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A374030
Number of decompositions of 2n-1 into sums of a prime and a primitive practical number (A267124).
1
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.
The graph of this sequence forms a comet that is analogous to Goldbach's comet. - Frank M Jackson, Sep 18 2024
LINKS
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
Sequence in context: A271608 A087740 A029439 * A225743 A218828 A075117
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jun 26 2024
STATUS
approved