A328237 -id:A328237 - cp's OEIS frontend

A328235 The least k > 0 such that the arithmetic derivative of n+k is a multiple of the arithmetic derivative of n.

1, 1, 4, 1, 15, 1, 12, 11, 8, 1, 4, 1, 13, 1, 12, 1, 20, 1, 24, 4, 23, 1, 56, 7, 10, 27, 36, 1, 28, 1, 44, 15, 114, 1, 76, 1, 84, 5, 56, 1, 48, 1, 20, 27, 53, 1, 80, 3, 36, 25, 76, 1, 81, 9, 4, 23, 26, 1, 116, 1, 64, 207, 80, 3, 52, 1, 40, 3, 82, 1, 232, 1, 205, 31, 36, 4, 27, 1, 92, 27, 88, 1, 160, 36, 130, 5, 12, 1, 81, 9, 52, 3

Offset: 2

Antti Karttunen, Oct 08 2019

nonn

			Arithmetic derivative of 6 is A003415(6) = 5. Not until at k=21 we find another number whose arithmetic derivative is a multiple of five (as A003415(21) = 10 = 2*5), thus a(6) = 21-6 = 15.

Antti Karttunen, Table of n, a(n) for n = 2..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A328236, A328237 (gives the quotient).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328235(n) = { my(d=A003415(n)); for(k=1,oo,if(!(A003415(n+k)%d), return(k))); };

A328238 Quotient A003415(n*m)/A003415(n) for the least m > 1 for which such a quotient is an integer. Here A003415(x) is the arithmetic derivative of x.

Original entry on oeis.org

4, 5, 3, 7, 12, 9, 16, 10, 20, 13, 7, 15, 28, 42, 6, 19, 16, 21, 18, 42, 44, 25, 48, 14, 52, 3, 26, 31, 60, 33, 8, 66, 68, 43, 9, 39, 56, 51, 80, 43, 84, 45, 19, 37, 92, 49, 13, 18, 24, 236, 46, 55, 16, 67, 112, 138, 88, 61, 116, 63, 124, 124, 12, 130, 132, 69, 82, 138, 140, 73, 28, 75, 119, 84, 108, 142, 156, 81, 44, 7, 164

Offset: 2

Antti Karttunen, Oct 08 2019

nonn

			Arithmetic derivative of 6 is 6' = A003415(6) = 5. Taking arithmetic derivatives of its successive multiples, we obtain 12' = 16, 18' = 21, 24' = 44, 30' = 31, and not until with A003415(6*6) = 36' = 60 we obtain a multiple of 5. Thus a(6) = 60/5 = 12.

Antti Karttunen, Table of n, a(n) for n = 2..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A328237.

Cf. A328236 (gives the corresponding m).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328238(n) = { my(d=A003415(n),t); for(m=2,oo,if(!((t=A003415(n*m))%d), return(t/d))); };

a(n) = A003415(n*A328236(n)) / A003415(n).

cp's OEIS Frontend

A328235 The least k > 0 such that the arithmetic derivative of n+k is a multiple of the arithmetic derivative of n.

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