A230353 -id:A230353 - cp's OEIS frontend

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

25, 575, 51175, 4554575, 405357175

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Oct 19 2013

This sequence is a subsequence of the sequence {b(n)} defined as follows:
"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).
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.
10^8 < a(6) <= 405357175. - Robert Israel, Jul 18 2025
If bigomega(a(7)) = 7 then a(7) > 10^12. - David A. Corneth, Jul 21 2025

Vladimir Shevelev, An idea of new 4 sequences

Cf. A000069, A001969, A227932, A230213, A230226, A230306, A230353, A230384, A230385, A230386, A230387.

f:= proc(n) # least k such that n is the product of k evil numbers
option remember;
local t,r,x;
if convert(convert(n,base,2),`+`)::even then return 1 fi;
t:= infinity;
for x in select(s -> s^2 <= n, numtheory:-divisors(n)) minus {1} do
  t:= min(t, procname(x) + procname(n/x))
od;
t
end proc:
V:= Array(1..5): count:= 0:
for n from 1 while count < 5 do
  v:= f(n);
  if v <= 5 and V[v] = 0 then V[v]:= n; count:= count+1; fi
od:
convert(V,list); # Robert Israel, Jul 18 2025

a(6) from David A. Corneth, Jul 21 2025

cp's OEIS Frontend

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

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