|
|
A355533
|
|
Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).
|
|
11
|
|
|
1, 2, 0, 3, 1, 4, 0, 0, 0, 2, 5, 0, 1, 6, 3, 1, 0, 0, 0, 7, 1, 0, 8, 0, 2, 2, 4, 9, 0, 0, 1, 0, 5, 0, 0, 0, 3, 10, 1, 1, 11, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 12, 7, 4, 0, 0, 2, 13, 1, 2, 14, 0, 4, 0, 1, 8, 15, 0, 0, 0, 1, 0, 2, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
COMMENTS
|
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 version where zero is prepended to the prime indices before taking differences is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.
|
|
LINKS
|
|
|
FORMULA
|
Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.
|
|
EXAMPLE
|
Triangle begins (showing n, prime indices, differences*):
2: (1) 1
3: (2) 2
4: (1,1) 0
5: (3) 3
6: (1,2) 1
7: (4) 4
8: (1,1,1) 0 0
9: (2,2) 0
10: (1,3) 2
11: (5) 5
12: (1,1,2) 0 1
13: (6) 6
14: (1,4) 3
15: (2,3) 1
16: (1,1,1,1) 0 0 0
For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[PrimeQ[n], {PrimePi[n]}, Differences[primeMS[n]]], {n, 2, 30}]
|
|
CROSSREFS
|
Crossrefs found in the link are not repeated here.
The version for prime indices prepended by 0 is A287352.
Constant rows have indices A325328.
Number of distinct terms in each row are 1 if prime, otherwise A355523.
The version with prime-indexed rows empty is A355536, Heinz number A325352.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|