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a(n) = 64 + A000720(n) - A365398(n).
3

%I #23 Sep 09 2023 06:48:00

%S 63,63,63,62,63,62,63,62,62,61,62,61,62,62,61,60,61,60,61,60,60,60,61,

%T 60,60,60,60,59,60,59,60,60,60,60,60,59,60,60,60,59,60,59,60,60,60,60,

%U 61,60,60,60,60,60,61,60,60,59,59,59,60,59,60,60,59,58,58

%N a(n) = 64 + A000720(n) - A365398(n).

%C It is conjectured that A365339(n) - PrimePi(n) = 64 for all n >= 31957 (Pollack et al.). Does a similar relation apply if one replaces Euler's totient by the sum of divisors function in A365339? In particular, note remark (4.4) by Terence Tao in the linked paper.

%C From _Chai Wah Wu_, Sep 08 2023: (Start)

%C a(n) seems to be decreasing for n=10^i:

%C a(1) = 63

%C a(10) = 61

%C a(100) = 58

%C a(1000) = 58

%C a(10^4) = 54

%C a(10^5) = 53

%C a(10^6) = 48

%C a(10^7) = 46

%C a(10^8) = 43

%C (End)

%H Paul Pollack, Carl Pomerance, and Enrique Treviño, <a href="https://math.dartmouth.edu/~carlp/MonotonePhi.pdf">Sets of monotonicity for Euler's totient function</a>, preprint. See M(n).

%H Paul Pollack, Carl Pomerance, and Enrique Treviño, <a href="https://doi.org/10.1007/s11139-012-9386-6">Sets of monotonicity for Euler's totient function</a>, Ramanujan J. 30 (2013), no. 3, pp. 379-398.

%H Terence Tao, <a href="https://arxiv.org/abs/2309.02325">Monotone non-decreasing sequences of the Euler totient function</a>, arXiv:2309.02325 [math.NT], 2023.

%F a(n)<=63. This is due to the fact that A000203(p) = p+1 for p prime, and therefore A365398(n) >= A000720(n)+1. - _Chai Wah Wu_, Sep 08 2023

%o (Python)

%o from bisect import bisect

%o from sympy import divisor_sigma, primepi

%o def A365397(n):

%o plist, qlist, c = tuple(divisor_sigma(i) for i in range(1,n+1)), [0]*(n+1), 0

%o for i in range(n):

%o qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i

%o c = max(c,a)

%o return 64+primepi(n)-c # _Chai Wah Wu_, Sep 08 2023

%Y Cf. A000720, A000203, A365398, A365400.

%K nonn

%O 1,1

%A _Peter Luschny_, Sep 08 2023