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A309769
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Even numbers m having at least one odd prime divisor p for which there exists a positive integer k < p-1 such that p-k|m-k.
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2
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20, 28, 42, 44, 50, 52, 66, 68, 70, 76, 78, 80, 88, 92, 102, 104, 110, 112, 114, 116, 124, 130, 136, 138, 140, 148, 152, 154, 156, 164, 170, 172, 174, 176, 182, 184, 186, 188, 190, 196, 200, 204, 208, 212, 222, 228, 230, 232, 236, 238, 242, 244, 246, 248, 252
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OFFSET
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1,1
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COMMENTS
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Complement in A005843 of A309239. Every odd number > 1 has the property mentioned in Name, but these are the only even numbers with this property. No term is either a power of 2 or a semiprime. A number m is a term if and only if m = 2rp, where r >= 2, and p is a prime > q, the smallest prime divisor of 2r-1 (k=p-q). For any given r, 2rz is the smallest multiple of 2r in this sequence, where z=nextprime(q). If m = 2rp is a term and 2r-1 is prime, then p is the greatest prime divisor of m (the converse is not true; e.g., m=70=10*7).
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LINKS
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EXAMPLE
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20 = 4*5 is a term because with k=2, 5-k|20-k.
66 = 6*11 is a term (k=6), although when expressed as 66=22*3 no k exists.
110 = 10*11 = 22*5 is a term for two reasons, since with both of its odd prime factors it has the required property; 5-2|110-2 and 11-8|110-8. This is the smallest term having two distinct odd prime factors, both of which have the above property (see A309780, A309781).
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MATHEMATICA
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kQ[n_, p_] := Module[{ans = False}, Do[If[Divisible[n - k, p - k], ans = True; Break[]], {k, 1, p - 2}]; ans]; aQ[n_] := EvenQ[n] && Length[(p = FactorInteger[ n][[2 ;; -1, 1]])] > 0 && AnyTrue[p, kQ[n, #] &]; Select[Range[252], aQ] (* Amiram Eldar, Aug 17 2019 *)
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PROG
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(PARI) getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)); ); }
isok(m) = {if ((m % 2) == 0, my(f = factor(m)[, 1]~); if (#f == 1, return (0)); for (i=2, #f, if (getk(f[i], m), return(1)); ); ); } \\ Michel Marcus, Aug 26 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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