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 A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308. 13
 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 * A327929 A327864 A235865 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 January 23 16:48 EST 2020. Contains 331173 sequences. (Running on oeis4.)