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A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

%I A034173 #47 Mar 17 2025 07:03:27
%S A034173 1,2,33,19940,204323,380480345,440738966073
%N A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.
%C A034173 a(8) > 10^13. - _Donovan Johnson_, Oct 20 2009
%C A034173 _Don Reble_ has shown that a(8) < 1.9*10^42, cf. link.
%C A034173 From _David Wasserman_, Jan 05 2019: (Start)
%C A034173 a(8) <= 108111092880293127811946663766147737122,
%C A034173 a(9) <= 6850672946809600696044301071559918192380244,
%C A034173 a(10) <= 96037988156124494415303285590850571857698741869620,
%C A034173 a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)
%H A034173 Don Reble, <a href="/A034173/a034173.txt">A034173(8) exists</a>, SeqFan list, Oct 23 2012.
%F A034173 a(n) = A034174(n) - n + 1. - _Max Alekseyev_, Nov 10 2009
%F A034173 a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - _M. F. Hasler_, Oct 28 2012
%e A034173 a(4) = 19940 because 19940, ..., 19943 all have the form p^2 q r.
%o A034173 (PARI) A034173(n)={my(f);for(k=1,oo,f=0;for(i=1,n, f==(f=vecsort(factor(k+n-i)[,2])) || i==1 || [k+=n-i; next(2)]);return(k))} \\ _M. F. Hasler_, Oct 23 2012
%Y A034173 Cf. A034174.
%Y A034173 Cf. A006558, A083790.
%Y A034173 Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - _M. F. Hasler_, Oct 28 2012
%K A034173 hard,nonn,more
%O A034173 1,2
%A A034173 _Dean Hickerson_, Oct 01 1998
%E A034173 a(7) from _Donovan Johnson_, Oct 20 2009
%E A034173 Don Reble link repaired by _N. J. A. Sloane_, Oct 24 2024