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A230454 Smallest odious number (A000069) that can be written as a product of n, but not fewer than n, evil numbers (A001969).

%I A230454 #36 Jul 23 2025 01:00:40
%S A230454 25,575,51175,4554575,405357175
%N A230454 Smallest odious number (A000069) that can be written as a product of n, but not fewer than n, evil numbers (A001969).
%C A230454 This sequence is a subsequence of the sequence {b(n)} defined as follows:
%C A230454 "Odious numbers which can be written as a product of evil numbers." It differs from A230213 only at the 56th term (b(56) = a(3) = 575).
%C A230454 An algorithm for calculation of {b(n)} is the following: Consider an odious number n. Let d_1 be the smallest evil divisor of n (if n does not have an evil divisor, then n is not in {b(n)}). If n/d_1 is either evil or odious but is already in {b(n)}, then n is in this sequence. If n/d_1 is odious and not in the sequence, then we consider the following evil divisor d_2 > d_1 (if d_2 does not exist, then n is not in {b(n)}). If n/d_2 is either evil or odious but already in this sequence, then n is in {b(n)}, etc. Formally, by a continuation of {b(n)} sufficiently far, we can calculate terms a(k), k=2,3,4,... A direct calculation for an upper limit of, say, a(4) is connected with the finding of 4 evil primes p,q,r,s with the smallest possible product, such that all 11 numbers p*q, p*r, p*s, q*r, q*s, r*s, p*q*r, p*q*s, p*r*s, q*r*s, p*q*r*s are odious. In this case we find p=5, q=5, r=23, s=89, such that a(4) = 5*5*23*89 = 51175.
%C A230454 10^8 < a(6) <= 405357175. - _Robert Israel_, Jul 18 2025
%C A230454 If bigomega(a(7)) = 7 then a(7) > 10^12. - _David A. Corneth_, Jul 21 2025
%H A230454 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-October/011790.html">An idea of new 4 sequences</a>
%p A230454 f:= proc(n) # least k such that n is the product of k evil numbers
%p A230454 option remember;
%p A230454 local t,r,x;
%p A230454 if convert(convert(n,base,2),`+`)::even then return 1 fi;
%p A230454 t:= infinity;
%p A230454 for x in select(s -> s^2 <= n, numtheory:-divisors(n)) minus {1} do
%p A230454   t:= min(t, procname(x) + procname(n/x))
%p A230454 od;
%p A230454 t
%p A230454 end proc:
%p A230454 V:= Array(1..5): count:= 0:
%p A230454 for n from 1 while count < 5 do
%p A230454   v:= f(n);
%p A230454   if v <= 5 and V[v] = 0 then V[v]:= n; count:= count+1; fi
%p A230454 od:
%p A230454 convert(V,list); # _Robert Israel_, Jul 18 2025
%Y A230454 Cf. A000069, A001969, A227932, A230213, A230226, A230306, A230353, A230384, A230385, A230386, A230387.
%K A230454 nonn,base,more
%O A230454 2,1
%A A230454 _Vladimir Shevelev_ and _Peter J. C. Moses_, Oct 19 2013
%E A230454 a(6) from _David A. Corneth_, Jul 21 2025