cp's OEIS Frontend

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

%I A223881 #62 May 01 2025 08:32:25
%S A223881 3,2,19,31,37,79,41,97,53,139,71,157,83,199,211,229,131,271,137,307,
%T A223881 331,337,173,367,379,197,439,227,499,263,547,281,577,293,197,199,601,
%U A223881 607,619,661,227,229,691,239,727,383,269,811,829,283,431,877,467,937,313
%N A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.
%C A223881 It appears that all terms are primes.
%C A223881 From _Alexander R. Povolotsky_, Apr 26 2025: (Start)
%C A223881 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.
%C A223881 Additionally, the approximation formulas for each of the four subsets were derived.
%C A223881 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.
%C A223881 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.)
%C A223881 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.
%C A223881 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.
%C A223881 Also it appears that:
%C A223881 1. The first group constitutes a sequence, such that for n>=1, a(n) = A005382(n+6).
%C A223881 2. The third group constitutes a sequence, such that for n>1, a(n) = A158015(n+20).
%C A223881 3. The fourth group constitutes a sequence, such that for n>=1, a(n) = A158016(n+32).
%C A223881 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)
%H A223881 Paolo P. Lava, <a href="/A223881/b223881.txt">Table of n, a(n) for n = 1..1000</a>
%H A223881 Alexander R. Povolotsky, Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/5030561/conjecture-when-fracm-textprimem-11-m1-is-not-an-integer-the">Conjecture regarding this sequence</a>, 2025.
%H A223881 Alexander R. Povolotsky, <a href="/A223881/a223881.txt">Reindexed group1 primes</a>, 2025.
%H A223881 Alexander R. Povolotsky, <a href="/A223881/a223881_1.txt">Reindexed group2 primes</a>, 2025.
%H A223881 Alexander R. Povolotsky, <a href="/A223881/a223881_2.txt">Reindexed group3 primes</a>, 2025.
%H A223881 Alexander R. Povolotsky, <a href="/A223881/a223881_3.txt">Reindexed group4 primes</a>, 2025.
%t A223881 Denominator[Select[Table[m!/(Prime[m - 1] + 1), {m, 2, 300}], ! IntegerQ[#] &]] (* _T. D. Noe_, May 03 2013 *)
%o A223881 (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
%Y A223881 Cf. A007528, A024898, A002476, A091178.
%K A223881 nonn
%O A223881 1,1
%A A223881 _Alexander R. Povolotsky_, May 01 2013