|
|
A369252
|
|
Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.
|
|
14
|
|
|
27, 39, 51, 55, 75, 71, 87, 75, 91, 111, 103, 123, 95, 119, 147, 131, 119, 151, 183, 151, 135, 195, 167, 155, 231, 147, 199, 191, 187, 255, 167, 267, 211, 291, 195, 215, 247, 191, 263, 215, 327, 251, 247, 363, 203, 375, 311, 271, 255, 239, 411, 231, 311, 343, 299, 231, 435, 359, 331, 447, 311, 263, 391, 483, 263
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The table showing the possible modulo 3 combinations for p, q, r and the sum ((p*q) + (p*r) + (q*r)):
| p | q | r | sum ((p*q) + (p*r) + (q*r)) (mod 3)
--+------+------+------+----------------------------------------
| 0 | 0 | 0 | 0, p=q=r=3, sum is 27.
--+------+------+------+----------------------------------------
| 0 | 0 | +/-1 | 0, p=q=3, r > 3.
--+------+------+------+----------------------------------------
| 0 | +1 | +1 | +1
--+------+------+------+----------------------------------------
| 0 | -1 | -1 | +1
--+------+------+------+----------------------------------------
| 0 | -1 | +1 | -1
--+------+------+------+----------------------------------------
| 0 | +1 | -1 | -1
--+------+------+------+----------------------------------------
| +1 | +1 | +1 | 0
--+------+------+------+----------------------------------------
| -1 | -1 | -1 | 0
--+------+------+------+----------------------------------------
| -1 | +1 | +1 | -1, regardless of the order, thus x3.
--+------+------+------+----------------------------------------
| +1 | -1 | -1 | -1, regardless of the order, thus x3.
--+------+------+------+----------------------------------------
Notably a(n) is a multiple of 3 only when A046316(n) is either a multiple of 9, or all primes p, q and r are either == +1 (mod 3) or all are == -1 (mod 3), and the case a(n) == +1 (mod 3) is only possible when A046316(n) is a multiple of 3, but not of 9, and furthermore, it is required that r == q (mod 3). See how these combinations affects sequences like A369241, A369245, A369450, A369451, A369452.
For n=1..9 the number of terms of the form 3k, 3k+1 and 3k+2 in range [1..10^n-1] are:
6, 2, 1,
39, 22, 38,
291, 209, 499,
2527, 1884, 5588,
23527, 17020, 59452,
227297, 156240, 616462,
2232681, 1453030, 6314288,
22119496, 13661893, 64218610,
220098425, 129624002, 650277572.
It seems that 3k+2 terms are slowly gaining at the expense of 3k+1 terms when n grows, while the density of the multiples of 3 might converge towards a limit.
|
|
LINKS
|
|
|
FORMULA
|
|
|
CROSSREFS
|
Cf. A369251 (same sequence sorted into ascending order, with duplicates removed).
Cf. A369464 (numbers that do not occur in this sequence).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|