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A281891
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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.
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5
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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, 1830, 1, 0, 1, 454, 115432, 2744088, 2822340, 24270, 1, 0, 1, 1394, 1827592, 319881696, 3913037880, 496348740, 418350, 1, 0, 1, 4246, 28390552, 35924741232, 5079363328560, 9082206410040, 147569907780, 8040810, 1
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OFFSET
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0,9
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COMMENTS
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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.
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.
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.
Intermediate values in the calculation of this sequence appear in A281890.
If n > 0, A(n,1) = A053144(n) in accordance with the comment on A053144 dated Apr 08 2010.
A(2,k) = A027649(k) = 2*(3^k) - 2^k.
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LINKS
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FORMULA
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A(n,0) = 1 for n >= 0, A(0,k) = 0 for k >= 1.
A(n,k) = prime(n)^k * A(n-1,k) + A281890(n,k) for n >= 1, k >= 1.
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EXAMPLE
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The table starts:
1 0 0 0 0 0 0 ...
1 1 1 1 1 1 1 ...
1 4 14 46 146 454 1394 ...
1 22 412 7072 115432 1827592 28390552 ...
1 162 22164 2744088 319881696 35924741232 ...
1 1830 2822340 3913037880 5079363328560 ...
1 24270 496348740 9082206410040 ...
...
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.
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CROSSREFS
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A079474 re-read as a square array gives values of primorial(n)^k = A002110(n)^k.
The values in the body of the factorization table described in the author's comments are in the irregular array A027746.
A096294 gives the equivalent array for integers expressed as a product of prime powers.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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