cp's OEIS Frontend

The sequence is Primorial rows of A308121.

Row n has length A005867(n).

Row n > 1 average value = A060753(n)/2.

Row n > 1 has sum = A002110(n-1)*A038110(n)/2.

First value on row(n) = A161527(n-1).

Last value on row(n) = A038110(n) for n > 2.

For n > 1, A060753(n) = Max(row) + Min(row).

For values x and y on row n > 1 at positions a and b on the row:

x + y = A060753(n), where a = A005867(n-1) - (b-1).

For n > 2 the penultimate value on row A002110(n) is given by

A038110(n)*A000040(n)-A060753(n).

Related identity:

A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

When the primorial base representation is expressed with decimal digits as here, the sequence stays unambiguous only up to the 317th prime, 2099, written as 96421, because after that primorial base digits larger than 9 would be needed.

By writing down terms from a(6) to a(46) (primes 13 .. 199):

201, 221, 301, 321, 421, 1001, 1101, 1121, 1201, 1221, 1321, 1421, 2001, 2101, 2121, 2201, 2301, 2321, 2421, 3101, 3121, 3201, 3221, 3301, 3321, 4101, 4121, 4221, 4301, 4421, 5001, 5101, 5201, 5221, 5321, 5421, 6001, 6121, 6201, 6221, 6301,

and then from a(48) to a(80) (primes 223 .. 409):

10201, 10221, 10301, 10321, 10421, 11001, 11121, 11221, 11321, 11421, 12001, 12101, 12121, 12201, 12321, 13101, 13121, 13201, 13221, 14001, 14101, 14221, 14301, 14321, 14421, 15101, 15201, 15301, 15321, 15421, 16101, 16121, 16301,

it is clearly seen that if n is a prime, then p+n is also likely to be prime, where p is the next higher primorial (A002110) > n. See also A324656.

Let primorial P(n) = A002110(n) and let r < P(n) be a number such that gcd(r, P(n)) = 1. Thus r is a residue in the reduced residue system (RRS) of P(n), and the number of r pertaining to P(n) is given by phi(P(n)) = A005867(n). We take the union of the first differences of the r in the RRS of P(n) to arrive at row n of this sequence.

Let L be the run length of numbers m in the cototient of a number k and let the first differences D in the RRS of k. The cototient includes any m such that at least 1 prime p | m also divides k, in other words, any m such that gcd(m, k) > 1. We note L = D - 1.

Row n of this sequence is the union of first differences of row n of A286941.

Let D be a primitive first difference as defined above. D is necessarily even since P(n) (for n > 0) is even and all r are odd.

Length of row n = A329815(n).

From Michael De Vlieger, May 18 2017: (Start)

Row n of a(n) is the list of numbers 1 <= k <= A002110(n) that are coprime to A002110(n-1).

A286941(n) and A279864(n) are subsets of a(n) such that the terms of the rows of each sequence combined and sorted comprise all the terms of a(n).

Row lengths = A005867(n) + A005867(n-1): {2, 3, 10, 56, 528, 6240, 97920, ...}.

1 is coprime to all n thus delimits the rows of a(n).

The smallest prime q in row n of a(n) is gpf(primorial(n)) = A006530(A002110(n)) = prime(n) by definition of primorial.

The smallest composite x in row n of a(n) is q^2 = A001248(n).

The Kummer number A057588(n) = A002110(n) - 1 is the largest term in row n of a(n). (End)

This sequence is the number of distinct terms in the first difference sequence for rows n in A286941 and A309497.

Number of distinct terms listed in row n of A331118. - Michael De Vlieger, Jul 11 2020

Sum of the integers up to A002110(n) and coprime to A002110(n).

The sequence gives the sum of row n of A286941(n).

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.