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.

Showing 1-6 of 6 results.

A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.

Original entry on oeis.org

6, 9, 10, 18, 21, 22, 25, 26, 30, 33, 34, 38, 42, 45, 49, 57, 58, 62, 63, 66, 69, 70, 74, 75, 78, 82, 85, 90, 93, 98, 102, 105, 106, 110, 114, 117, 118, 121, 126, 129, 130, 133, 134, 142, 145, 147, 150, 153, 154, 161, 165, 166, 169, 170, 171, 174, 175, 177, 178, 182, 185, 186, 190, 195, 198, 201, 202, 205, 206, 209, 210, 213
Offset: 1

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

Sequence A328393 without primes.
No multiples of 4 because this is a subsequence of A048103.
All terms are cubefree, but being a cubefree non-multiple of 4 doesn't guarantee a membership, as for example 99 = 3^2 * 11 has an arithmetic derivative 11*(2*3) + 3^2 = 75 = 5^2 * 3, and thus is not included in this sequence. (See e.g., A328305).

Links

Crossrefs

Cf. A003415, A005117, A068328, A068719, A235991, A327862, A328239, A328242, A328244, A328245, A328246, A328247.
Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).
Subsequence of following sequences: A004709, A048103, A328393.
Complement of the union of A000040 and A328303, i.e., complement of A328303, but without primes.
Cf. also A328248, A328250, A328305.

Programs

  • Mathematica
    arthD[n_]:=Module[{fi=FactorInteger[n]},n Total[(fi[[;;,2]]/fi[[;;,1]])]]; Select[Range[300],arthD[#]>1&&SquareFreeQ[arthD[#]]&] (* Harvey P. Dale, Dec 01 2024 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328234(n) = { my(u=A003415(n)); (u>1 && issquarefree(u)); };

A328244 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117).

Original entry on oeis.org

6, 9, 10, 14, 18, 21, 22, 25, 30, 34, 38, 42, 46, 50, 57, 58, 62, 65, 66, 69, 70, 77, 78, 82, 85, 86, 93, 94, 99, 105, 114, 118, 121, 122, 125, 126, 130, 133, 134, 138, 142, 145, 146, 150, 154, 161, 165, 166, 169, 170, 174, 177, 182, 185, 186, 198, 201, 202, 206, 207, 209, 213, 214, 217, 221, 222, 230, 231, 237, 238, 242, 246, 253, 254, 255
Offset: 1

Views

Author

Antti Karttunen, Oct 11 2019

Keywords

Comments

Numbers n for which A008966(A003415(A003415(n))) = 1.
Numbers whose first, second or third arithmetic is prime (A157037, A192192, A328239) are all included in this sequence, because: (1) taking arithmetic derivative of a prime gives 1, which is squarefree, (2) primes themselves are squarefree, and (3) only squarefree numbers may have arithmetic derivative that is a prime.

Examples

			For n=6, its first arithmetic derivative is A003415(6) = 5, and its second derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 6 is included in this sequence.
For n=9, A003415(9) = 6, A003415(6) = 5, and 5, like all prime numbers, is squarefree, thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6 = 2*3, and as 6 is squarefree, 14 is included in this sequence.

Links

Crossrefs

Cf. A003415, A005117, A008966, A068346, A328234, A328246.
Cf. A157037, A192192, A328239, A328245, A328253 (subsequences).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328244(n) = { my(u=A003415(A003415(n))); (u>0 && issquarefree(u)); };

A328253 Nonsquarefree numbers whose first arithmetic derivative (A003415) is not squarefree, but the second derivative (A068346) is.

Original entry on oeis.org

50, 99, 125, 207, 343, 375, 531, 686, 725, 747, 750, 819, 875, 931, 1083, 1175, 1331, 1375, 1750, 1775, 1899, 2057, 2058, 2075, 2197, 2250, 2299, 2331, 2367, 2499, 2525, 2625, 2750, 2853, 3250, 3425, 3430, 3577, 3610, 3771, 3789, 3843, 3875, 4059, 4149, 4250, 4311, 4394, 4459, 4475, 4626, 4693, 4750, 4775, 4875, 4913, 4998, 5145
Offset: 1

Views

Author

Antti Karttunen, Oct 11 2019

Keywords

Examples

			50 (= 2 * 5^2) is not squarefree, and its first derivative A003415(50) = 45 = 3^2 * 5 also is not squarefree, but taking derivative yet again, gives A003415(45) = 39 = 3*13, which is squarefree, thus 50 is included in this sequence.

