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a(n) is the product of (1+A328311(x)) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until 1 is encountered, or 0 if no 1 is ever encountered (in which case such a product would be infinite).

Antti Karttunen, <a href="/A328312/b328312.txt">Table of n, a(n) for n = 1..65537</a>

Cf. also A328248, A328314.

a(n) is the product of (1+A328311(x)) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until 1 is encountered, or 0 if no 1 is ever encountered (in which case such a product would be infinite).

1, 1, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 1, 6, 0, 0, 1, 4, 1, 0, 4, 2, 1, 0, 2, 0, 0, 0, 1, 2, 1, 0, 12, 2, 0, 0, 1, 8, 0, 0, 1, 2, 1, 0, 0, 6, 1, 0, 6, 0, 0, 0, 1, 0, 0, 0, 4, 2, 1, 0, 1, 24, 0, 0, 12, 2, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 12, 2, 1, 0, 0, 2, 1, 0, 4, 0, 0, 0, 1, 0, 0, 0, 4, 18, 0, 0, 1, 12, 0, 0, 1, 0, 1, 0, 2

1,6

A328312(n) = { my(m=1); while(n>1, m *= (1+A328311(n)); n = A003415checked(n)); (n*m); };

a(n) is the sum of A328311(x) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until 1 is encountered, or -1 if no 1 is ever encountered (in which case such sum would be infinite).

a(n) is

0, 0, 0, -1, 0, 1, 0, -1, 1, 1, 0, -1, 0, 3, -1, -1, 0, 2, 0, -1, 2, 1, 0, -1, 1, -1, -1, -1, 0, 1, 0, -1, 4, 1, -1, -1, 0, 3, -1, -1, 0, 1, 0, -1, -1, 3, 0, -1, 3, -1, -1, -1, 0, -1, -1, -1, 2, 1, 0, -1, 0, 5, -1, -1, 4, 1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 4, 1, 0, -1, -1, 1, 0, -1, 2, -1, -1, -1, 0, -1, -1, -1, 2, 5, -1, -1, 0, 4, -1, -1, 0, -1, 0, -1, 1

0, 0, 0

1,14

1

a(62) = 5 because 62 = 2*31 (highest exponent 1), A003415(62) = 33 = 3*11 (highest exponent 1, add (1-1)+1 = 1 to sum), A003415(33) = 14 = 2*7 (highest exponent 1, add (1-1)+1 = 1 to sum), A003415(14) = 9 = 3^2 (highest exponent 2, add (2-1)+1 = 2 to sum), A003415(9) = 6 = 2*3 (highest exponent 1, add (1-2)+1 = 0 to sum), A003415(6) = 5 (highest exponent 1, add (1-1)+1 = 1 to sum), and with A003415(5) = 1, the iteration is terminated and the total sum collected is 1+1+2+0+1 = 5.

A328312(n) = { my(s=0); while(n>1, s += A328311(n); n = A003415checked(n)); if(n, s, -1); };

Cf. A008578 (positions of 01's), A099309 (of -10's).

a(n) is the sum of A328311(x) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until zero 1 is encountered, or -1 if no zero 1 is ever encountered (in which case such sum would be infinite).

a(62) = 5 asbecause 62 = 2*31 (highest exponent 1), A003415(62) = 33 = 3*11 (highest exponent 1, add (1-1)+1 = 1 to sum), A003415(33) = 14 = 2*7 (highest exponent 1, add (1-1)+1 = 1 to sum), A003415(14) = 9 = 3^2 (highest exponent 2, add (2-1)+1 = 2 to sum), A003415(9) = 6 = 2*3 (highest exponent 1, add (1-2)+1 = 0 to sum), A003415(6) = 5 (highest exponent 1, add (1-1)+1 = 1 to sum), and with A003415(5) = 1, the iteration is terminated and the total sum collected is 1+1+2+0+1 = 5.

Cf. A099309 A008578 (positions of 0's), A099309 (of -1's).

Cf. also A328248, A328314.

a(n) is the sum of A328311(x) applied over all values x obtained when arithmetic derivative (A003415) is iterated, starting from x=n, until zero is encountered, or -1 if no zero is ever encountered (in which case such sum would be infinite).

allocated for Antti Karttunena(n) is the sum of A328311(x) applied over all values x obtained when arithmetic derivative (A003415) is iterated, until zero is encountered, or -1 if no zero is ever encountered (in which case such sum would be infinite).

0, 0, 0, -1, 0, 1, 0, -1, 1, 1, 0, -1, 0, 3, -1, -1, 0, 2, 0, -1, 2, 1, 0, -1, 1, -1, -1, -1, 0, 1, 0, -1, 4, 1, -1, -1, 0, 3, -1, -1, 0, 1, 0, -1, -1, 3, 0, -1, 3, -1, -1, -1, 0, -1, -1, -1, 2, 1, 0, -1, 0, 5, -1, -1, 4, 1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 4, 1, 0, -1, -1, 1, 0, -1, 2, -1, -1, -1, 0, -1, -1, -1, 2, 5, -1, -1, 0, 4, -1, -1, 0, -1, 0, -1, 1

1,14

a(62) = 5 as

(PARI)

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

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]));

A328311(n) = if(n<=1, 0, 1+(A051903(A003415(n)) - A051903(n)));

A328312(n) = { my(s=0); while(n>1, s += A328311(n); n = A003415checked(n)); if(n, s, -1); };

Cf. A003415, A051903, A328311.

Cf. A099309 (positions of -1's).

Cf. also A328248.

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Antti Karttunen, Oct 13 2019

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