A361372 Lexicographically earliest sequence of distinct positive numbers such that the number of occurrences of each prime number in the factorization of all terms a(1)..a(n) is at most one more than the number of occurrences of the next most frequently occurring prime.

1, 2, 3, 5, 6, 7, 10, 9, 11, 13, 17, 19, 23, 29, 31, 35, 4, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 15, 26, 79, 83, 89, 91, 21, 34, 97, 101, 103, 107, 109, 113, 121, 25, 57, 38, 127, 131, 137, 139, 143, 49, 14, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Author

Scott R. Shannon, Mar 27 2023

Keywords

• nonn
• look

Comments

The sequence generally consists of runs of prime 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.

Crossrefs

Cf. A027746, A011371, A000040.

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.