cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Antti Karttunen, Jan 19 2024

Keywords

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

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.

A368704 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th primorial, and 0 if no such k exists.

Original entry on oeis.org

0, 9, 221, 11021, 1333349, 225450221, 65155115009, 23520996509141, 12442607161209161, 10464232622576957201, 10056127550296456854221, 13766838616355849433396389, 23142055714094182897602595769, 42789661015360144177667200022669, 94522361182930558488466844910827309, 265513312562851938794103367354849976069
Offset: 1

Views

Author

Antti Karttunen, Jan 16 2024

Keywords

Comments

a(n) is the greatest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.
See also comments in A116979.

Examples

			a(1) = 0 as there are no number k such that A003415(k) = A002110(1) = 2.
a(3) = 221 as A003415(221) = A003415(13*17) = A003415(13)*17 + 13*A003415(17) = 1*17 + 13*1 = 30 which is A002110(3) and no k>221 has arithmetic derivative 30. - _David A. Corneth_, Jan 17 2024

Links

Crossrefs

Cf. A002110, A003415, A116979, A351029, A368703, A369059 (an upper bound).
Cf. also A369244.

Programs

  • PARI
    a(n) = {if(n==1,return(0)); pr = vecprod(primes(n)); prover2 = pr/2; forprime(p = prover2, oo, if(isprime(pr - p), return(p*(pr-p))))} \\ David A. Corneth, Jan 17 2024

Formula

For n >= 1, A368703(n) <= a(n) <= A369059(n).
For n > 1, A003415(a(n)) = A002110(n).
For n > 1, a(n) = p*q, where p, q are primes, p+q = A002110(n) and q >= p and q - p is minimal. - David A. Corneth, Jan 17 2024 [This depends on Goldbach's conjecture being valid, at least on primorials, for which there is strong empirical evidence though.] - Antti Karttunen, Jan 19 2024

Extensions

More terms from David A. Corneth, Jan 17 2024

A369243 a(n) is the least 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

2, 0, 14, 45, 198, 5114, 65174, 1086194, 20485574, 465779078, 0, 293420849770, 318745032938881
Offset: 0

Views

Author

Antti Karttunen, Jan 19 2024

Keywords

Comments

a(n) = the smallest integer k for which A003415(k) = A143293(n), and 0 if no such k exists.

Examples

			a(0) = 2 as the least number k such that A003415(k) = A143293(0) = 1 is 2.
a(1) = 0 as there is no number k such that A003415(k) = A143293(1) = 3.

Links

Crossrefs

Cf. A003415, A143293, A369244.
Cf also A368703.

Formula

a(n) <= A369244(n).
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