A374030 - OEIS

%I #16 Sep 18 2024 08:42:24

%S 0,1,1,1,2,1,2,1,1,2,1,2,3,1,1,3,4,3,3,3,2,4,3,4,5,2,3,2,2,4,4,2,3,3,

%T 3,4,6,2,3,4,4,4,5,2,6,5,2,6,3,4,5,6,2,6,8,4,4,5,4,5,4,2,5,4,4,5,6,3,

%U 5,5,4,4,6,4,4,6,2,5,6,5,3,4,3,5,7,4,3,4,5,6,5,4,4,6

%N Number of decompositions of 2n-1 into sums of a prime and a primitive practical number (A267124).

%C It has been conjectured in comments for A267124 that every odd number, beginning with 3, is the sum of a prime number and a primitive practical number. That is a(n) > 0 for n >= 2. This sequence is analogous to A045917 for the Goldbach conjecture.

%C The graph of this sequence forms a comet that is analogous to Goldbach's comet. - _Frank M Jackson_, Sep 18 2024

%H Frank M Jackson, <a href="/A374030/b374030.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 2 because 9 can be decomposed twice as 2+7, 6+3 with 3, 7 prime and 2, 6 primitive practical.

%t PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];

%t DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[!PracticalQ[n/plst[[m]]], ok=True, ok=False; Break[]], {m, 1, Length@plst}]; ok];

%t PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);

%t f[n_] := Select[2n-1-Prime[Range[PrimePi[2n-1]]], PPracticalQ]; Table[Length@f[n], {n, 1, 200}]

%o (PARI) a(n) = my(nn=2*n-1); sum (i=1, nn, isprime(i) && is_A267124(nn-i)); \\ _Michel Marcus_, Jun 27 2024

%Y Cf. A045917, A267124.

%K nonn

%O 1,5

%A _Frank M Jackson_, Jun 26 2024