A368703
a(n) is the least integer k whose arithmetic derivative is equal to the n-th primorial, or 0 if no such k exists.
Original entry on oeis.org
2, 0, 9, 161, 2189, 29861, 510221, 1547371, 79332523, 9592991561, 265257420749, 1102527599503
Offset: 0
a(0) = 2 as the least number k such that A003415(k) = A002110(0) = 1 is 2.
a(1) = 0 as there is no number k such that A003415(k) = A002110(1) = 2.
a(7) = 1547371 as it is the least number k such that A003415(k) = A002110(7) = 510510. See also A366890.
A369240
Irregular triangle read by rows, where row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th partial sum of primorials, A143293(n). Rows of length zero are simply omitted, i.e., when A369239(n) = 0.
Original entry on oeis.org
14, 45, 74, 198, 5114, 10295, 65174, 1086194, 40354813, 20485574, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253, 307379277253, 321317084917, 342666536237, 348440115979
Offset: 1
Row 1 has no terms because there are no numbers whose arithmetic derivative is equal to 3 = A143293(1).
Row 2 has just one term: 14 (= 2 * 7), with A003415(14) = 2+7 = 9 = A143293(2).
Row 3 has two terms: 45 (= 3^2 * 5) and 74 (= 2 * 37), with A003415(3*3*5) = (3*3) + (3*5) + (3*5) = 39, and A003415(2*37) = 2+37 = 39 = A143293(3).
Row 4 has one term: 198 (= 2 * 3^2 * 11).
Row 5 has two terms: 5114 (= 2 * 2557) and 10295 (= 5 * 29 * 71).
Row 6 has one term: 65174 (= 2 * 32587).
Row 7 has two terms: 1086194 (= 2 * 543097) and 40354813 (= 97 * 541 * 769).
Row 8 has one term: 20485574 (= 2 * 10242787).
Row 9 has 27 terms:
465779078 (= 2 * 1049 * 222011),
12101385979 (= 79 * 151 * 1014451),
15237604243 (= 67 * 2659 * 85531),
18046312939 (= 79 * 3931 * 58111),
29501083259 (= 179 * 431 * 382391),
52467636437 (= 233 * 8501 * 26489),
65794608773 (= 449 * 761 * 192557),
86725630997 (= 449 * 2213 * 87281),
87741700037 (= 449 * 2381 * 82073),
131833085077 (= 613 * 12241 * 17569),
etc., up to the last one of them:
680909375411 (= 8171 * 8219 * 10139).
Row 10 has no terms.
Row 11 has 319 terms, beginning as:
293420849770 (= 2 * 5 * 157 * 186892261),
414527038034 (= 2 * 207263519017),
12092143168139 (= 59 * 5231 * 39180191),
16359091676491 (= 79 * 91291 * 2268319),
20784361649963 (= 167 * 251 * 495845639),
etc., up to the last one of them:
17866904665985941 (= 224869 * 248041 * 320329).
Row 12 has just one term: 318745032938881 (= 71 * 173 * 307 * 1259 * 67139).
Row 13 probably has thousands of terms. Interestingly, many of them appear in clusters that share a smallest prime factor. For example the following five:
390120053091860677 (= 1321 * 23563 * 12533283799),
407566547631686353 (= 1321 * 121687 * 2535429439),
410999481465461617 (= 1321 * 547999 * 567752023),
411668623600396429 (= 1321 * 1701571 * 183144919),
411913933485848977 (= 1321 * 8787799 * 35483263),
and also these:
3846842704473466739 (= 20231 * 31601 * 6017086469),
4300947161911032233 (= 20231 * 43319 * 4907590697),
4437898843097002379 (= 20231 * 47969 * 4572980861),
6130224093530040341 (= 20231 * 692459 * 437587529),
6210584908378844243 (= 20231 * 1275569 * 240664037).
Cf.
A328243 (same sequence sorted into ascending order).
Cf.
A369239 (number of terms on row n),
A369243 (the first element of each row),
A369244 (the last element of each row).
A369244
a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th partial sum of primorials, and 0 if no such k exists.
Original entry on oeis.org
0, 14, 74, 198, 10295, 65174, 40354813, 20485574, 680909375411, 0, 17866904665985941, 318745032938881
Offset: 1
Showing 1-3 of 3 results.
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