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%I #13 Jun 21 2023 06:41:12
%S 1,1,3,8,108,1107,15788,252603,5121763
%N Number of primitive practical numbers (PPNs)(A267124) between successive primorial numbers (A002110) where the PPNs q are in the range A002110(n-1) < q <= A002110(n).
%C The sequence of primorial numbers is a subset of the sequence of PPNs. Note that the sequence A002110 has an offset of 0 and A002110(0) = 1.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Practical_number">Practical number</a> and <a href="http://en.wikipedia.org/wiki/Primorial">Primorial</a>
%e a(4) = 8, because between successive primorials 30 and 210 (that includes 210) is the sequence {42, 66, 78, 88, 104, 140, 204, 210} of PPNs. It contains 8 members.
%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1);
%t pracQ[fct_] := (ind=Position[fct[[;; , 1]]/(1+FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)])=={};
%t pracTestQ[fct_, k_] := Module[{f=fct}, f[[k, 2]]-= 1; pracQ[f]];
%t primPracQ[n_] := Module[{fct=FactorInteger[n]}, pracQ[fct]&&AllTrue[Range@Length[fct], fct[[#, 2]]==1||!pracTestQ[fct, #] &]];
%t pri[n_] := Module[{m}, If[n==1, 1, Product[Prime[m], {m, 1, n-1}]]];
%t plst=Join[{1}, Select[Range[2, 10^9, 2], primPracQ]]; pasc=<||>;
%t Do[AppendTo[pasc, <|plst[[n]]->n|>], {n, 1, Length@plst}]; Table[pasc[pri[n+1]]-pasc[pri[n]], {n, 1, 9}]
%o (PARI)
%o f(n) = factorback(primes(n)); \\ A002110
%o a(n) = sum(k=f(n-1)+1, f(n), is_A267124(k)); \\ _Michel Marcus_, Mar 28 2023
%Y Cf. A002110, A267124.
%K nonn,more
%O 1,3
%A _Frank M Jackson_, Mar 27 2023
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