login
A366890
Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.
8
1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403
OFFSET
1,1
COMMENTS
For n > 0, numbers k such that A003415(k) = A002110(n) and A001222(k) > 2.
Sequence as a whole is not listed in ascending order, even though each batch of solutions for each n for which A369000(n) > 0 are. For example, we have a(14) < a(13) because A003415(22101822021783739) = A002110(12), while A003415(4958985803436403) = A002110(13). See the examples.
Question: Are there any common terms with A036785, that is, with A368697?
LINKS
Antti Karttunen, PARI program
EXAMPLE
For rows n=1..6, 9 & 10 nothing is listed, as those rows are empty.
Row for n=7 has just one term: 1547371 (= 7^2 * 23 * 1373). Note that A003415(1547371) = 510510 = A002110(7).
Row for n=8 has just one term: 79332523 (= 17^2 * 277 * 991).
Row for n=11 has two terms:
1102527599503 (= 11^2 * 11071 * 823033),
25336943536819 (= 157 * 743 * 5749 * 37781).
Row for n=12 has nine terms:
25962012375103 (= 7^2 * 8597 * 61630451),
25970380120783 (= 7^2 * 41387 * 12806141),
66702554987143 (= 19^2 * 167 * 1106416889),
526285951027003 (= 73 * 3919 * 7013 * 262313),
927949814519899 (= 269 * 271 * 1697 * 7501033),
7777707036642079 (= 2203 * 2791 * 7349 * 172127),
9584173681667203 (= 2131 * 5953 * 7901 * 95621),
13082430772438171 (= 3109 * 5861 * 24421 * 29399),
22101822021783739 (= 8783 * 11777 * 13921 * 15349).
Row for n=13 has 18 terms, and begins with:
4958985803436403 (= 37^2 * 137 * 26440450451),
and ends with:
3206697143570677543 (= 36899 * 41983 * 45233 * 45763).
Note that A003415(3206697143570677543) = 304250263527210 = A002110(13).
PROG
(PARI) \\ See the attached PARI-program
CROSSREFS
When the whole sequence is sorted into ascending order, equal to A327978 without any semiprime solutions (solutions in A001358), and also a subsequence of following sequences: A004709, A327862, A328234.
Sequence in context: A364701 A236117 A255694 * A089492 A253034 A328563
KEYWORD
nonn,hard,tabf
AUTHOR
Antti Karttunen, Jan 09 2024
STATUS
approved