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.

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

Author

Antti Karttunen, Jan 19 2024

Keywords

• nonn
• tabf

Comments

Only two nonsquarefree terms are currently known: 45, 198.

See comments in A369239 for an explanation why rows with an odd n generally have more terms than those with an even n.

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..357; all terms up to the row 12 of the table.
• Antti Karttunen, PARI program for computing terms of this and related sequences.

Crossrefs

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).

Cf. A003415, A143293.

Cf. also A366890.