%I #43 Feb 04 2014 00:46:40
%S 210,630,1050,1470,1890,2310,2730,3150,3570,3990,4410,4830,5250,5670,
%T 6090,6510,6930,7350,7770,8190,8610,9030,9450,9870,10290,10710,11130,
%U 11550,11970,12390,12810,13230,13650,14070,14490,14910,15330,15750,16170,16590,17010
%N Numbers n such that smallest number not dividing n^2 (A236454) is different from smallest prime not dividing n (A053669).
%C 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)}.
%C All terms are multiples of 210 = 2*3*5*7, the fourth primorial, A002110(4).
%C 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.
%C Note that a more comprehensive description for which terms are included is still lacking. Compare for example to the third definition of A055926.
%C At least we know the following:
%C 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:
%C 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.
%C 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.
%C If the least nondividing prime is 7, but 2, 3 and 5 are present, then we have A053669(n) = A236454(n) = 7.
%C 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.
%C 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.
%C 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.
%C 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.
%H Antti Karttunen, <a href="/A235921/b235921.txt">Table of n, a(n) for n = 1..10000</a>
%H Antti Karttunen, <a href="/A235921/a235921.txt">C-program for computing the first few terms empirically</a>
%H <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%e 210 (= 2*3*5*7) is a member, because A053669(210)=11, while A236454(210) = A007978(210*210) = A007978(44100) = 8.
%e 446185740 (= 2*2*3*5*7*11*13*17*19*23) is a member, because A053669(446185740) = 29, while A236454(446185740) = 27, as there is only one 3 present in 446185740, which means that its square is only divisible by 9, but not by 27 = 3^3.
%o (Scheme, with _Antti Karttunen_'s IntSeq-library, two variants):
%o (define A235921 (MATCHING-POS 1 1 (lambda (n) (not (= (A236454 n) (A053669 n))))))
%o (define A235921 (MATCHING-POS 1 1 (lambda (n) (not (= (A071222 (- n 1)) (A055874 (A000290 n)))))))
%Y Cf. A053669, A071222, A236454, A055874, A235918, A236432, A002110, A034386, A232099, A055926.
%K nonn
%O 1,1
%A _Antti Karttunen_ and _Michel Marcus_, Jan 17 2014