divisors of 221760. Records appear to occur at n=10 and n in A002182, the highly composite numbers.
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LINKS
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EXAMPLE
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a(4) = 4 because the divisors (1,2,4) produce 4 primes (2,1+2,1+4,1+2+4).
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MATHEMATICA
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CountPrimes[n_] := Module[{d=Divisors[n], t, lim, x}, t=CoefficientList[Product[1+x^d[[i]], {i, Length[d]}], x]; lim=PrimePi[Length[t]-1]; Plus@@t[[1+Prime[Range[lim]]]]]; Table[CountPrimes[n], {n, 100}]
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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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