A076745 a(n) = the least positive integer k such that b(k) = n, where b(k) (A076526) is defined by b(k) = r * max{e_1,...,e_r} if k = p_1^e_1 *...* p_r^e_r is the canonical prime factorization of k.

2, 4, 8, 12, 32, 24, 128, 48, 120, 96, 2048, 192, 8192, 384, 480, 768, 131072, 960, 524288, 3072, 1920, 6144, 8388608, 3840, 36960, 24576, 7680, 13440, 536870912, 15360, 2147483648, 26880, 30720, 393216, 147840, 53760, 137438953472, 1572864, 122880, 107520, 2199023255552

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Carlos Rivera, Puzzle 201: The Arithmetic Function A(n), The Prime Puzzles and Problems Connection.

Formula

From Amiram Eldar, Sep 08 2024: (Start)

a(n) = Min_{d|n} (2^d * Product_{i=1..n/d-1} prime(i+1)).

a(p) = 2^p for a prime p.

a(2*p) = 3*2^p for a prime p.

a(3*p) = 15*2^p for a prime p > 2. (End)

Example

a(12) = 2 * max{1,2} = 4 since 12 = 2^2 * 3^1 and 12 is the least k for which b(k) = 4. Hence a(4) = 12.

Mathematica

a[n_] := Min[Table[2^d*Times @@ Prime[Range[2, n/d]], {d, Divisors[n]}]]; Array[a, 50] (* Amiram Eldar, Sep 08 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n), nd = numdiv(f), v = vector(nd), k = 0); fordiv(f, d, k++; v[k] = 2^d * prod(i = 1, n/d-1, prime(i+1))); vecmin(v); } \\ Amiram Eldar, Sep 08 2024

Crossrefs

Cf. A076526.

Sequence in context: A204088 A187941 A085083 * A007374 A105207 A202148

Adjacent sequences: A076742 A076743 A076744 * A076746 A076747 A076748

Keyword

nonn,easy,changed

Author

Joseph L. Pe, Nov 11 2002

Extensions

More terms from Amiram Eldar, Sep 08 2024

Status

approved