A351255 -id:A351255 - cp's OEIS frontend

A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

0, 1, 2, 3, 0, 5, 5, 7, 0, 5, 7, 11, 0, 13, 5, 0, 0, 17, 7, 19, 0, 7, 13, 23, 0, 7, 0, 0, 0, 29, 31, 31, 0, 5, 19, 0, 0, 37, 7, 0, 0, 41, 41, 43, 0, 0, 7, 47, 0, 5, 0, 0, 0, 53, 0, 0, 0, 13, 31, 59, 0, 61, 5, 0, 0, 7, 61, 67, 0, 0, 59, 71, 0, 73, 0, 0, 0, 7, 71, 79, 0, 0, 43, 83, 0, 13, 0, 0, 0, 89, 0, 0, 0, 19, 5

Offset: 0

Antti Karttunen, Feb 11 2022

Primes of A189483 occur only once, on the corresponding indices, while A189441 may also occur in other positions.

There are interesting white "filament-like regions" in the scatter plot.

			For n = 15, if we iterate with A003415, we get a path 15 -> 8 -> 12 -> 16 -> 32 -> 80 -> 176 -> 368 -> ..., where the terms just keep on growing without ever reaching a prime or 1, therefore a(15) = 0.
For n = 18, its path down to zero, when iterating A003415 is: 18 -> 21 -> 10 -> 7 -> 1 -> 0, and the first noncomposite term on the path is prime 7, therefore a(18) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A189441, A189483, A351079, A351259 [= a(A351255(n))].
Cf. A099309 (positions of zeros after the initial one at a(0)=0), A328115 (positions of 5's), A328117 (positions of 7's).
Cf. also A327968.

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));
A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };

For all n, a(4*n) = a(27*n) = a((p^p)*n) = a(A099309(n)) = 0.

a(p) = p for all primes p.

A351079 a(n) is the largest term encountered on the path from n to 0 when iterating the map x -> x', or 0 if 0 cannot be reached from n (or if n is 0). Here x' is the arithmetic derivative of x, A003415.

Original entry on oeis.org

0, 1, 2, 3, 0, 5, 6, 7, 0, 9, 10, 11, 0, 13, 14, 0, 0, 17, 21, 19, 0, 21, 22, 23, 0, 25, 0, 0, 0, 29, 31, 31, 0, 33, 34, 0, 0, 37, 38, 0, 0, 41, 42, 43, 0, 0, 46, 47, 0, 49, 0, 0, 0, 53, 0, 0, 0, 57, 58, 59, 0, 61, 62, 0, 0, 65, 66, 67, 0, 0, 70, 71, 0, 73, 0, 0, 0, 77, 78, 79, 0, 0, 82, 83, 0, 85, 0, 0, 0, 89

Offset: 0

Antti Karttunen, Feb 11 2022

nonn

Question: Is there any good upper bound for ratio a(n)/n? See also comments in A351261.

			For n = 15, if we iterate with A003415, we get a path 15 -> 8 -> 12 -> 16 -> 32 -> 80 -> 176 -> 368 -> ..., where the terms just keep on growing without ever reaching zero, therefore a(15) = 0.
For n = 18, its path down to zero, when iterating A003415 is: 18 -> 21 -> 10 -> 7 -> 1 -> 0, and the largest term is 21, therefore a(18) = 21.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A099307, A099309, A351078, A351261 [= a(A351255(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));
A351079(n) = { my(m=n); while(n>1, n = A003415checked(n); m = max(m,n)); if(n,m); };

For n > 0, a(n) = 0 if A099307(n) = 0, otherwise a(n) = max(n, a(A003415(n))).

a(0) = 0 and a(A099309(n)) = 0 for all n.

A351072 a(n) is the number of prime(1+n)-smooth numbers larger than one whose k-th arithmetic derivative is zero for some k.

Original entry on oeis.org

1, 5, 13, 57, 273, 1747, 12877, 105366

Offset: 0

Antti Karttunen, Feb 14 2022

Number of prime(1+n)-smooth terms > 1 in A099308 and in A351255.

See comments in A351071 about extending this sequence.

Partial sums of A351071.

Cf. A003415, A002110, A099308, A328307, A351255.

a(n) = A328307(A002110(1+n))-2.

cp's OEIS Frontend

A351078 First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351079 a(n) is the largest term encountered on the path from n to 0 when iterating the map x -> x', or 0 if 0 cannot be reached from n (or if n is 0). Here x' is the arithmetic derivative of x, A003415.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351072 a(n) is the number of prime(1+n)-smooth numbers larger than one whose k-th arithmetic derivative is zero for some k.

Views

Author

Keywords

Comments

Crossrefs

Formula