What is the maximal number of semiprimes among 100 consecutive numbers? 38?
We might never know. Considering only divisibility by small semiprimes, one can find a sequence of only 69 consecutive numbers, 39 of which are not divisible by any semiprime <= 69. For examples, 42117702927300*N + 13684292903, 05, 06, 07, 09, 11, 13, 14, 15, 17, 18, 21, 23, 26, 27, 29, 31, 33, 35, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 59, 61, 62, 63, 66, 67, 69, 71. Maybe for some choice of N, all of these are semiprime. But I despair of finding such an N. The minimum span (last - first + 1) of N semiprimes goes 1,2,3,5,6,7,9,11,13,14,16,17,19,21,... or does it? If that's correct, it (or the incremented sequence) would surely be in the OEIS: where did I go wrong? How does the density of semiprimes vary with the size? -- Don Reble djr@nk.ca
Paraphrasing Hardy & Wright, section 22.18, Thm 437: The percentage of numbers less than X which have K+1 prime factors is asymptotic to (loglog X)^K ------------------ K! log X Here, repeated primes are counted as additional prime factors, so 12 = 2 * 2 * 3 has three prime factors. Fine point: The asymptotic percentage is the same if you don't include the numbers with square factors. K=0 is the regular Prime Number Theorem. Memory aid: The sum over K is the expansion for e^loglogX, minus 1, divided by logX. For large X, the percentage is nearly the same for X and cX, so we can loosely read this as saying that the probability that X has K prime factors is the same expression. The relative likelihood of two prime factors versus prime is a factor of loglogX. A third prime factor multiplies the likelihood by another factor of loglogX / 2. And so on. H & W devotes 2.5 pages to the proof, but I think it boils down to writing out the sum of 1, taken over the appropriate set of N <= X, and approximating with the appropriate integral based on PNT. I assume H & W is online somewhere. I'm working from the hardcopy 4th edition (1960), price 42 shillings, net. (!) I paid about $15; my local bookstore ordered it for me. Rich ----------- Quoting Don Reble <djr@nk.ca>:
How does the density of semiprimes vary with the size?
On Oct 19, 2016, at 5:56 PM, rcs@xmission.com wrote:
Paraphrasing Hardy & Wright, section 22.18, Thm 437:
The percentage of numbers less than X which have K+1 prime factors is asymptotic to
(loglog X)^K ------------------ K! log X
Here, repeated primes are counted as additional prime factors, so 12 = 2 * 2 * 3 has three prime factors.
Fine point: The asymptotic percentage is the same if you don't include the numbers with square factors.
K=0 is the regular Prime Number Theorem.
Memory aid: The sum over K is the expansion for e^loglogX, minus 1, divided by logX. Or, think of the "excess" count of prime factors K being Poisson-distributed with mean loglog X.
Fun fact: loglog autocorrects to dogleg. -Veit
For large X, the percentage is nearly the same for X and cX, so we can loosely read this as saying that the probability that X has K prime factors is the same expression. The relative likelihood of two prime factors versus prime is a factor of loglogX. A third prime factor multiplies the likelihood by another factor of loglogX / 2. And so on.
H & W devotes 2.5 pages to the proof, but I think it boils down to writing out the sum of 1, taken over the appropriate set of N <= X, and approximating with the appropriate integral based on PNT.
I assume H & W is online somewhere. I'm working from the hardcopy 4th edition (1960), price 42 shillings, net. (!) I paid about $15; my local bookstore ordered it for me.
Rich
----------- Quoting Don Reble <djr@nk.ca>:
How does the density of semiprimes vary with the size?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
* rcs@xmission.com <rcs@xmission.com> [Oct 20. 2016 13:37]:
[...]
I assume H & W is online somewhere. I'm working from the hardcopy 4th edition (1960), price 42 shillings, net. (!) I paid about $15; my local bookstore ordered it for me.
Yes it is: https://archive.org/details/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.Har... Best regards, jj
Rich
----------- Quoting Don Reble <djr@nk.ca>:
How does the density of semiprimes vary with the size?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Don Reble -
Joerg Arndt -
rcs＠xmission.com -
Veit Elser