A361872 - OEIS

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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).

1, 1, 3, 8, 108, 1107, 15788, 252603, 5121763

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..9.

Wikipedia, Practical number and Primorial

EXAMPLE

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.

MATHEMATICA

f[p_, e_] := (p^(e + 1) - 1)/(p - 1);

pracQ[fct_] := (ind=Position[fct[[;; , 1]]/(1+FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)])=={};

pracTestQ[fct_, k_] := Module[{f=fct}, f[[k, 2]]-= 1; pracQ[f]];

primPracQ[n_] := Module[{fct=FactorInteger[n]}, pracQ[fct]&&AllTrue[Range@Length[fct], fct[[#, 2]]==1||!pracTestQ[fct, #] &]];

pri[n_] := Module[{m}, If[n==1, 1, Product[Prime[m], {m, 1, n-1}]]];

plst=Join[{1}, Select[Range[2, 10^9, 2], primPracQ]]; pasc=<||>;

Do[AppendTo[pasc, <|plst[[n]]->n|>], {n, 1, Length@plst}]; Table[pasc[pri[n+1]]-pasc[pri[n]], {n, 1, 9}]

PROG

(PARI)

f(n) = factorback(primes(n)); \\ A002110

a(n) = sum(k=f(n-1)+1, f(n), is_A267124(k)); \\ Michel Marcus, Mar 28 2023