A361289 - OEIS

A361289

For the odd numbers 2n + 1, the least practical number r such that 2n + 1 = r + p where p is prime.

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 8, 16, 12, 18, 2, 4, 2, 4, 6, 2, 2, 4, 6, 8, 12, 2, 2, 4, 6, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4, 6, 8, 12

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OFFSET

1,2

COMMENTS

Conjecture by Hal M. Switkay: every odd number, beginning with 3, is the sum of a prime number and a practical number. Note that this conjecture occupies the space between the unproven Goldbach conjecture and the theorem that every even number, beginning with 2, is the sum of two practical numbers (Melfi's 1996 proof of Margenstern's conjecture).

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

a(61) = 16, because the 61st odd number is 123 = {(10+113), (14+109), (16+107), ...} and 16 is the least practical number. 10 and 14 are not 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]]];

part[n_, m_] := Module[{p=NextPrime[n, -m], d}, d=n-p; {d, p}];

find[n_] := Module[{m=1}, While[!PracticalQ[part[n, m][[1]]], m++]; part[n, m]];

Table[find[2 n + 1][[1]], {n, 1, 1000}]

PROG

(Python)

from sympy import factorint, prevprime

# Stale copy of code for is_A005153 deleted, please see there. Code duplication potentially creates important problems. - M. F. Hasler, Jun 19 2023

A361289 = []

for odds in range(3, 192, 2):

prime = prevprime(odds)

while not is_A005153(odds - prime): prime = prevprime(prime)

A361289.append(odds - prime)