A051191 -id:A051191 - cp's OEIS frontend

A386260 Maximum exponent in the prime factorization of the exponent of the highest power of 2 dividing 2*n.

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^k-2) = A051191(k).

The asymptotic density of the occurrences of 1 in this sequence is 4 * Sum_{k squarefree > 1} (1/2^k - 1/2^(k+1)) = 0.862752712766... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001511, A051191, A051903.

a[n_] := Module[{v = IntegerExponent[n, 2] + 2}, If[v == 1, 0, Max[FactorInteger[v][[;;, 2]]]]]; Array[a, 100]

a(n) = my(v = valuation(4*n, 2)); if(v == 1, 0, vecmax(factor(v)[,2]));

a(n) = A051903(A001511(2*n)).
A051903(A001511(2*n-1)) = 0 for all n >= 1, and therefore the odd-indexed terms of A051903(A001511(n)) are omitted.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} A051903(k)/2^(k-1) = 1.14512095789925078232... . If the odd-indexed zero terms had not been omitted, the asymptotic mean would be half this value, 0.57256047894962539116... .

A353264 a(n) is the least number k such that A018804(k)/k = n.

Original entry on oeis.org

1, 4, 15, 64, 48, 60, 144, 16384, 240, 1300, 1296, 960, 1008, 3564, 3840, 1073741824, 6000, 14580, 7056, 20800, 11520, 25500, 944784, 245760, 13104, 24948, 34560, 57024, 750000, 16380, 156816, 4611686018427387904, 102000, 364500, 46800, 233280, 134064, 174636

Offset: 1

Amiram Eldar, Apr 09 2022

nonn

a(n) exist for all n>=1.
The solution k to A018804(k)/k = n is unique if and only if n is a power of 2: a(2^m) = 2^(2^(m+1)-2) = A051191(m+1).
The questions of existence and uniqueness are a part of a problem that was proposed during the Forty-Fifth International Mathematical Olympiad in Athens, Greece, July 7-19, 2004.

			a(2) = 4 since A018804(4)/4 = 8/4 = 2 and 4 is the least number with this property.

Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović, The IMO Compendium, A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004, Springer, New York, 2006. See Problem 25, pp. 331, 726-727.

Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović, IMO Shortlist 2004, 2005, Problem 25, pp. 5-6, 18-19.
R. E. Woodrow, Problem N2, The Olympiad Corner, Crux Mathematicorum, Vol. 34, No. 7 (2008), pp. 419-421.
Index to sequences related to Olympiads.

Cf. A018804, A051191.

Subsequence of A066862.

f[p_, e_] := (e*(p - 1)/p + 1); r[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Module[{k = 1}, While[r[k] != n, k++]; k]; Array[a, 15]

cp's OEIS Frontend

A386260 Maximum exponent in the prime factorization of the exponent of the highest power of 2 dividing 2*n.

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