A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.

3, 2, 19, 31, 37, 79, 41, 97, 53, 139, 71, 157, 83, 199, 211, 229, 131, 271, 137, 307, 331, 337, 173, 367, 379, 197, 439, 227, 499, 263, 547, 281, 577, 293, 197, 199, 601, 607, 619, 661, 227, 229, 691, 239, 727, 383, 269, 811, 829, 283, 431, 877, 467, 937, 313

Offset: 1

Author

Alexander R. Povolotsky, May 01 2013

Keywords

• nonn

Comments

It appears that all terms are primes.

From Alexander R. Povolotsky, Apr 26 2025: (Start)

The scatter plot reveals four distinct, well-separated, monotonically increasing curves. It became possible to extract the integers (all conjectured to be primes) corresponding to each of the four subsets.

Additionally, the approximation formulas for each of the four subsets were derived.

These four approximation formulas, given in the exponential form y=C_k*x^m were found to have a common slope: m=1.197311990 while their displacement coefficients are: C_1≈6.86845, C_2≈3.42058, C_3≈2.28335, C_4≈1.70460.

Notably, these displacement coefficients values exhibit a clear pattern: C_2≈C_1/2, C_3≈C_1/3, C_4≈C_1/4. (For instance, 3.42058≈6.86845/2, and so on.)

Above approximations were derived using general separation and approximation methods and do not specifically account for the fact that these values correspond to the prime numbers.

It appears that all primes in the groups 4, 2 and 1 are generated by the 6*k+1 formula, and so primes in the above groups constitute three subsets of A002476 terms, while all primes in the group 3 are generated by the 2*k+1 formula, and so primes in that group constitute a subset of the terms presented in A000040.

Also it appears that:

1. The first group constitutes a sequence, such that for n>=1, a(n) = A005382(n+6).

2. The third group constitutes a sequence, such that for n>1, a(n) = A158015(n+20).

3. The fourth group constitutes a sequence, such that for n>=1, a(n) = A158016(n+32).

The text files containing the primes, corresponding to the above discussed four groups, where primes are indexed against their position in the complete primes listing (see OEIS's A000040), are viewable and downloadable at the below links section. (End)

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..1000
• Alexander R. Povolotsky, Mathematics StackExchange, Conjecture regarding this sequence, 2025.
• Alexander R. Povolotsky, Reindexed group1 primes, 2025.
• Alexander R. Povolotsky, Reindexed group2 primes, 2025.
• Alexander R. Povolotsky, Reindexed group3 primes, 2025.
• Alexander R. Povolotsky, Reindexed group4 primes, 2025.

Crossrefs

Cf. A007528, A024898, A002476, A091178.

Programs

Mathematica

Denominator[Select[Table[m!/(Prime[m - 1] + 1), {m, 2, 300}], ! IntegerQ[#] &]] (* T. D. Noe, May 03 2013 *)

PARI

m=M=1;forprime(p=2,1e5,M*=m++;t=denominator(M/(p+1)); if(t>1, print1(t", "))) \\ Charles R Greathouse IV, May 08 2013