A328253 -id:A328253 - cp's OEIS frontend

A328248 a(n) = 1 if n is a squarefree number (A005117), otherwise a(n) = 1 + number of iterations of arithmetic derivative (A003415) needed to reach a squarefree number, or 0 if no such number is ever reached.

1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 3, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 3, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

			For n = 9, it itself is not a squarefree number, while its arithmetic derivative A003415(9) = 6 is, so it took just one iteration to find a squarefree number, thus a(9) = 1+1 = 2.
For n = 50, which is not squarefree, and its first derivative A003415(50) = 45 also is not squarefree, but taking derivative yet again, gives A003415(45) = 39 = 3*13, which is squarefree, thus a(50) = 2+1 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A008966, A256750.

Cf. A328251, A005117, A328252, A328253 (indices of terms k=0, 1, 2, 3).

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); };

a(4*n) = a(27*n) = 0 and in general, a(m * p^p) = 0, for any m >= 1 and any prime p.

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

Antti Karttunen, Oct 11 2019

nonn

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.

			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.

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

Cf. A003415, A005117, A008966, A068346, A328234, A328246.

Cf. A157037, A192192, A328239, A328245, A328253 (subsequences).

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)); };

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));

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

Antti Karttunen, Oct 13 2019

nonn

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

			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.

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

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

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));

A328245 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.

Original entry on oeis.org

14, 46, 50, 65, 77, 86, 94, 99, 122, 125, 138, 146, 207, 230, 302, 334, 343, 346, 375, 426, 531, 546, 554, 581, 590, 626, 662, 682, 686, 710, 717, 718, 725, 734, 747, 750, 819, 842, 869, 875, 931, 965, 1002, 1041, 1083, 1130, 1145, 1146, 1166, 1175, 1202, 1241, 1265, 1310, 1331, 1337, 1349, 1375, 1418, 1461, 1466, 1469, 1501, 1529, 1541

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

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

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

Cf. A003415, A005117, A008966, A068346.
Setwise difference A328244 \ A328234.
Cf. A328253 (a subsequence, nonsquarefree terms).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328245(n) = { my(u=A003415(n)); (!issquarefree(u) && issquarefree(A003415(u))); }; \\ issquarefree(0) returns 0 as zero is not considered as a squarefree number.

A328250 Square array A(n,k) read by descending antidiagonals where A(n,k) is the k-th solution x to A328248(x) = n-1.

Original entry on oeis.org

4, 8, 1, 12, 2, 9, 16, 3, 18, 50, 20, 5, 25, 99, 306, 24, 6, 45, 125, 549, 5831, 27, 7, 49, 207, 1611, 6849, 20230, 28, 10, 63, 343, 2662, 14225, 33026, 52283, 32, 11, 75, 375, 2842, 16299, 47107, 225998, 286891, 36, 13, 90, 531, 2891, 19431, 49806, 1336047, 1292750, 10820131, 40, 14, 98, 686, 4575, 21231, 117649, 1422275, 2886982, 21628098, 38452606

Offset: 1

Antti Karttunen, Oct 12 2019

Row 1 of the array is reserved for numbers for which no squarefree number is ever reached, and from then on, each row n > 1 of array gives in ascending order all natural numbers that require n-2 iterations of arithmetic derivative (A003415) to reach a squarefree number. Squarefree numbers (A005117) thus occupy the row 2, as they require no iterations.

			The upper left corner of the array:
         4,        8,       12,       16,       20,       24,       27,       28,
         1,        2,        3,        5,        6,        7,       10,       11,
         9,       18,       25,       45,       49,       63,       75,       90,
        50,       99,      125,      207,      343,      375,      531,      686,
       306,      549,     1611,     2662,     2842,     2891,     4575,     4802,
      5831,     6849,    14225,    16299,    19431,    21231,    22638,    24010,
     20230,    33026,    47107,    49806,   117649,   121671,   145386,   162707,
     52283,   225998,  1336047,  1422275,  1500759,  1576899,  2309503,  3023398,
    286891,  1292750,  2886982,  3137526,  6882453,  8703459, 15358457, 16777114,
  10820131, 21628098, 23934105, 24332763, 46295435, 51320698, 52320191, 56199375,
  38452606, ...
  ...

Index entries for sequences that are permutations of the natural numbers

Cf. A003415, A328248.
Column 1: A328302.
Rows 1 - 4 are: A328251, A005117, A328252, A328253.

up_to = 45; \\ 10585 = binomial(145+1,2)
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); };
memoA328250sq = Map();
A328250sq(n, k) = { my(v=0); if(!mapisdefined(memoA328250sq,[n,k-1],&v),if(1==k, v=0, v = A328250sq(n, k-1))); for(i=1+v,oo,if((1+A328248(i))==n,mapput(memoA328250sq,[n,k],i); return(i))); };
A328250list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328250sq(col,(a-(col-1))))); (v); };
v328250 = A328250list(up_to);
A328250(n) = v328250[n];

A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

Original entry on oeis.org

50, 99, 207, 306, 531, 549, 725, 747, 819, 931, 1083, 1175, 1611, 1775, 1899, 2057, 2075, 2299, 2331, 2367, 2499, 2525, 2842, 2853, 2891, 3425, 3577, 3610, 3771, 3789, 3843, 4059, 4149, 4311, 4475, 4575, 4626, 4693, 4775, 4998, 5239, 5274, 5341, 5547, 5634, 5706, 5715, 5746, 5819, 5949, 6147, 6223, 6275, 6381, 6413, 6475, 6575

Offset: 1

Antti Karttunen, Oct 13 2019

nonn

Numbers n for which A051903(n) = 2 and A328248(n) > 2.

			50 is not squarefree, as 50 = 2 * 5^2, and neither its arithmetic derivative A003415(50) = 45 = 3^2 * 5 is squarefree, but its second derivative A003415(45) = 39 = 3*13 is, thus 50 is included in this sequence.

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

Cf A003415, A051903, A328248, A328253.

Subsequence of A067259, A328303, A328304 and A328321.

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));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA067259(n) = (2==A051903(n));
isA328305(n) = (isA067259(n)&&(A328248(n)>2));

cp's OEIS Frontend

A328248 a(n) = 1 if n is a squarefree number (A005117), otherwise a(n) = 1 + number of iterations of arithmetic derivative (A003415) needed to reach a squarefree number, or 0 if no such number is ever reached.

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

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

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A328303 Numbers whose arithmetic derivative is not squarefree.

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A328245 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.

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A328250 Square array A(n,k) read by descending antidiagonals where A(n,k) is the k-th solution x to A328248(x) = n-1.

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A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

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