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 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 (list; graph; refs; listen; history; text; internal format)
 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 Robert Israel, Table of n, a(n) for n = 1..10000 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: 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 *) PROG (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 CROSSREFS Cf. A002808, A018252, A120628, A334026. Sequence in context: A134928 A279040 A141109 * A289426 A186331 A258036 Adjacent sequences: A333194 A333195 A333196 * A333198 A333199 A333200 KEYWORD nonn,changed AUTHOR Michel Lagneau, Mar 11 2020 STATUS approved

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Last modified April 14 23:31 EDT 2024. Contains 371667 sequences. (Running on oeis4.)