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A252493 Numbers n such that n(n+1) is 13-smooth. (Related to the abc conjecture.)

%I A252493 #18 Jan 01 2019 10:22:41
%S A252493 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,20,21,24,25,26,27,32,35,39,44,48,
%T A252493 49,54,55,63,64,65,77,80,90,98,99,104,120,125,143,168,175,195,224,242,
%U A252493 324,350,351,363,384,440,539,624,675,728,1000,1715,2079,2400,3024,4095,4224,4374,6655,9800,10647,123200
%N A252493 Numbers n such that n(n+1) is 13-smooth. (Related to the abc conjecture.)
%C A252493 Equivalently: Numbers n such that all prime factors of n and n+1 are <= 13, i.e., both are in A080197.
%C A252493 This sequence is complete by a theorem of Stormer, cf. A002071.
%C A252493 This is the 6th row of the table A138180. It has 68=A002071(6)=A145604(1)+...+ A145604(6) terms and ends with A002072(6)=123200. It is the union of all terms in rows 1 through 6 of the table A145605.
%C A252493 Contains A085152, A085153, A252494 as subsequences.
%H A252493 Abderrahmane Nitaj, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html">The ABC conjecture homepage</a>
%H A252493 <a href="/index/Ab#abc">OEIS Index entries for sequences related to the abc conjecture</a>
%p A252493 N:= 130000: # to get all entries <= N
%p A252493 f:= proc(n)
%p A252493 uses padic;
%p A252493 evalb(2^ordp(n,2)*3^ordp(n,3)*5^ordp(n,5)*7^ordp(n,7)*11^ordp(n,11)*13^ordp(n,13) = n)
%p A252493 end proc:
%p A252493 L:= map(f, [$1..N+1]):
%p A252493 select(t -> L[t] and L[t+1], [$1..N]); # _Robert Israel_, Jan 16 2015
%t A252493 Select[Range[123456], FactorInteger[ # (# + 1)][[ -1,1]] <= 13 &]
%o A252493 (PARI) for(n=1,123456, vecmax(factor(n++,13)[,1])<17 && vecmax(factor(n--+(n<2),13))<17 && print1(n",")) \\ Skips the next n if n+1 is not 13-smooth: Twice as fast as the naïve version. Instead of vecmax(.)<17 one could use is_A080197().
%Y A252493 Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492.
%K A252493 nonn,fini,full
%O A252493 1,2
%A A252493 _M. F. Hasler_, Jan 16 2015