OFFSET

1,2

COMMENTS

The sequence generally consists of runs of primes numbers followed by a series of much smaller values; see the linked image. It is unknown if all numbers eventually appear. If they do it is likely that the powers of 2 take an extremely large number of terms before appearing. For example assuming the number 2^k was to be a term this would require one or more primes to have the same number of appearances as 2 while the current number of appearances of 2 plus k would have to be no more than 1 more than the number of occurrences of the most frequently occurring prime. In the first 5000 terms 8 = 2^3 has not appeared which suggests 2^k for very large values of k may never appear.

LINKS

Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.

FORMULA

a(4) = 5 as a(1) = 1, a(2) = 2, a(3) = 3 and the number of occurrences of both 2 and 3 in the factorization of these terms is one. Therefore a(4) cannot be 4 = 2*2 as the number of occurrences of 2 would then be three, which would be more than one more than the number of occurrences of 3.

a(5) = 6 = 2*3, and as a(4) = 5 the prime 5 has appeared once, so the number of occurrences of both 2 and 3 can increase by one.

a(16) = 35 = 5*7 as in the factorization of all terms up to a(15) the prime 2 has occurred three times, 3 has appeared four times, 5 has appeared two times while the primes between 7 to 31 inclusive have appeared once. Therefore 35 can be the next term as this adds one to both the occurrence counts of 5 and 7; 7 now a appears two times while 5 appears three times. Note that, example, 25 = 5*5 could not be chosen as this would add 2 to the appearance count of 5, leaving no prime appearing two times.

CROSSREFS

KEYWORD

nonn,look

AUTHOR

Scott R. Shannon, Mar 27 2023

STATUS

approved