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%I #34 Jun 21 2023 12:48:09
%S 1,2,2,2,6,2,2,6,2,2,6,2,20,6,2,2,6,6,2,28,2,2,6,2,20,6,2,20,6,2,2,6,
%T 6,2,28,2,2,6,6,2,30,2,20,6,2,20,6,30,2,28,2,2,6,2,2,6,2,20,6,20,20,
%U 28,20,2,28,2,28,6,2,2,6,6,20,42,2,2,6,6,2,30,6,2,28,2,20,6,2,20,6,2,2,6,6,88,28
%N For the odd number 2n + 1, the least primitive practical number r such that 2n + 1 = r + p where p is prime.
%C Conjecture: every odd number, beginning with 3, is the sum of a prime number and a primitive practical number. This is a tighter conjecture than that posed by _Hal M. Switkay_ (see comments of A005153).
%e a(61) = 20, because 61st odd number is 123 = {(10+113), (14+109), (16+107), (20+103), ...} and 20 is the least primitive practical number. 10 and 14 are not practical numbers and 16 is practical but not primitive.
%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 part[n_, m_] := Module[{p = NextPrime[n, -m], d}, d = n - p; {d, p}];
%t find[n_] := Module[{m=1}, While[!PPracticalQ[part[n, m][[1]]], m++]; part[n, m]];
%t Table[find[2 n + 1][[1]], {n, 1, 1000}]
%o (Python)
%o from sympy import prevprime, factorint
%o # See A005153 for is_A005153(). - _M. F. Hasler_, Jun 19 2023
%o def is_primitive(n):
%o for i in range(0, len(list(factorint(n)))):
%o if list(factorint(n).values())[i] > 1:
%o if is_A005153(n//list(factorint(n))[i]): return False
%o return True
%o def is_A267124(n):
%o if is_A005153(n) and is_primitive(n) : return True
%o A361301 = []
%o for odds in range(3, 192, 2):
%o prime = prevprime(odds)
%o while not is_A267124(odds - prime): prime = prevprime(prime)
%o A361301.append(odds - prime)
%o print(A361301) # _Karl-Heinz Hofmann_, Mar 10 2023
%Y Cf. A005153, A267124.
%K nonn
%O 1,2
%A _Frank M Jackson_, Mar 08 2023
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