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A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

%I A342456 #18 Mar 20 2021 10:55:49
%S A342456 2,3,5,9,7,25,35,15,11,49,117649,625,717409,1225,55,225,13,121,
%T A342456 1771561,2401,36226650889,184877,1127357,875,902613283,514675673281,
%U A342456 3780549773,1500625,83852850675321384784127,3025,62004635,21,17,169,4826809,14641,8254129,143,2924207,77,8223741426987700773289,59797108943,546826709
%N A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.
%C A342456 This sequence (which could be viewed as a binary tree, like the underlying A005940 and A329886) is similar to A324289, but unlike its underlying tree A283477 that generates only numbers that are products of distinct primorial numbers (i.e., terms of A129912), here the underlying tree A329886 generates all possible products of primorial numbers, i.e., terms of A025487, but in different order.
%H A342456 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A342456 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H A342456 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A342456 a(n) = A276086(A329886(n)) = A324886(A005940(1+n)).
%F A342456 For all n >= 0, gcd(a(n), A329886(n)) = 1.
%F A342456 For all n >= 1, A055396(a(n))-1 = A061395(A329886(n)) = A290251(n) = 1+A080791(n).
%F A342456 For all n >= 0, a(2^n) = A000040(2+n).
%t A342456 Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] :=
%t A342456 Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[f@ a[#] &, 43, 0]] (* _Michael De Vlieger_, Mar 17 2021 *)
%o A342456 (PARI)
%o A342456 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A342456 A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
%o A342456 A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
%o A342456 A342456(n) = A276086(A329886(n));
%Y A342456 Cf. A005940, A025487, A108951, A129912, A276086, A283980, A324886, A342457 [= 2*A246277(a(n))], A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342463 [= A342001(a(n))], A342464 [= A051903(a(n))].
%Y A342456 Cf. A324289 (a subset of these terms, in different order).
%K A342456 nonn
%O A342456 0,1
%A A342456 _Antti Karttunen_ and _Michael De Vlieger_, Mar 14 2021