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

Author

T. D. Noe, Oct 12 2004

Keywords

• nonn

Comments

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

References

• See A003415.

Links

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

Crossrefs

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).

Programs

PARI

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