cp's OEIS Frontend

A361872 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).

%I A361872 #13 Jun 21 2023 06:41:12
%S A361872 1,1,3,8,108,1107,15788,252603,5121763
%N A361872 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 A361872 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 A361872 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 A361872 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 A361872 f[p_, e_] := (p^(e + 1) - 1)/(p - 1);
%t A361872 pracQ[fct_] := (ind=Position[fct[[;; , 1]]/(1+FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)])=={};
%t A361872 pracTestQ[fct_, k_] := Module[{f=fct}, f[[k, 2]]-= 1; pracQ[f]];
%t A361872 primPracQ[n_] := Module[{fct=FactorInteger[n]}, pracQ[fct]&&AllTrue[Range@Length[fct], fct[[#, 2]]==1||!pracTestQ[fct, #] &]];
%t A361872 pri[n_] := Module[{m}, If[n==1, 1, Product[Prime[m], {m, 1, n-1}]]];
%t A361872 plst=Join[{1}, Select[Range[2, 10^9, 2], primPracQ]]; pasc=<||>;
%t A361872 Do[AppendTo[pasc, <|plst[[n]]->n|>], {n, 1, Length@plst}]; Table[pasc[pri[n+1]]-pasc[pri[n]], {n, 1, 9}]
%o A361872 (PARI)
%o A361872 f(n) = factorback(primes(n)); \\ A002110
%o A361872 a(n) = sum(k=f(n-1)+1, f(n), is_A267124(k)); \\ _Michel Marcus_, Mar 28 2023
%Y A361872 Cf. A002110, A267124.
%K A361872 nonn,more
%O A361872 1,3
%A A361872 _Frank M Jackson_, Mar 27 2023