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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A327862 Numbers whose arithmetic derivative is of the form 4k+2, cf. A003415.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 65, 69, 77, 85, 93, 121, 129, 133, 135, 141, 145, 161, 169, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 289, 301, 305, 309, 315, 321, 329, 341, 351, 361, 365, 375, 377, 381, 393, 413, 417, 437, 445, 453, 459, 469, 473, 481, 485, 489, 493, 495, 497, 501, 505, 517, 529, 533, 537
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

All terms are odd because the terms A068719 are either multiples of 4 or odd numbers.
Odd numbers k for which A064989(k) is one of the terms of A358762. - Antti Karttunen, Nov 30 2022
The second arithmetic derivative (A068346) of these numbers is odd. See A235991. - Antti Karttunen, Feb 06 2024

Links

Crossrefs

Setwise difference A235992 \ A327864.
Setwise difference A046337 \ A360110.
Union of A369661 (k' has an even number of prime factors) and A369662 (k' has an odd number of prime factors).
Subsequences: A001248 (from its second term onward), A108181, A327978, A366890 (when sorted into ascending order), A368696, A368697.
Cf. A003415, A064989, A068346, A068719, A327863, A327865, A353495 (characteristic function).
Cf. also A235991, A358762, A358772, A358774.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
isA327862(n) = (2==(A003415(n)%4));
k=1; n=0; while(k<105, if(isA327862(n), print1(n, ", "); k++); n++);

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.

Original entry on oeis.org

1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2024

Keywords

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?

Examples

			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).

Links

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.

Programs

  • PARI
    \\ See the attached PARI-program

A368696 Numbers whose arithmetic derivative (A003415) is a squarefree number of the form 4k+2.

Original entry on oeis.org

9, 21, 25, 33, 49, 57, 69, 85, 93, 121, 129, 133, 145, 161, 169, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 289, 305, 309, 315, 321, 341, 361, 365, 377, 381, 393, 413, 417, 437, 445, 453, 469, 485, 489, 493, 495, 497, 501, 505, 517, 529, 537, 545, 553, 565, 573, 597, 633, 649, 669, 681, 685, 689
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2024

Keywords

Crossrefs

Intersection of A327862 and A328393.
Cf. A003415, A005117, A368697 (subsequence).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA368696(n) = { my(d=A003415(n)); ((2==(d%4))&&issquarefree(d)); };
Showing 1-3 of 3 results.

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