[seqfan] A partition of Euler totient function
Vladimir Shevelev
shevelev at bgu.ac.il
Thu Oct 10 20:24:49 CEST 2013
Dear SeqFans,
Let A,B be two disjoint positive sequences such that A+B=N. Consider two sequences which give a partition of values of Euler totient function (A000010): 1) number of integers from A not exceeding n and respectively prime to n and 2) number of integers from B not exceeding n and respectively prime to n. Probably, earlier already there considered (A,B)-partitions of
{phi(n)} with different A,B. For example, in a trivial case, A, B are sequences of odd and even numbers respectively. Much more intersting case we obtain, if A=A000069 (odious numbers) and B=A001969 (evil numbers). Then we obtain
two sequences:
{a(n)}: 1, 1, 2, 2, 3, 1, 3, 2, 5, 2, 5, 3, 6, 3, 8, 4, 9, 4, 9, 5, 8, 5, 12, 5, 12, 6, 13, 5, 15, 5, 15, 8, 14,... (submitted as A230070);
{b(n)}: 0, 0, 0, 2, 1, 1, 3, 2, 1, 2, 5, 1, 6, 3, 0, 4, 7, 2, 9, 3, 4, 5, 10, 3, 8, 6, 5, 7, 13, 3, 15, 8, 6,... (submitted as A230120).
Naturally at least two questions arise: (1) For which n, a(n) < b(n)? The first such number is 28: a(28): a(28)=5 < b(28)=7; (2) For n=1,2,3,15, we have a(n)=phi(n). What other n's for which b(n)=0?
Best regards,
Vladimir
Shevelev Vladimir
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