A369240 - OEIS

%I #19 Jan 22 2024 06:00:03

%S 14,45,74,198,5114,10295,65174,1086194,40354813,20485574,465779078,

%T 12101385979,15237604243,18046312939,29501083259,52467636437,

%U 65794608773,86725630997,87741700037,131833085077,168380217557,176203950283,177332276971,226152989747,292546582253,307379277253,321317084917,342666536237,348440115979

%N 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.

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

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

%H Antti Karttunen, <a href="/A369240/b369240.txt">Table of n, a(n) for n = 1..357; all terms up to the row 12 of the table</a>.

%H Antti Karttunen, <a href="/A369239/a369239.txt">PARI program for computing terms of this and related sequences</a>.

%e Row 1 has no terms because there are no numbers whose arithmetic derivative is equal to 3 = A143293(1).

%e Row 2 has just one term: 14 (= 2 * 7), with A003415(14) = 2+7 = 9 = A143293(2).

%e 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).

%e Row 4 has one term: 198 (= 2 * 3^2 * 11).

%e Row 5 has two terms: 5114 (= 2 * 2557) and 10295 (= 5 * 29 * 71).

%e Row 6 has one term: 65174 (= 2 * 32587).

%e Row 7 has two terms: 1086194 (= 2 * 543097) and 40354813 (= 97 * 541 * 769).

%e Row 8 has one term: 20485574 (= 2 * 10242787).

%e Row 9 has 27 terms:

%e   465779078 (= 2 * 1049 * 222011),

%e   12101385979 (= 79 * 151 * 1014451),

%e   15237604243 (= 67 * 2659 * 85531),

%e   18046312939 (= 79 * 3931 * 58111),

%e   29501083259 (= 179 * 431 * 382391),

%e   52467636437 (= 233 * 8501 * 26489),

%e   65794608773 (= 449 * 761 * 192557),

%e   86725630997 (= 449 * 2213 * 87281),

%e   87741700037 (= 449 * 2381 * 82073),

%e   131833085077 (= 613 * 12241 * 17569),

%e   etc., up to the last one of them:

%e   680909375411 (= 8171 * 8219 * 10139).

%e Row 10 has no terms.

%e Row 11 has 319 terms, beginning as:

%e   293420849770 (= 2 * 5 * 157 * 186892261),

%e   414527038034 (= 2 * 207263519017),

%e   12092143168139 (= 59 * 5231 * 39180191),

%e   16359091676491 (= 79 * 91291 * 2268319),

%e   20784361649963 (= 167 * 251 * 495845639),

%e   etc., up to the last one of them:

%e   17866904665985941 (= 224869 * 248041 * 320329).

%e Row 12 has just one term: 318745032938881 (= 71 * 173 * 307 * 1259 * 67139).

%e 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:

%e   390120053091860677 (= 1321 * 23563 * 12533283799),

%e   407566547631686353 (= 1321 * 121687 * 2535429439),

%e   410999481465461617 (= 1321 * 547999 * 567752023),

%e   411668623600396429 (= 1321 * 1701571 * 183144919),

%e   411913933485848977 (= 1321 * 8787799 * 35483263),

%e   and also these:

%e   3846842704473466739 (= 20231 * 31601 * 6017086469),

%e   4300947161911032233 (= 20231 * 43319 * 4907590697),

%e   4437898843097002379 (= 20231 * 47969 * 4572980861),

%e   6130224093530040341 (= 20231 * 692459 * 437587529),

%e   6210584908378844243 (= 20231 * 1275569 * 240664037).

%Y Cf. A328243 (same sequence sorted into ascending order).

%Y Cf. A369239 (number of terms on row n), A369243 (the first element of each row), A369244 (the last element of each row).

%Y Cf. A003415, A143293.

%Y Cf. also A366890.

%K nonn,tabf

%O 1,1

%A _Antti Karttunen_, Jan 19 2024