A083544 -id:A083544 - cp's OEIS frontend

A225420 a(n) is the least number k such that the sum of the n Moebius function values beginning at k reaches the maximum value A083544(n).

1, 14, 33, 32, 91, 141, 213, 212, 213, 3090, 3093, 3090, 3090, 3090, 38405, 38404, 3090, 3090, 38401, 38400, 294581, 294581, 39569681, 5571498, 68780189, 294577, 68780189, 53758490, 92636277, 456742389, 1176172581, 880346227, 3953000577, 13821836609, 948481781, 948481780, 948481781, 5332819926, 35398246981, 35398246979

Offset: 1

T. D. Noe, May 07 2013

			For n = 2, the 14 sums are 0, -2, -1, -1, 0, 0, -1, 0, 1, 0, -1, -1, 0, 2.

Cf. A008683 (Moebius function), A083544 (maximum values of sums).

mu = Table[MoebiusMu[i], {i, 1000000}]; t = Table[s = Total /@ Partition[mu, n, 1]; mx = Max[s]; pos = Position[s, mx, 1, 1][[1, 1]]; {mx, pos}, {n, 22}]; Transpose[t][[2]]

a(23)-a(24) added by T. D. Noe, May 08 2013

More terms from Don Reble, Apr 21 2021

A343172 Least number k such that the sum of the n Moebius function values beginning at k reaches the minimum value -A083544(n).

Original entry on oeis.org

2, 2, 29, 2, 101, 281, 429, 428, 2081, 6298, 30089, 30088, 143491, 567354, 693677, 693676, 8229, 693674, 1432677, 1123291, 1432677, 2156853, 25085909, 2156851, 25085909, 2156849, 24771577, 24771576, 126398226, 126398226, 3349160985, 389565283, 2928714078, 10441021690, 1353696733

Offset: 1

Don Reble, Apr 21 2021

nonn

Cf. A083544, A225420.

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 2, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 12, 12, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 12, 40, 24, 30, 30, 24, 56, 56, 24, 32, 224, 210, 570, 532, 28, 36, 60, 480, 672, 70, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36, 900

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=33: cototient(33) = 33-20 = 13, cototient(34) = 34-16 = 18;
lcm(13,18) = 234, gcd(13,18) = 1, so a(34) = 234.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953, A083538, A083539, A083540, A083541, A083542, A083543, A083544, A083545, A083546, A083547, A083548, A083549, A083550, A083551, A083552, A083553, A083554, A083555, A049586.

f[x_] := x-EulerPhi[x]; Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 69}]
(* Second program: *)
Map[Apply[LCM, #]/Apply[GCD, #] &@ Map[# - EulerPhi@ # &, #] &, Partition[Range[69], 2, 1]] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = lcm(A051953(n), A051952(n+1))/gcd(A051953(n), A051952(n+1)) = lcm(cototient(n+1), cototient(n))/A049586(n).

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A225420 a(n) is the least number k such that the sum of the n Moebius function values beginning at k reaches the maximum value A083544(n).

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A343172 Least number k such that the sum of the n Moebius function values beginning at k reaches the minimum value -A083544(n).

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A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

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