A328250 -id:A328250 - cp's OEIS frontend

A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212, 216, 220, 224, 225, 228, 232, 236, 240, 243, 244, 248, 250, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

This probably is NOT an intersection of A013929 and A099309.

			We see that 225 = 3^2 * 5^2 is not squarefree, and then when starting iterating with A003415, we obtain --> 240 --> 608 --> 1552 --> ... which is a trajectory that will never reach neither a prime nor any squarefree number at all, because already 240 = 2^4 * 3 * 5 is a member of A100716, whose terms all belong into A099309, as any divisor of the form p^p of n will be always present when taking its successive arithmetic derivatives. Thus 225 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Row 1 of array A328250. Indices of zeros in A328248.
Cf. A003415, A099309, A327929, A327934.
Cf. A013929, A100716 (a subsequence).
Differs from A100716 and A276079 for the first time at a(63) = 225, the term which is not present in them.

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); };
isA328251(n) = (0==A328248(n));

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

Antti Karttunen, Oct 10 2019

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 1..10000

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.

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 *)

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)); };

A328252 Numbers that are not squarefree, but whose arithmetic derivative (A003415) is.

Original entry on oeis.org

9, 18, 25, 45, 49, 63, 75, 90, 98, 117, 121, 126, 147, 150, 153, 169, 171, 175, 198, 234, 242, 245, 261, 279, 289, 294, 315, 325, 333, 338, 342, 350, 361, 363, 369, 387, 414, 423, 425, 450, 475, 477, 490, 495, 507, 522, 529, 539, 550, 558, 575, 578, 603, 605, 630, 637, 639, 650, 657, 666, 711, 722, 726, 735, 738, 774, 775, 801

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

			18 = 2 * 3^2 is not squarefree, but its arithmetic derivative A003415(18) = 21 = 3*7 is, thus 18 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A328253.
Row 3 of array A328250. Positions of 2's in A328248.
Setwise difference A328234 \ A005117. Intersection of A013929 and A328234.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328252(n) = (!issquarefree(n) && issquarefree(A003415(n)));

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); };
isA328252(n) = (2==A328248(n));

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

Antti Karttunen, Oct 11 2019

nonn

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

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))));

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));

A328302 For n > 1, a(n) is the least number > 0 for which it takes n-2 steps to reach a squarefree number by applying arithmetic derivative (A003415) zero or multiple times. a(1) = 4 is the least number for which no squarefree number is ever reached.

Original entry on oeis.org

4, 1, 9, 50, 306, 5831, 20230, 52283, 286891, 10820131, 38452606

Offset: 1

Antti Karttunen, Oct 12 2019

The least number k such that A328248(k) = n. After the initial two terms, probably also the positions of records in A328248, that is, it is conjectured that the records in A328248 appear in order, with each new record one larger than previous.

No other terms below 2^30.

			a(2) = 1 is the least number that is squarefree already at the "zeroth derivative".
52283 = 7^2 * 11 * 97 is not squarefree, and applying A003415 successively 1-6 times yields numbers 20230, 19431, 14250, 21175, 15345, 15189. Only the last one of these 15189 = 3*61*83 is squarefree, and there are no numbers < 52283 that would produce as long (6) finite chain of nonsquarefree numbers, thus a(2+6) = 52283.

The leftmost column in A328250.

Cf. A003415, A328248, A328320, A328321.

cp's OEIS Frontend

A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A328252 Numbers that are not squarefree, but whose arithmetic derivative (A003415) is.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

A328302 For n > 1, a(n) is the least number > 0 for which it takes n-2 steps to reach a squarefree number by applying arithmetic derivative (A003415) zero or multiple times. a(1) = 4 is the least number for which no squarefree number is ever reached.

Views

Author

Keywords

Comments

Examples

Crossrefs