A318954 -id:A318954

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A318953

Maximum Heinz number of a strict factorization of n into factors > 1.

+10
9

1, 3, 5, 7, 11, 15, 17, 21, 23, 33, 31, 39, 41, 51, 55, 57, 59, 69, 67, 87, 85, 93, 83, 111, 97, 123, 115, 129, 109, 165, 127, 159, 155, 177, 187, 195, 157, 201, 205, 231, 179, 255, 191, 237, 253, 249, 211, 285, 227, 319, 295, 303, 241, 345, 341, 357, 335, 327

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The Heinz number of a factorization (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

LINKS

Table of n, a(n) for n=1..58.

EXAMPLE

The strict factorizations of 80 are (2*4*10), (2*5*8), (2*40), (4*20), (5*16), (8*10), (80), with Heinz numbers 609, 627, 519, 497, 583, 551, 409 respectively, so a(80) = 627.

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];

Table[Max[Times@@Prime/@#&/@Select[facs[n], UnsameQ@@#&]], {n, 100}]

CROSSREFS

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318954.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 05 2018

STATUS

approved

A319057

Minimum sum of a strict factorization of n into factors > 1.

+10
9

0, 2, 3, 4, 5, 5, 7, 6, 9, 7, 11, 7, 13, 9, 8, 10, 17, 9, 19, 9, 10, 13, 23, 9, 25, 15, 12, 11, 29, 10, 31, 12, 14, 19, 12, 11, 37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 12, 49, 15, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 14, 18, 16, 67, 21, 26

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) >= A001414(n), with equality iff n is squarefree or four times a squarefree number (i.e., A000188(n) <= 2). - Charlie Neder, Sep 10 2018

LINKS

Charlie Neder, Table of n, a(n) for n = 1..1000

EXAMPLE

The strict factorizations of 48 are (48), (2*24), (3*16), (4*12), (6*8), (2*3*8), (2*4*6), with sums 48, 26, 19, 16, 14, 13, 12 respectively, so a(48) = 12.

MATHEMATICA

strfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&)/@Select[strfacs[n/d], Min@@#>d&], {d, Rest[Divisors[n]]}]];

Table[Min[Total/@strfacs[n]], {n, 100}]