A366890 - OEIS

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.

1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403

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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, Table of n, a(n) for n = 1..31

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.

Cf. A001222, A002110, A003415, A116979, A351029, A368703.

Cf. also A036785, A368697, A328243, A369240.

Sequence in context: A364701 A236117 A255694 * A089492 A253034 A328563

Adjacent sequences: A366887 A366888 A366889 * A366891 A366892 A366893

KEYWORD

nonn,hard,tabf

AUTHOR

Antti Karttunen, Jan 09 2024

STATUS

approved