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A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308. 8
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 (list; graph; refs; listen; history; text; internal format)
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).

Sequence in context: A248159 A190679 A009023 * A235865 A073432 A133803

Adjacent sequences:  A099306 A099307 A099308 * A099310 A099311 A099312

KEYWORD

nonn

AUTHOR

T. D. Noe, Oct 12 2004

STATUS

approved

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Last modified November 18 10:38 EST 2017. Contains 294887 sequences.