Brian Darrow, Jr. - cp's OEIS frontend

User: Brian Darrow, Jr.

Brian Darrow, Jr. has authored 2 sequences.

A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.

2, 10, 60, 396, 2976, 25056, 234720, 2423520, 27371520, 335819520, 4449150720, 63318931200, 963548006400, 15614378035200, 268480048435200, 4882321001779200, 93627018326016000, 1888394741194752000, 39963486306078720000, 885457095215616000000

Offset: 2

Brian Darrow, Jr. and Joe Fields, Feb 20 2024

nonn

These are the numerators in calculating an expected value. The expectation of the number of steps one takes in marking the elements of a predetermined list before reaching a state where only isolated unmarked entries remain.

			For n=5, an example vector of isolated 0's is 01011, which has k=3 1's.
For n=3, the following paths (from 000 to 111) reach isolated 0's at k=1 many 1's (010):
  000,010,011,111
  000,010,110,111
The following paths reach isolated 0's only at k=2 1's:
  000,100,110,111
  000,100,101,111
  000,001,101,111
  000,001,011,111
So 2 paths of k=1 and 4 paths of k=2 are weighted total a(3) = 2*1 + 4*2 = 10.

Cf. A067331.

a(n) = sum(k=n\2, n-1, k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!) \\ Andrew Howroyd, Feb 23 2024

k, n = var('k,n')
sum((binomial(k+1,n-k)-binomial(k-1,n-k))*factorial(k)*factorial(n-k), k, floor(n/2),n-1)

a(n) = Sum_{k=floor(n/2)..n-1} k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!.

A363992 The number of ways 2n can be expressed as the sum of an odd prime number and an odd nonprime, both of which are relatively prime to n.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 3, 3, 1, 6, 3, 1, 8, 4, 2, 6, 6, 3, 5, 7, 4, 8, 8, 2, 12, 7, 3, 13, 6, 6, 11, 9, 4, 12, 12, 4, 13, 13, 3, 14, 14, 8, 17, 11, 7, 15, 15, 10, 14, 13, 7, 16, 18, 3, 22, 18, 7, 24, 14, 11, 20, 20, 14, 17, 18, 10, 22, 22, 8

Offset: 0

Brian Darrow, Jr., Jun 30 2023

nonn

			For n=24 (2n=48), we have a(24)=3 since 48=1+47, 48=13+35, and 48=23+25. These are the only sums containing one prime and one nonprime, both of which are relatively prime to n.

Cf. A002375, A141095.

f:= proc(n) local k;
   nops(select(k -> igcd(n,k) = 1 and igcd(n,2*n-k) = 1 and isprime(k) and not isprime(2*n-k), [seq(k,k=1..2*n-1,2)]))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 03 2023

def d(a):
    """
    This function returns the number of ways n=2a can be expressed as the sum of one prime number and an odd composite that are relatively prime to n
    """
    d=0
    for i in range(1,a+1):
        if ((is_prime(i) and not is_prime(2*a-i) and gcd(i,2*a-i) == 1)) or ((not is_prime(i) and is_prime(2*a-i) and gcd(i,2*a-i) == 1)):
            d=d+1
    return d

cp's OEIS Frontend

User: Brian Darrow, Jr.

A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.

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