|
| |
|
|
A330998
|
|
Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.
|
|
9
|
|
|
|
1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
This is the sorted list of positions of first appearances in A318284 of each element of the range A045782.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.
|
|
|
LINKS
|
Table of n, a(n) for n=1..67.
|
|
|
EXAMPLE
|
Factorizations of the inverse prime shadows of the initial terms:
4 8 12 16 36 24 60 48
2*2 2*4 2*6 2*8 4*9 3*8 2*30 6*8
2*2*2 3*4 4*4 6*6 4*6 3*20 2*24
2*2*3 2*2*4 2*18 2*12 4*15 3*16
2*2*2*2 3*12 2*2*6 5*12 4*12
2*2*9 2*3*4 6*10 2*3*8
2*3*6 2*2*2*3 2*5*6 2*4*6
3*3*4 3*4*5 3*4*4
2*2*3*3 2*2*15 2*2*12
2*3*10 2*2*2*6
2*2*3*5 2*2*3*4
2*2*2*2*3
The corresponding multiset partitions:
{11} {111} {112} {1111} {1122} {1112}
{1}{1} {1}{11} {1}{12} {1}{111} {1}{122} {1}{112}
{1}{1}{1} {2}{11} {11}{11} {11}{22} {11}{12}
{1}{1}{2} {1}{1}{11} {12}{12} {2}{111}
{1}{1}{1}{1} {2}{112} {1}{1}{12}
{1}{1}{22} {1}{2}{11}
{1}{2}{12} {1}{1}{1}{2}
{2}{2}{11}
{1}{1}{2}{2}
|
|
|
MATHEMATICA
|
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
nds=Table[Length[facs[Times@@Prime/@nrmptn[n]]], {n, 50}];
Table[Position[nds, i][[1, 1]], {i, First/@Gather[nds]}]
|
|
|
CROSSREFS
|
Taking n instead of the inverse prime shadow of n gives A330972.
Factorizations are A001055, with image A045782, with complement A330976.
Factorizations of inverse prime shadows are A318284.
Cf. A025487, A033833, A045778, A045783, A181819, A181821, A318286, A325238, A330973, A330989, A330990, A330993, A330997.
Sequence in context: A125236 A137624 A238246 * A099467 A054353 A284555
Adjacent sequences: A330995 A330996 A330997 * A330999 A331000 A331001
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Gus Wiseman, Jan 07 2020
|
|
|
STATUS
|
approved
|
| |
|
|