Links

Crossrefs

Cf. A003415, A328251.
Row 4 of array A328250. Indices of 3's in A328248.
Setwise difference A328245 \ A005117. Intersection of A013929 and A328245.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328253(n) = if(issquarefree(n), 0, my(u=A003415(n)); if(issquarefree(u),0, issquarefree(A003415(u))));
  • PARI
    A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA328253(n) = (3==A328248(n));

A328303 Numbers whose arithmetic derivative is not squarefree.

Original entry on oeis.org

0, 1, 4, 8, 12, 14, 15, 16, 20, 24, 27, 28, 32, 35, 36, 39, 40, 44, 46, 48, 50, 51, 52, 54, 55, 56, 60, 64, 65, 68, 72, 76, 77, 80, 81, 84, 86, 87, 88, 91, 92, 94, 95, 96, 99, 100, 104, 108, 111, 112, 115, 116, 119, 120, 122, 123, 124, 125, 128, 132, 135, 136, 138, 140, 141, 143, 144, 146, 148, 152, 155, 156, 158, 159, 160, 162, 164, 168, 172, 176, 180, 183, 184, 187, 188, 189, 192, 194, 196, 200
Offset: 1

Views

Author

Antti Karttunen, Oct 13 2019

Keywords

Comments

Numbers n for which either A003415(n) = 0 or A051903(A003415(n)) > 1.

Examples

			Arithmetic derivative of 1 is A003415(1) = 0, which is not a squarefree number (not in A005117), thus 1 is included in this sequence. Ditto for 0, as A003415(0) = 0.
Arithmetic derivative of 8 is A003415(8) = 12 = 2^2 * 3, which is not squarefree, thus 8 is included in this sequence.
Arithmetic derivative of 15 is A003415(15) = 8 = 2^3, which is not squarefree, thus 15 is included in this sequence.

Links

Crossrefs

Cf. A003415, A005117, A008966, A051903.
Complement of the union of A000040 and A328234.
Cf. A328245, A328251, A328253, A328304, A328305 (subsequences).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328303(n) = !issquarefree(A003415(n));

A328246 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117).

Original entry on oeis.org

9, 14, 18, 21, 25, 33, 38, 46, 49, 57, 65, 77, 85, 93, 98, 121, 126, 129, 134, 138, 141, 145, 150, 161, 166, 177, 185, 186, 194, 201, 205, 206, 209, 217, 221, 237, 242, 249, 253, 258, 262, 265, 266, 289, 290, 301, 305, 306, 315, 322, 326, 333, 334, 338, 341, 342, 350, 361, 365, 369, 375, 377, 381, 393, 398, 402, 413, 414
Offset: 1

Views

Author

Antti Karttunen, Oct 11 2019

Keywords

Comments

Numbers n for which A008966(A003415(A003415(A003415(n)))) = 1.

Examples

			For n=9, its first arithmetic derivative is A003415(9) = 6, its second derivative is A003415(6) = 5, and its third derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6, A003415(6) = 5, and 5, like all primes, is a squarefree number, thus 14 is included in this sequence.
For n=49, A003415(49) = 14, A003415(14) = 9, A003415(9) = 6 = 2*3, and 6 is a squarefree number, thus 49 is included in this sequence.

Links

Crossrefs

Cf. A003415, A005117, A008966, A099306, A328234, A328244, A328245.
Cf. A192192, A328239, A328247 (subsequences).

Programs

A328247 Numbers whose third arithmetic derivative (A099306) is a squarefree number (A005117), but the second derivative (A068346) is not.

Original entry on oeis.org

33, 49, 98, 129, 141, 194, 205, 249, 301, 306, 445, 481, 493, 529, 549, 553, 589, 615, 681, 741, 746, 913, 917, 946, 949, 962, 973, 993, 1010, 1106, 1273, 1386, 1397, 1417, 1430, 1518, 1561, 1611, 1633, 1761, 1802, 1842, 1849, 1858, 1870, 1946, 1957, 1977, 2030, 2049, 2078, 2105, 2139, 2166, 2170, 2173, 2175, 2209, 2223, 2330
Offset: 1

Views

Author

Antti Karttunen, Oct 11 2019

Keywords

Examples

			For n=33, its first arithmetic derivative is A003415(33) = 14, its second derivative is A003415(14) = 9 = 3^2 (which is not squarefree) and its third derivative is A003415(9) = 6 = 2*3, which is, thus 33 is included in this sequence.

Links

Crossrefs

Cf. A003415, A005117, A008966, A068346, A099306, A328234, A328244, A328245.
Setwise difference A328246 \ A328244.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328247(n) = { my(u=A003415(A003415(n))); (!issquarefree(u) && issquarefree(A003415(u))); };
Showing 1-6 of 6 results.

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