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A141829
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a(n) = the number of positive divisors of (p(n)-1) that are each <= p(n+1)-p(n), where p(n) is the n-th prime.
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3
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1, 2, 2, 3, 2, 4, 2, 3, 2, 2, 5, 4, 2, 3, 2, 3, 2, 6, 3, 2, 5, 3, 2, 4, 4, 2, 3, 2, 4, 6, 3, 3, 2, 4, 2, 5, 5, 3, 2, 3, 2, 8, 2, 4, 2, 6, 7, 3, 2, 4, 3, 2, 8, 3, 3, 2, 2, 5, 4, 2, 4, 3, 3, 2, 4, 3, 5, 7, 2, 4, 3, 2, 4, 5, 3, 2, 3, 4, 5, 6, 2, 8, 2, 5, 3, 2, 5, 4, 2, 3, 2, 2, 3, 4, 3, 2, 3, 2, 6, 6, 5, 3, 2, 2, 5
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OFFSET
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1,2
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COMMENTS
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a(n) also equals the number of positive integers k, k <= p(n+1)-p(k), that divide (p(n)+k-1).
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LINKS
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EXAMPLE
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The 16th prime is 53 and the 17th prime is 59. So the divisors of 53-1=52 that are <= 59-53=6 are 1,2,4. There are three such divisors.
Also, 53 is divisible by 1. 54 is divisible by 2. 55 is not divisible by 3. 56 is divisible by 4. 57 is not divisible by 5. And 58 is not divisible 6. So in the span of integers p(16)=53 to p(17)-1=58, there are 3 integers k where k divides (p(16)+k-1). So a(16) = 3.
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MAPLE
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A141829 := proc(n) local p, q, a, d ; p := ithprime(n) ; q := nextprime(p) ; a := 0 ; for d in numtheory[divisors](p-1) do if d <= q-p then a :=a+1 ; fi; od: RETURN(a) ; end: for n from 1 to 200 do printf("%a, ", A141829(n)) ; od: # R. J. Mathar, Aug 08 2008
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MATHEMATICA
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Table[Function[{p, q}, DivisorSum[p - 1, 1 &, # <= q - p &]] @@ {Prime@ n, Prime[n + 1]}, {n, 105}] (* Michael De Vlieger, Oct 25 2017 *)
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PROG
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(PARI) a(n) = #select(x->(x <= prime(n+1)-prime(n)), divisors(prime(n)-1)); \\ Michel Marcus, Oct 26 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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