

A333197


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


2



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
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OFFSET

1,2


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


LINKS



EXAMPLE

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.


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:


MATHEMATICA

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


PROG

(PARI) isok(m) = !isprime(m) && (sumdiv(m, d, !isprime(d) && (isprime(d+1)  ((d>1) && isprime(d1)))) == sumdiv(m, d, !isprime(d))); \\ Michel Marcus, Mar 11 2020


CROSSREFS



KEYWORD

nonn,changed


AUTHOR



STATUS

approved



