cp's OEIS Frontend

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.

%I A366890 #40 Jan 19 2024 23:55:18
%S A366890 1547371,79332523,1102527599503,25336943536819,25962012375103,
%T A366890 25970380120783,66702554987143,526285951027003,927949814519899,
%U A366890 7777707036642079,9584173681667203,13082430772438171,22101822021783739,4958985803436403,32006922970429003,32076018550175863,49806227168831659,84682266449971639,97995266657958403
%N 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.
%C A366890 For n > 0, numbers k such that A003415(k) = A002110(n) and A001222(k) > 2.
%C A366890 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.
%C A366890 Question: Are there any common terms with A036785, that is, with A368697?
%H A366890 Antti Karttunen, <a href="/A366890/b366890.txt">Table of n, a(n) for n = 1..31</a>
%H A366890 Antti Karttunen, <a href="/A351029/a351029.txt">PARI program</a>
%e A366890 For rows n=1..6, 9 & 10 nothing is listed, as those rows are empty.
%e A366890 Row for n=7 has just one term: 1547371 (= 7^2 * 23 * 1373). Note that A003415(1547371) = 510510 = A002110(7).
%e A366890 Row for n=8 has just one term: 79332523 (= 17^2 * 277 * 991).
%e A366890 Row for n=11 has two terms:
%e A366890   1102527599503 (= 11^2 * 11071 * 823033),
%e A366890   25336943536819 (= 157 * 743 * 5749 * 37781).
%e A366890 Row for n=12 has nine terms:
%e A366890   25962012375103 (= 7^2 * 8597 * 61630451),
%e A366890   25970380120783 (= 7^2 * 41387 * 12806141),
%e A366890   66702554987143 (= 19^2 * 167 * 1106416889),
%e A366890   526285951027003 (= 73 * 3919 * 7013 * 262313),
%e A366890   927949814519899 (= 269 * 271 * 1697 * 7501033),
%e A366890   7777707036642079 (= 2203 * 2791 * 7349 * 172127),
%e A366890   9584173681667203 (= 2131 * 5953 * 7901 * 95621),
%e A366890   13082430772438171 (= 3109 * 5861 * 24421 * 29399),
%e A366890   22101822021783739 (= 8783 * 11777 * 13921 * 15349).
%e A366890 Row for n=13 has 18 terms, and begins with:
%e A366890   4958985803436403 (= 37^2 * 137 * 26440450451),
%e A366890 and ends with:
%e A366890   3206697143570677543 (= 36899 * 41983 * 45233 * 45763).
%e A366890 Note that A003415(3206697143570677543) = 304250263527210 = A002110(13).
%o A366890 (PARI) \\ See the attached PARI-program
%Y A366890 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.
%Y A366890 Cf. A001222, A002110, A003415, A116979, A351029, A368703.
%Y A366890 Cf. also A036785, A368697, A328243, A369240.
%K A366890 nonn,hard,tabf
%O A366890 1,1
%A A366890 _Antti Karttunen_, Jan 09 2024