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 A365400 a(n) = 64 + A000720(n) - A365339(n). 10
 63, 63, 63, 62, 62, 62, 62, 62, 61, 61, 61, 61, 61, 61, 60, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 58, 58, 58, 58, 58, 58, 58, 58, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 56 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is conjectured that A365339(n) = PrimePi(n) + 64 for all n >= 31957 (Pollack et al.). Assuming this conjecture a(n) = 0 for n > 31956. a is not monotonically decreasing. LINKS Table of n, a(n) for n=1..65. Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M(n). Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, pp. 379-398. Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023. PROG (Julia) # Computes the first N terms of the sequence. using Nemo function A365400List(N) phi = Int64[i for i in 1:N + 1] for i in 2:N + 1 if phi[i] == i for j in i:i:N + 1 phi[j] -= div(phi[j], i) end end end lst = zeros(Int64, N) dyn = zeros(Int64, N) pi = 64 for n in 1:N p = phi[n] nxt = dyn[p] + 1 while p <= N && dyn[p] < nxt dyn[p] = nxt p += 1 end pi += is_prime(n) ? 1 : 0 lst[n] = pi - dyn[n] end return lst end println(A365400List(32000)) (Python) from bisect import bisect from sympy import totient, primepi def A365400(n): plist, qlist, c = tuple(totient(i) for i in range(1, n+1)), [0]*(n+1), 0 for i in range(n): qlist[a:=bisect(qlist, plist[i], lo=1, hi=c+1, key=lambda x:plist[x])]=i c = max(c, a) return 64+primepi(n)-c # Chai Wah Wu, Sep 06 2023 CROSSREFS Cf. A000720, A365339, A365474. Sequence in context: A086859 A278827 A125638 * A365397 A332926 A112816 Adjacent sequences: A365397 A365398 A365399 * A365401 A365402 A365403 KEYWORD nonn AUTHOR Peter Luschny, Sep 06 2023 STATUS approved

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Last modified March 3 07:55 EST 2024. Contains 370499 sequences. (Running on oeis4.)