A329622 - OEIS

%I #9 Nov 18 2019 22:04:38

%S -1,0,1,0,2,1,3,1,2,2,4,0,5,3,-1,0,6,1,7,1,0,4,8,1,0,5,4,2,9,0,10,3,1,

%T 6,-3,-2,11,7,2,4,12,5,13,3,1,8,14,2,0,-5,3,4,15,3,2,-1,4,9,16,1,17,

%U 10,2,2,-5,-4,18,5,5,-2,19,1,20,11,-2,6,-9,-3,21,3,0,12,22,4,-2,13,6,0,23,-4,-8,7,7,14,3

%N a(n) = A056239(n) - A324888(n) = A001222(A108951(n)) - A001222(A324886(n)).

%H Antti Karttunen, <a href="/A329622/b329622.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F a(n) = A056239(n) - A324888(n) = A001222(A108951(n)) - A001222(A324886(n)).

%t With[{b = MixedRadix[Reverse@ Prime@ Range@ 500]}, Array[Subtract @@ PrimeOmega@ {#, Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[#, b]} &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 95]] (* _Michael De Vlieger_, Nov 18 2019 *)

%o (PARI)

%o A034386(n) = prod(i=1, primepi(n), prime(i));

%o A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A329622(n) = { my(u=A108951(n)); (bigomega(u) - bigomega(A276086(u))); };

%Y Cf. A001222, A034386, A056239, A108951, A276086, A324886, A324888, A329619, A329621.

%K sign

%O 1,5

%A _Antti Karttunen_, Nov 18 2019