OFFSET
1,1
COMMENTS
Equivalent definition is: numbers n such that {the largest m such that 1, 2, ..., m divide n^2 = A055874(n^2) = A235918(n)} is different from {the smallest k such that gcd(n-1,k) = gcd(n,k+1) = A071222(n-1)}.
All terms are multiples of 210 = 2*3*5*7, the fourth primorial, A002110(4).
The first term which is an even multiple of 210 (i.e., 210 times an even number), is 446185740 = 2124694 * 210 = 2*223092870 = 2*A002110(9) = 2*A034386(23). Note that 23 is the 9th prime, and 223092870 is its primorial. Thus this sequence differs from its subsequence, A236432, "the odd multiples of 210" = (2n-1)*210, for the first time at n = 1062348, where a(n) = 446185740, while A236432(n) = 446185950.
Note that a more comprehensive description for which terms are included is still lacking. Compare for example to the third definition of A055926.
At least we know the following:
If a number is not divisible by 210, then it cannot be a member, as then it is "missing" (i.e., not divisible by) one of those primes, 2, 3, 5 or 7, and thus its square is also "missing" the same prime. In more detail, this follows because:
If the least nondividing prime is 2, then A053669(n) = A236454(n) = 2. If the least nondividing prime is 3, then A053669(n) = A236454(n) = 3.
If the least nondividing prime is 5 (so 2 and 3 are present), then as 2|n and 4|(n^2), we have A053669(n) = A236454(n) = 5.
If the least nondividing prime is 7, but 2, 3 and 5 are present, then we have A053669(n) = A236454(n) = 7.
On the other hand, when n is an odd multiple of 210 (= 2*3*5*7), i.e., (2k+1)*210, so that its prime factorization is of the form 2*3*5*7*{zero or more additional odd prime factors}, then A053669(n) must be at least 11, the next prime after 7, which is certainly different from A236454(n) = A007978(n^2) which must be 8, as then 4 is the highest power of 2 dividing n^2.
In contrast to that, when n is an even multiple of 210, so that its prime factorization is of the form 2*2*3*5*7*{zero or more additional prime factors}, then also all the composites 8, 9, 10, 12, 14, 15, 16, 18 and 20 divide n^2, thus if A053669(n) is any prime from 11 to 19, A236454(n) will return the same result.
However, if n is of the form k*446185740, where k is not a multiple of 3, so that the prime factorization of n is 2*2*3*5*7*11*13*17*19*23*{zero or more additional prime factors, all different from 3}, then A053669(n) must be at least 29 (next prime after 23), but A236454(n) = 27, because then 9 is the highest power of 3 dividing n^2.
The pattern continues indefinitely: If n is of the form (2k+1)*2*3*200560490130, where 200560490130 = A002110(11), so that n has a prime factorization of the form 2*2*3*3*5*7*11*13*17*19*23*29*31*{zero or more additional odd prime factors}, then A053669(n) must be at least 37, while A236454(n) = 32 = 2^5, because then 16 is the highest power of 2 dividing n^2.
LINKS
EXAMPLE
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen and Michel Marcus, Jan 17 2014
STATUS
approved