cp's OEIS Frontend

Let p(i) be primes with p(1)=2, p(n)# the n-th primorial number, and h(n) the Jacobsthal function for primorial p(n)#. Conjecture: gcd(h(n), p(n+1)) = 1.

For a multiple m of a prime n, terms in this sequence give the number of contiguous numbers starting at m+1 which have at least one prime factor < n.

Consider a range s of the first n + 1 primes. Let p be the largest of these primes, i.e., A000040(n+1). Let P be the product of the first n primes, i.e., the primorial A002110(n), and let Q be the product of all the primes in s, i.e., the primorial A002110(n+1). Consider the reduced residue system R of primorial P, that is, those numbers 1 <= r < P such that gcd(r, P) = 1; therefore R = row n of A286941. For each n, we generate the multiples k = j*p, with 1 <= j <= P. For each k, we find the smallest residue r in R that exceeds k and take the difference m = r - k - 1. If no value in R exceeds k, then we use Q + 1 (which is also coprime to Q). Row n is thus a list of these m.

Alternatively, consider a multiple k = j*p, with 1 <= j <= P. We can compute m by iterating i such that the sum (i + k) is coprime to Q and subtracting 1. This technique is more efficient in terms of memory, as it does not require storing the reduced residue system of Q.

For n > 1: The penultimate value m on row n = A040976(n). The number of values m on row n is given by the sequence: 1,1,2,2,10,22,500,...

For n > 3: For any even x = m in row n, the number of x in row n is equal to the count of y in row n where y = x + 1. If x = 0, the count of x and y in row n = A000010(A002110(n-1)). For example, on row 4, A000010(A002110(4-1)) = 8, as 0 and 1 each occur 8 times on row 4. The sequence of counts of x and x+1 pairs on consecutive rows is given by the sequence A059861. For example, for x=0 and y=1 occurring 8 times on row 4, x=2 and y=3 occur 8-3=5 times on row 4 given by the value 3 in A059861. For example, for row 8, x=0 and y=1 occur A000010(A002110(8-1)) = 92160 times on row 8, and x=2 and y=3 occur 92160-22275=69885 times on row 8 given by the value 22275 in A059861.

For 3 < n < 9: The largest value on row n occurs twice, the pattern of occurrence is shown in table 1 of Ziller & Morack in the Links section.

Looking for prime factors > 5=prime(3) in 8=A005867(3) candidates mod 30=A002110(3) two candidates in the form 30k +- 7 with k > 0 never belong to a twin prime pair. Twin primes can be (30k-13, 30k-11) A331840, (30k-1, 30k +1) A176114, or (30k+11, 30k+13) A089160.

All terms are odd.

All terms except a(0) = 0 are odd.

Each prime number sets a new record.

This sequence is a repeating cycle 12, 7, 4, 7, 4, 7, 12, 3 of length A005867(4) = 8 = (prime(1)-1)*(prime(2)-1)*(prime(3)-1).

The mean of the cycle is prime(4) = 7.

The cycle is constructed from the sieve of Eratosthenes as follows.

In the first 2 steps of the sieve, the gaps between the deleted numbers are constant: gaps of 2 in step 1 when we delete multiples of 2, and gaps of 3 in step 2 when we delete multiples of 3.

In step 3, when we delete all multiples of 5, the gaps are alternately 7 and 3 (i.e., cycle [7,3]).

For this sequence, we look at the interesting cycle from step 4 (multiples of 7).

Excluding the final 3, the cycle has reflective symmetry: 12, 7, 4, 7, 4, 7, 12. This is true for every subsequent step of the sieve too.

The central element is 7 (BUT not all steps have their active prime number as the central element).

a(1) is A054272(4).

a(8) = 3, the first appearance of the last element of the cycle, corresponds to deletion of 217 = A002110(4)+7.

This sequence is an example demonstrating how an integer sequence (thus a rational number sequence) converges to distinct limits in all p-adic systems; that is, converges to 1 in 2-adic, to 2 in 3-adic, to k in prime(k)-adic, and so on.

Moreover, the rational number sequence a(n) / prime(n+1) ^ (primorial(n)^(n-1) * A005867(n)) converges to distinct limits in all p-adic systems as well as the real number system, with limit zero in real numbers, and limit k in prime(k)-adic, where k is any positive integer.

There is a simple relationship between this sequence and the number of primes < primorial(p), as given by A000849 and sequence A005867 which gives the number of composites in primorial(p+1) having (p+1) as their lowest prime factor: a(n) = n + A005867(n) - A000849(n) - 1. - Dennis Martin (dennis.martin(AT)dptechnology.com), Apr 15 2007

The sequence is strongly associated with A072752.

In A072752, you are looking for a maximum sized set of consecutive numbers where none are counted by Euler's phi(3*5*7*...*Pn); this sequence looks for a minimum sized set of consecutive numbers where all the numbers counted by Euler are included.

One candidate for A (not necessarily of minimum size) is the set {1, 2, 3,..., 3*5*..*Pn}, which has the requested number of coprime elements. This yields the simple upper bound a(n) <= A070826(n+1). - R. J. Mathar, May 03 2017