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A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

%I A327978 #69 Jan 20 2024 09:17:27
%S A327978 9,161,209,221,2189,2561,3281,3629,5249,5549,6401,7181,7661,8321,8909,
%T A327978 9089,9869,10001,10349,10541,10961,11009,11021,29861,38981,52601,
%U A327978 66149,84101,93029,97481,132809,150281,158969,163301,197669,214661,227321,235721,285449,321989,338021,357881,369701,381449,385349,416261,420089,442889
%N A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.
%C A327978 Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime.
%C A327978 Composite terms in A328232.
%C A327978 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.
%C A327978 Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present.
%C A327978 Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9.
%C A327978 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
%C A327978 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.
%C A327978 Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - _Antti Karttunen_, Jan 17 2024
%H A327978 Giovanni Resta, <a href="/A327978/b327978.txt">Table of n, a(n) for n = 1..10000</a> (first 1078 terms from Antti Karttunen)
%H A327978 Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
%H A327978 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A327978 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A327978 A327969(a(n)) = 4 for all n.
%t A327978 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 *)
%o A327978 (PARI)
%o A327978 A002620(n) = ((n^2)>>2);
%o A327978 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A327978 A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
%o A327978 isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow!
%o A327978 k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));
%Y A327978 Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.
%Y A327978 Cf. A351029 (number of k for which k' = A002110(n)).
%Y A327978 Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).
%Y A327978 Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).
%Y A327978 Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.
%K A327978 nonn
%O A327978 1,1
%A A327978 _Antti Karttunen_, Oct 09 2019