cp's OEIS Frontend

A281891 Square array A(n,k): number of integers having k or more factors less than prime(n+1) in their prime factorization, within any interval of primorial(n)^k positive integers.

%I A281891 #19 Apr 15 2017 11:07:46
%S A281891 1,0,1,0,1,1,0,1,4,1,0,1,14,22,1,0,1,46,412,162,1,0,1,146,7072,22164,
%T A281891 1830,1,0,1,454,115432,2744088,2822340,24270,1,0,1,1394,1827592,
%U A281891 319881696,3913037880,496348740,418350,1,0,1,4246,28390552,35924741232,5079363328560,9082206410040,147569907780,8040810,1
%N A281891 Square array A(n,k): number of integers having k or more factors less than prime(n+1) in their prime factorization, within any interval of primorial(n)^k positive integers.
%C A281891 Square array read by descending antidiagonals; A(n,k) with rows n >= 0, columns k >= 0. Prime factors are counted with multiplicity. Primorial(n) = A002110(n): product of first n primes.
%C A281891 Visualize the prime factorizations of the positive integers as a table with row headings giving each successive integer, and the primes of which the row heading is the product listed across the columns in nondecreasing order, repeated when necessary. Except for 1, which lacks prime factors, column 1 has the row heading's least prime factor, column 2 has a value for composite numbers but is blank for primes, and so on. This sequence measures precisely how frequently values up to and including the various primes occur in each column. This is possible because any given prime occurs cyclically in any given column, for the reason following.
%C A281891 The occurrence pattern of up to k factors of prime(n) in such prime factorizations has a fundamental period over the positive integers of prime(n)^k. The least common period for up to k factors of each of the first n primes is primorial(n)^k, and this covers everything that can affect the occurrence of prime(n) in the least k factors. Thus prime(n) is k-th least prime factor of integer m if and only if it is k-th least prime factor of m + primorial(n)^k.
%C A281891 Intermediate values in the calculation of this sequence appear in A281890.
%C A281891 If n > 0, A(n,1) = A053144(n) in accordance with the comment on A053144 dated Apr 08 2010.
%C A281891 A(2,k) = A027649(k) = 2*(3^k) - 2^k.
%F A281891 A(n,0) = 1 for n >= 0, A(0,k) = 0 for k >= 1.
%F A281891 A(n,k) = prime(n)^k * A(n-1,k) + A281890(n,k) for n >= 1, k >= 1.
%e A281891 The table starts:
%e A281891    1     0         0             0             0           0        0 ...
%e A281891    1     1         1             1             1           1        1 ...
%e A281891    1     4        14            46           146         454     1394 ...
%e A281891    1    22       412          7072        115432     1827592 28390552 ...
%e A281891    1   162     22164       2744088     319881696 35924741232    ...
%e A281891    1  1830   2822340    3913037880 5079363328560      ...
%e A281891    1 24270 496348740 9082206410040       ...
%e A281891    ...
%e A281891 Primes less than prime(2+1)=5 occur as second least factor 14 times in the prime factorizations of every interval of 36 = primorial(2)^2 positive integers (cf. A014673). Therefore, A(2,2) = 14.
%Y A281891 A079474 re-read as a square array gives values of primorial(n)^k = A002110(n)^k.
%Y A281891 The values in the body of the factorization table described in the author's comments are in the irregular array A027746.
%Y A281891 A096294 gives the equivalent array for integers expressed as a product of prime powers.
%Y A281891 Cf. A014673, A027649, A053144, A281890.
%K A281891 nonn,tabl
%O A281891 0,9
%A A281891 _Peter Munn_, Feb 08 2017
%E A281891 Edited by _M. F. Hasler_, Apr 14 2017