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 A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1. 9
 9, 161, 209, 221, 2189, 2561, 3281, 3629, 5249, 5549, 6401, 7181, 7661, 8321, 8909, 9089, 9869, 10001, 10349, 10541, 10961, 11009, 11021, 29861, 38981, 52601, 66149, 84101, 93029, 97481, 132809, 150281, 158969, 163301, 197669, 214661, 227321, 235721, 285449, 321989, 338021, 357881, 369701, 381449, 385349, 416261, 420089, 442889 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime. Composite terms in A328232. Although it first might seem that the numbers whose arithmetic derivative is A002110(k) all appear before any of those whose arithmetic derivative is A002110(k+1), that is not true, as for example, we have a(56) = 570149, and A003415(570149) = 2310, a(57) = 570209, and A003415(570209) = 30030, but then a(58) = 573641 with A003415(573641) = 2310 again. Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present. Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9. Of the first 10000 terms, all others are semiprimes (with 9 the only square one), except 1547371 = 7^2 * 23 * 1373 and 79332523 = 17^2 * 277 * 991, the latter being the only known term whose decimal expansion ends with 3. If all solutions were semiprimes p*q such that p+q = A002110(k) for some k > 1 (see A002375), it would be a sufficient reason for the above conjecture to hold. - David A. Corneth and Antti Karttunen, Oct 11 2019 In any case, the solutions have to be of the form "odd numbers with an even number of prime factors with multiplicity" (see A235992), and terms must also be cubefree (A004709), as otherwise the arithmetic derivative would not be squarefree. LINKS Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen) Victor Ufnarovski and Bo Ã…hlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4. FORMULA A327969(a(n)) = 4 for all n. MATHEMATICA ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;; , 2]]] == 1 && PrimePi[f[[-1, 1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *) PROG (PARI) A002620(n) = ((n^2)>>2); A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow! k=0; for(n=1, A002620(30030), if(isA327978flat(n), k++; write("b327978.txt", k, " ", n))); CROSSREFS Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243. Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234. Sequence in context: A309472 A060348 A217392 * A062232 A020523 A023039 Adjacent sequences:  A327975 A327976 A327977 * A327979 A327980 A327981 KEYWORD nonn AUTHOR Antti Karttunen, Oct 09 2019 STATUS approved

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)