A374030 Number of decompositions of 2n-1 into sums of a prime and a primitive practical number (A267124).

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, 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, 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

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..94.

Example

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

Mathematica

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]]];
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];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
f[n_] := Select[2n-1-Prime[Range[PrimePi[2n-1]]], PPracticalQ]; Table[Length@f[n], {n, 1, 200}]

Prog

(PARI) a(n) = my(nn=2*n-1); sum (i=1, nn, isprime(i) && is_A267124(nn-i)); \\ Michel Marcus, Jun 27 2024

Crossrefs

Cf. A045917, A267124.

Sequence in context: A271608 A087740 A029439 * A225743 A218828 A075117

Adjacent sequences: A374027 A374028 A374029 * A374031 A374032 A374033

Keyword

nonn

Author

Frank M Jackson, Jun 26 2024

Status

approved