A328312 -id:A328312 - cp's OEIS frontend

A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308.

4, 8, 12, 15, 16, 20, 24, 26, 27, 28, 32, 35, 36, 39, 40, 44, 45, 48, 50, 51, 52, 54, 55, 56, 60, 63, 64, 68, 69, 72, 74, 75, 76, 80, 81, 84, 86, 87, 88, 90, 91, 92, 95, 96, 99, 100, 102, 104, 106, 108, 110, 111, 112, 115, 116, 117, 119, 120, 122, 123, 124, 125, 128, 132

Offset: 1

T. D. Noe, Oct 12 2004

nonn

Numbers of the form n = m*p^p (where p is prime), i.e., multiples of some term in A051674, have n' = (m + m')*p^p, which is again of the same form, but strictly larger iff m > 1. Therefore successive derivatives grow to infinity in this case, and they are constant when m = 1. There are other terms in this sequence, but I conjecture that they all eventually lead to a term of this form, e.g., 26 -> 15 -> 8 etc. - M. F. Hasler, Apr 09 2015

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099308 (numbers whose k-th arithmetic derivative is zero for some k).
Cf. A341999 (characteristic function),
Positions of zeros in A256750, A351078, A351079 (after their initial zeros), also in A328308, A328312.
Subsequences include: A100716, A327929, A327934, A328251, A359547 (intersection with A048103).

is(n)=until(4>n=factorback(n~)*sum(i=1,#n,n[2,i]/n[1,i]), for(i=1,#n=factor(n)~,n[1,i]>n[2,i]||return(1))) \\ M. F. Hasler, Apr 09 2015

A328311 a(n) = 1 + A051903(A003415(n)) - A051903(n), a(1) = 0 by convention.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 13 2019

nonn

All terms are nonnegative because taking the arithmetic derivative (A003415) of n may decrease its "degree" (i.e., its maximal exponent, A051903) by at most one, and in many cases may also increase it, or keep it same.

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

One more than A328310.
Cf. A003415, A051903, A328312.
Cf. A328320 (indices of zeros), A328321 (of nonzero terms).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));

a(1) = 0, for n > 1, a(n) = 1 + A051903(A003415(n)) - A051903(n).

For n > 1, a(n) = 1 + A328310(n).

cp's OEIS Frontend

A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308.

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