A318954 -id:A318954 - cp's OEIS frontend

A319057 Minimum sum of a strict factorization of n into factors > 1.

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

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

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

			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.

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

Cf. A001055, A007716, A045778, A056239, A162247, A215366, A246868, A318871, A318953, A318954.

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}]

A318953 Maximum Heinz number of a strict factorization of n into factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

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

			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.

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

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}]

cp's OEIS Frontend

A319057 Minimum sum of a strict factorization of n into factors > 1.

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