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A141829 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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) also equals the number of positive integers k, k <= p(n+1)-p(k), that divide (p(n)+k-1).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MAPLE

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

MATHEMATICA

Table[Function[{p, q}, DivisorSum[p - 1, 1 &, # <= q - p &]] @@ {Prime@ n, Prime[n + 1]}, {n, 105}] (* Michael De Vlieger, Oct 25 2017 *)

PROG

(PARI) a(n) = #select(x->(x <= prime(n+1)-prime(n)), divisors(prime(n)-1)); \\ Michel Marcus, Oct 26 2017

CROSSREFS

Cf. A141830, A141831.

Sequence in context: A083901 A274517 A038148 * A111336 A083902 A205562

Adjacent sequences:  A141826 A141827 A141828 * A141830 A141831 A141832

KEYWORD

nonn

AUTHOR

Leroy Quet, Jul 09 2008

EXTENSIONS

Extended beyond a(17) by R. J. Mathar, Aug 08 2008

STATUS

approved

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Last modified September 21 23:55 EDT 2019. Contains 327286 sequences. (Running on oeis4.)