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A137678 Consider the consecutive composite numbers between prime(n) and prime(n+1). Letting k=prime(n+1)-prime(n)-1, a(n) is the number of these numbers that have all primes factors less than k. 1
0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 1, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Grimm's conjecture states that for k consecutive composite numbers there are k distinct prime numbers such that each prime divides one of the composite numbers. As pointed out by Grimm, for a composite number c having a prime factor p>=k, we associate p with c. Hence there are only a(n) numbers remaining between prime(n) and prime(n+1) for which we need to associate a distinct prime. Puzzle 430 gives a heuristic algorithm for finding those a(n) primes. According to Puzzle 430, the largest known value of a(n) is 10, which occurs between the primes 31397 and 31469.

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. Monthly, 76 (1969), 1126-1128.

Carlos Rivera, Puzzle 430: Grimm's Conjecture

EXAMPLE

a(4)=1 because of the 3 numbers between 7 and 11 (8=2^3, 9=3^2, 10=2*5), only 8 has all of its prime factors less than 3. Similarly, a(9)=2 because of the 5 numbers between 23 and 29, only 24 and 27 have all prime factors less than 5.

MATHEMATICA

Table[p1=Prime[i]; p2=Prime[i+1]; cnt=0; k=p2-p1-1; Do[If[FactorInteger[n][[ -1, 1]]<k, cnt++ ], {n, p1+1, p2-1}]; cnt, {i, 150}]

PROG

(PARI) a(n) = my(p = prime(n), q = nextprime(p+1)); sum(c=p+1, q-1, vecmax(factor(c)[, 1]) < q-p-1); \\ Michel Marcus, Sep 30 2017

CROSSREFS

Cf. A059686, A101083.

Sequence in context: A256812 A292251 A231715 * A218855 A069517 A193526

Adjacent sequences:  A137675 A137676 A137677 * A137679 A137680 A137681

KEYWORD

nonn

AUTHOR

T. D. Noe, Feb 04 2008

STATUS

approved

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Last modified October 24 19:46 EDT 2020. Contains 338009 sequences. (Running on oeis4.)