The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A361301 For the odd number 2n + 1, the least primitive practical number r such that 2n + 1 = r + p where p is prime. 1
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
LINKS
EXAMPLE
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.
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]);
part[n_, m_] := Module[{p = NextPrime[n, -m], d}, d = n - p; {d, p}];
find[n_] := Module[{m=1}, While[!PPracticalQ[part[n, m][[1]]], m++]; part[n, m]];
Table[find[2 n + 1][[1]], {n, 1, 1000}]
PROG
(Python)
from sympy import prevprime, factorint
# See A005153 for is_A005153(). - M. F. Hasler, Jun 19 2023
def is_primitive(n):
for i in range(0, len(list(factorint(n)))):
if list(factorint(n).values())[i] > 1:
if is_A005153(n//list(factorint(n))[i]): return False
return True
def is_A267124(n):
if is_A005153(n) and is_primitive(n) : return True
A361301 = []
for odds in range(3, 192, 2):
prime = prevprime(odds)
while not is_A267124(odds - prime): prime = prevprime(prime)
A361301.append(odds - prime)
print(A361301) # Karl-Heinz Hofmann, Mar 10 2023
CROSSREFS
Sequence in context: A270380 A241460 A167568 * A240501 A278251 A286847
KEYWORD
nonn
AUTHOR
Frank M Jackson, Mar 08 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 30 18:48 EDT 2024. Contains 372974 sequences. (Running on oeis4.)