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User:Peter Munn

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BSc Mathematics, University of Warwick, 1977, with a subsequent career in IT.

Main current interest is in the statistics of prime factorisations. A few years ago, looking at the prime factorisations of an aliquot sequence, factors listed in nondecreasing size order, I became interested in how the magnitude of the factors grew "on average" within each factorisation.

In 2015 I started looking at what could be determined about this for prime factorisations of integers in general, and realised it is analisable with simple methods if (for the statistics) a null "n-th least prime factor" is ordered above the other values. Given that the proportion of integers in 1..m that don't have such a factor tends to zero as m goes to infinity, this seems a sensible choice.

My interest focussed on statistical points, starting with medians, of the n-th least factors (in the sense operands) in the prime factorisations - that is the nth least list members in the prime factorization as the OEIS wiki defines it. Having only multiplication operators, the prime factorisation feels purer and more fundamental than the canonical prime power factorisation, and A281889 (3, 7, 433, 9257821) is the result.

When I found a straightforward way to define this sequence without even a reference to primes, this felt like it justified my preference, but the statistics of least factors in the "purer" prime factorisation also looks to be a less studied area. [I guess this is because the prime power factorisation has more application in number theory, in which case my preference is also "purer" in the sense pure vs. applied.]

Through Google I found De Koninck effectively publishes the equivalent medians for prime power factorisations under the entry for 37 in _Those Fascinating Numbers_, but I have yet to find the higher numbers of A281889 elsewhere. (Those prime power medians I have added as A284411.)

I pondered whether to have "2" or "3" as the first term. "3" seemed to fit best with my particular analytical approach, and the definition I devised (with respect to prime factorisations) appeared robust to several small changes. A first term of "2" seems to fit most easily with a more empirical approach, looking at bounded sets for the n-th least prime factor of only those integers that have n or more factors in their prime factorisation. ______

I first became interested in contributing to OEIS because of a simple variation on Eratosthenes sieve that I programmed in Fortran II whilst learning programming in 1974. I initially called the resulting sequence "herd numbers" on the basis of their distribution after hand-calculating those no greater than 100, but I now prefer the term "clumpy numbers".

However, I got side-tracked by interest in the statistics of prime factorisations, and then life intervened. When I next had time for OEIS I thought of submitting my herd/clumpy numbers sequence as a simpler start as a sequence author. Except, guess what, someone had submitted it (A270877) in the meantime. This encouraged me to submit, as my first sequence, a list of runs of herd numbers. So my authorship started January 2017 with A281256.