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

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

Offset: 1

Author

Antti Karttunen, Oct 09 2019

Keywords

• nonn

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.

Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - Antti Karttunen, Jan 17 2024

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen)
• Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.

Cf. A351029 (number of k for which k' = A002110(n)).

Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).

Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).

Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.

Programs

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 *)

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)));

Formula

A327969(a(n)) = 4 for all n.