A309780 -id:A309780 - cp's OEIS frontend

A309769 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.

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

Offset: 1

David James Sycamore, Aug 16 2019

nonn

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).

			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).

Cf. A005843, A309239, A309155, A307390, A309780, A309781.

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 *)

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

A309781 a(n) is the smallest even number m having n distinct odd prime divisors p_1, p_2, ..., p_n, each of which (p_i; i=1..n) has the property that there exists a k_i (0 < k_i < p_i-1) such that p_i - k_i | m - k_i.

Original entry on oeis.org

20, 110, 2926, 43010, 704990, 37461710, 859382810, 48530806610, 2383068532130, 139761750534406, 6586251483915290, 302528651777276210, 37556939168033169170, 2727217723862008961870, 222939356264469226235810

Offset: 1

David James Sycamore, Aug 17 2019

Subsequence of A309769 and A309780. a(1) is the only nonsquarefree known term. Conjecture: For n > 1 the minimal condition requires m to be squarefree and for every odd prime divisor p of m to be such that m/p - 1 is composite with least prime divisor q < p (k=p-q). No term is divisible by 3.

Not all terms are coprime to 7 and the terms aren't all 97-smooth. For n = 16..18, there are upper bounds on a(n): 16808251841257353520347590, 1627869069994521415245268370, 202089221222977079276742661490. - David A. Corneth, Sep 26 2019

			a(2) = 110 = (2*5)*11; q = 3 < 11; also 110=(2*11)*5; q = 3 < 5.

Cf. A309769, A309780.

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 && AllTrue[p, kQ[n, #] &]; oddomega[n_] := PrimeNu[n / 2^IntegerExponent[n, 2]]; s = {}; om = 1; Do[If[oddomega[n] == om && aQ[n], AppendTo[s, n]; om++], {n, 2, 10^16, 2}]; s (* Amiram Eldar, Aug 17 2019 *)

getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)); ); }
isok1(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(0)); ); return (1); ); }
isok(k, n) = (omega(k/(2^valuation(k, 2))) == n) && isok1(k);
a(n) = {my(k=2*prod(k=2, n+1, prime(k))); while (!isok(k, n), k+=2); k;} \\ Michel Marcus, Aug 27 2019

a(8) from Michel Marcus, Sep 25 2019

a(9)-a(15) from David A. Corneth, Sep 26 2019

cp's OEIS Frontend

A309769 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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