|
|
A340101
|
|
Number of factorizations of 2n + 1 into odd factors > 1.
|
|
28
|
|
|
1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 1, 2, 1, 2, 4, 2, 2, 2, 3, 1, 2, 1, 2, 7, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 1, 2, 2, 1, 5, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 2, 7, 1, 1, 5, 1, 1, 2, 2, 2, 4, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
27 45 135 225 315 405 1155
3*9 5*9 3*45 3*75 5*63 5*81 15*77
3*3*3 3*15 5*27 5*45 7*45 9*45 21*55
3*3*5 9*15 9*25 9*35 15*27 33*35
3*5*9 15*15 15*21 3*135 3*385
3*3*15 5*5*9 3*105 5*9*9 5*231
3*3*3*5 3*3*25 5*7*9 3*3*45 7*165
3*5*15 3*3*35 3*5*27 11*105
3*3*5*5 3*5*21 3*9*15 3*5*77
3*7*15 3*3*5*9 3*7*55
3*3*5*7 3*3*3*15 5*7*33
3*3*3*3*5 3*11*35
5*11*21
7*11*15
3*5*7*11
|
|
MAPLE
|
g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
d=numtheory[divisors](n) minus {1, n}))
end:
a:= n-> g(2*n+1$2):
|
|
MATHEMATICA
|
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ[Times@@#]&]], {n, 1, 100, 2}]
|
|
PROG
|
(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
|
|
CROSSREFS
|
A316439 counts factorizations by product and length.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|