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

1,1

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

Prog

(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

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).

Sequence in context: A248159 A190679 A009023 * A327929 A327864 A235865

Adjacent sequences: A099306 A099307 A099308 * A099310 A099311 A099312

Keyword

nonn

Author

T. D. Noe, Oct 12 2004

Status

approved