A022328 -id:A022328 - cp's OEIS frontend

A033031 Squarefree kernels of 3-smooth numbers.

1, 2, 3, 2, 6, 2, 3, 6, 2, 6, 6, 3, 2, 6, 6, 6, 2, 6, 3, 6, 6, 2, 6, 6, 6, 6, 3, 2, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 2, 3, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 3, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 3, 6, 2, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Jun 21 2003

nonn

			A003586(17) = 64 = 2^6 -> a(17) = 2,
A003586(18) = 72 = 2^3 * 3^2 -> a(18) = 2*3 = 6,
A003586(19) = 81 = 3^4 -> a(19) = 3.

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

Cf. A003586, A007947, A022328, A022329.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); rad /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A007947(A003586(n)).
a(n) = (2*0^(A022328(n)-1)) * (3*0^(A022329(n)-1)) for n>1. - Reinhard Zumkeller, Jul 18 2003
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6. - Amiram Eldar, Jul 13 2023

A372744 If the n-th 3-smooth number, A003586(n), equals 2^i * 3^j for some i, j >= 0, then the a(n)-th 3-smooth number, A003586(a(n)), equals 2^j * 3^i.

Original entry on oeis.org

1, 3, 2, 7, 5, 12, 4, 10, 19, 8, 16, 6, 27, 14, 24, 11, 37, 21, 9, 33, 18, 49, 30, 15, 44, 26, 13, 62, 40, 23, 57, 36, 20, 77, 52, 32, 17, 71, 47, 29, 93, 66, 43, 25, 87, 60, 39, 111, 22, 81, 55, 35, 104, 75, 51, 131, 31, 98, 69, 46, 123, 28, 91, 64, 152, 42

Offset: 1

Rémy Sigrist, May 12 2024

nonn

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points (A202821).

			A003586(8) = 12 = 2^2 * 3^1, A003586(10) = 18 = 2^1 * 3^2, so a(8) = 10 and.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A003586, A022328, A022329, A202821 (fixed points).

PARI
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A022328(a(n)) = A022329(n).
A022329(a(n)) = A022328(n).
a(n) = n iff n belongs to A202821.
sign(a(n) - n) = sign(A022328(n) - A022329(n)).

A379258 a(n) is the number of iterations of the Euler phi function needed to reach 1 starting at the n-th 3-smooth number.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 5, 4, 5, 5, 6, 5, 6, 5, 7, 6, 6, 7, 6, 8, 7, 6, 8, 7, 7, 9, 8, 7, 9, 8, 7, 10, 9, 8, 8, 10, 9, 8, 11, 10, 9, 8, 11, 10, 9, 12, 9, 11, 10, 9, 12, 11, 10, 13, 9, 12, 11, 10, 13, 10, 12, 11, 14, 10, 13, 12, 11, 14, 10, 13, 12, 15, 11, 14, 11

Offset: 1

Amiram Eldar, Dec 19 2024

			a(6) = 4 because the 6th 3-smooth number is A003586(6) = 8, and 4 iterations of phi are needed to reach 1: 8 -> 4 -> 2 -> 1.

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

Cf. A000010, A003586, A049108, A086420, A022328, A022329, A022330, A022331, A202821.

f[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, e2 + e3 + 1 + Boole[e2 == 0]]; f[1] = 1; With[{max = 3*10^4}, f /@ Sort[Flatten[Table[2^i*3^j, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]]]

list(lim) = {my(e2, e3); print1(1, ", "); for(k = 2, lim, e2 = valuation(k, 2); e3 = valuation(k, 3); if(k == (1 << e2) * 3^e3, print1(e2 + e3 + 1 + (e2 == 0), ", ")));}

a(n) = A049108(A003586(n)).
a(n) = valuation(A003586(n), 2) + valuation(A003586(n), 3) + 1 + [valuation(A003586(n), 2) == 0] for n > 1, where [] is the Iverson bracket.
a(n) = A022328(n) + A022329(n) + 1 + [n is in A022330], for n > 1.
a(A022330(n)) = n + 2 for n >= 1.
a(A022331(n)) = n + 1 for n >= 0.
a(A202821(n)) = 2*n + 1, for n >= 0.

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A033031 Squarefree kernels of 3-smooth numbers.

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A372744 If the n-th 3-smooth number, A003586(n), equals 2^i * 3^j for some i, j >= 0, then the a(n)-th 3-smooth number, A003586(a(n)), equals 2^j * 3^i.

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A379258 a(n) is the number of iterations of the Euler phi function needed to reach 1 starting at the n-th 3-smooth number.

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