A068346 -id:A068346 - cp's OEIS frontend

A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

9, 21, 25, 57, 85, 93, 121, 126, 145, 161, 185, 201, 206, 209, 221, 237, 242, 253, 265, 289, 305, 315, 326, 333, 341, 365, 369, 377, 381, 413, 417, 437, 453, 458, 490, 495, 497, 517, 537, 542, 545, 565, 566, 575, 578, 597, 605, 633, 637, 638, 649, 666, 685

Offset: 1

Vladimir Shevelev, Jun 25 2011

nonn

The fourth A003415-iteration of a(n) is the first to be 0.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (the first 1000 terms from T. D. Noe)

Cf. A003415, A038554, A068346, A192082, A192189, A192190, A327969, A328240.
Cf. A157037, A328239 (the first and third derivative is prime).
Subsequence of following sequences: A328234, A328244, A328246.

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Select[Range[1000], dn[dn[dn[#]]] == 1&] (* T. D. Noe, Mar 07 2013 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA192192(n) = isprime(A003415(A003415(n))); \\ Antti Karttunen, Oct 19 2019

For all n, A327969(a(n)) <= 4. - Antti Karttunen, Oct 19 2019

More terms from Olivier Gérard, Jul 04 2011

New primary definition added to the name by Antti Karttunen, Oct 19 2019

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)); };

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));

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.

A192016 Second arithmetic derivative of prime powers: a(n) = A068346(A000961(n)).

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 16, 5, 0, 0, 80, 0, 0, 0, 7, 27, 0, 0, 176, 0, 0, 0, 0, 9, 0, 0, 0, 640, 0, 0, 0, 0, 216, 0, 0, 0, 0, 0, 0, 0, 0, 13, 55, 0, 1408, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 621, 0, 5120, 0, 0, 0, 0, 0, 0, 0, 19

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040. A000961, A002808, A003415, A068346, A192015.

a192016 = a068346 . a000961  -- Reinhard Zumkeller, Apr 16 2014

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; f[n_] := If[n == 1, 0, If[PrimePowerQ[n], {p, e} = FactorInteger[n][[1]]; e*p^(e-1), Nothing]]; ad /@ Array[f, 300] (* Amiram Eldar, Apr 11 2025 *)

a(n) = A003415(A192015(n)).

a(A000040(n)) = 0; a(A002808(n)) > 0.

A192084 Second arithmetic derivative of squares of prime powers: a(n)=A068346(A056798(n)).

Original entry on oeis.org

0, 4, 5, 80, 7, 9, 640, 216, 13, 15, 5120, 19, 21, 25, 800, 3645, 31, 33, 26624, 39, 43, 45, 49, 1960, 55, 61, 63, 167936, 69, 73, 75, 81, 67068, 85, 91, 99, 103, 105, 109, 111, 115, 6776, 34375, 129, 819200, 133, 139, 141, 151, 153, 159, 165, 169, 10816

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

a(n) = A003415(A192083(n));

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Antti Karttunen, Oct 11 2019

nonn

			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.

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

Cf. A003415, A005117, A008966, A068346, A099306, A328234, A328244, A328245.

Setwise difference A328246 \ A328244.

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))); };

A368921 a(n) = n' - n'', where n' is the arithmetic derivative of n, A003415(n) and n'' is its second arithmetic derivative, A068346(n).

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 4, 1, -4, 1, 6, 1, -16, 1, 3, -4, -48, 1, 11, 1, -20, 3, 12, 1, -4, 3, 7, 0, -48, 1, 30, 1, -96, 5, 18, -4, -32, 1, 11, -16, -4, 1, 40, 1, -64, 23, 15, 1, -128, 5, 6, -4, -36, 1, -27, -16, -4, 9, 30, 1, -4, 1, 19, 31, -448, -3, 60, 1, -84, 11, 58, 1, -64, 1, 23, 39, -96, -3, 70, 1, -192, -108, 42

Offset: 0

Antti Karttunen, Jan 10 2024

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A068346, A348329 (positions of 0's), A368922.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A368921(n) = { my(u=A003415(n)); (u-A003415(u)); };

a(n) = A003415(n) - A068346(n).

a(n) = A368922(n) + A068346(n).

A368922 a(n) = n' - n''*2, where n' is the arithmetic derivative of n, A003415(n) and n'' is the second arithmetic derivative, A068346(n).

Original entry on oeis.org

0, 0, 1, 1, -4, 1, 3, 1, -20, -4, 5, 1, -48, 1, -3, -16, -128, 1, 1, 1, -64, -4, 11, 1, -52, -4, -1, -27, -128, 1, 29, 1, -272, -4, 17, -20, -124, 1, 1, -48, -76, 1, 39, 1, -176, 7, 5, 1, -368, -4, -33, -28, -128, 1, -135, -48, -100, -4, 29, 1, -100, 1, 5, 11, -1088, -24, 59, 1, -240, -4, 57, 1, -284, 1, 7, 23, -272

Offset: 0

Antti Karttunen, Jan 10 2024

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A068346, A334261 (positions of -4's), A368701, A368921.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A368922(n) = { my(u=A003415(n)); (u-2*A003415(u)); };

a(n) = A003415(n) - 2*A068346(n).

a(n) = A368921(n) - A068346(n).

A370123 Numbers whose second arithmetic derivative (A068346) is a multiple of 3.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 13, 14, 15, 17, 19, 23, 24, 27, 29, 31, 33, 37, 40, 41, 43, 47, 48, 49, 50, 51, 53, 54, 56, 59, 60, 61, 65, 67, 68, 69, 71, 73, 77, 79, 81, 83, 86, 89, 91, 97, 98, 101, 103, 104, 107, 108, 109, 113, 122, 123, 127, 131, 132, 133, 135, 136, 137, 139, 140, 141, 149, 150, 151, 152, 155, 157, 158

Offset: 1

Antti Karttunen, Feb 10 2024

nonn

Cf. A068346, A370119 (subsequence), A370122 (characteristic function).

PARI
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\\ See A370122.
```

cp's OEIS Frontend

A192192 Numbers whose second arithmetic derivative (A068346) is prime; Polynomial-like numbers of degree 3.

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

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

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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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A192016 Second arithmetic derivative of prime powers: a(n) = A068346(A000961(n)).

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A192084 Second arithmetic derivative of squares of prime powers: a(n)=A068346(A056798(n)).

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

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A368921 a(n) = n' - n'', where n' is the arithmetic derivative of n, A003415(n) and n'' is its second arithmetic derivative, A068346(n).

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A368922 a(n) = n' - n''*2, where n' is the arithmetic derivative of n, A003415(n) and n'' is the second arithmetic derivative, A068346(n).

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