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%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
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