cp's OEIS Frontend

A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

%I A380476 #41 Feb 19 2025 10:25:45
%S A380476 4686,32406,184866,209166,388086,1099626,1714866,2111406,2166846,
%T A380476 2356206,3081606,3303366,6445806,11366106,21621606,23022366,39824466,
%U A380476 39826986,42882846,43197846,46043826,58216686,61265886,63603546,66496506,66611166,87941706,88968246,92086746,97117026,101108706,103367886,118743306,119658066
%N A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.
%C A380476 Numbers m with four or more distinct prime factors such that their arithmetic derivative (A003415) can be formed as a carryless (or "carry-free") sum (in the primorial base, A049345) of the respective summands. See the example.
%C A380476 The terms are all squarefree and even (see A380468 and A380478 to find out why). Moreover, they are all multiples of six, because A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) applied to a product of 2*p*q*r, with p, q, r three odd primes > 3 would yield three subproducts which would be multiples of 3 (consider A047247), so the 3-adic valuation of the whole product would be >= 3; hence the second smallest prime factor must be 3. For a similar reason, with terms that are product of four primes, the two remaining prime factors are either both of the form 6m+1 (A002476), or they are both of the form 6m-1 (A007528).
%C A380476 It is conjectured that there are no terms with more than four prime factors. See A380475 and A380528, A380530, also A380526.
%H A380476 Antti Karttunen, <a href="/A380476/b380476.txt">Table of n, a(n) for n = 1..433; all terms <= 2^35</a>
%H A380476 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%e A380476 4686 = 2*3*11*71 and taking subproducts of three primes at time, we obtain 2*3*11 = 66, 2*3*71 = 426, 2*11*71 = 1562, 3*11*71 = 2343. Then A380459(4686) = A276086(66) * A276086(426) * A276086(1562) * A276086(2343) = 1622849599205985150 = 2^1 * 3^2 * 5^2 * 7^6 * 11^9 * 13^1, and because all the exponents are less than the corresponding primes, the product is in A048103.
%e A380476 Considering the primorial base expansions of the same summands (subproducts), we obtain
%e A380476     2100  = A049345(66)
%e A380476    20100  = A049345(426)
%e A380476    73010  = A049345(1562)
%e A380476   101011  = A049345(2343)
%e A380476   ------
%e A380476   196221  = A049345(A003415(4686)), with the summands adding together cleanly without any carries.
%e A380476 Note how the primorial base digits at the bottom are the exponents in the product A380459(4686) given above, read from the largest to the smallest prime factor
%o A380476 (PARI) is_A380476(n) = (issquarefree(n) && (omega(n)>=4) && A380467(n)); \\ Note that issquarefree here is just an optimization as A380467(n) = 1 implies squarefreeness of n.
%Y A380476 Intersection of A033987 and A380468.
%Y A380476 Subsequence of A005117, A358673, A380478.
%Y A380476 Conjectured to be a subsequence of A046386.
%Y A380476 Cf. A003415, A001222, A048103, A276086, A349394, A380459, A380467.
%Y A380476 Cf. A002476, A007528, A047247, A047257.
%Y A380476 Cf. also A317836, A358235, A380475, A380526, A380528, A380530.
%K A380476 nonn
%O A380476 1,1
%A A380476 _Antti Karttunen_, Feb 04 2025