A333197 Nonprime numbers k such that each nonprime divisor of k is 1 away from a prime number.

1, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 28, 32, 38, 40, 44, 46, 48, 58, 62, 74, 80, 82, 88, 96, 106, 148, 158, 164, 166, 178, 194, 212, 226, 262, 278, 314, 316, 332, 346, 358, 382, 388, 398, 422, 458, 466, 478, 502, 524, 542, 556, 562, 586, 614, 632, 662, 674

Offset: 1

Author

Michel Lagneau, Mar 11 2020

Keywords

• nonn

Comments

Let {d(i), i = 1..q} be the set of the q nonprime divisors of a number m. The sequence lists the nonprime numbers such that |d(i) - p(i)| = 1 for all i, where p(i) is prime.

Conjecture: except for a(n) = 4, 8, 16 and 32, a(n) is of the form 2^i*p^j with p = 3, 5, 7, 11, 19, 23, 29, 31, ... ({A120628} minus {2}).

Consequence: 2 * A120628(k) is in the sequence for k >= 1.

Note that all nonprime divisors of a term of the sequence must be 1 or even. Thus a term of the sequence can have at most one odd prime divisor, i.e., it is a power of 2 or 2^i*p where p is an odd prime. In the latter case, since 2*p is a nonprime divisor, p must be in A120628. - Robert Israel, Apr 12 2020

Examples

			48 is in the sequence because the nonprime divisors of 48 are {1, 4, 6, 8, 12, 16, 24, 48} and:
|1 - 2| = 1,
|4 - 5| = 1 (or |4 - 3| = 1),
|6 - 7| = 1 (or |6 - 5| = 1),
|8 - 7| = 1,
|12 - 13| = 1 (or |12 - 11| = 1),
|16 - 17| = 1,
|24 - 23| = 1,
|48 - 47| = 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002808, A018252, A120628, A334026.

Programs

Maple

with(numtheory):
for n from 1 to 50 do:
if type(n,prime)=false
  then
d:=divisors(n):n0:=nops(d):it:=0:
  for k from 1 to n0 do :
   if nextprime(d[k])- d[k]= 1
      or
      d[k] - prevprime(d[k])= 1
      or
      isprime(d[k])
       then
       it:=it+1:
       eles
    fi:
   od:
   if it=n0
   then
   printf(`%d, `,n):
     else fi:
  fi:
od:
# Alternative:
N:= 1000: # for terms <= N
P,NP:= selectremove(isprime, [$1..N]):
P:= convert(P,set):
P1:= P union map(`+`,P,1) union map(`-`,P,1):
filter:= proc(n) numtheory:-divisors(n) subset P1 end proc:
select(filter, NP); # Robert Israel, Apr 12 2020

Mathematica

seqQ[n_] := !PrimeQ[n] &&  AllTrue[Divisors[n], AnyTrue[# + {-1,0,1}, PrimeQ] &]; Select[Range[700], seqQ] (* Amiram Eldar, Mar 11 2020 *)

PARI

isok(m) = !isprime(m) && (sumdiv(m, d, !isprime(d) && (isprime(d+1) || ((d>1) && isprime(d-1)))) == sumdiv(m, d, !isprime(d))); \\ Michel Marcus, Mar 11 2020