A230454
Smallest odious number (A000069) that can be written as a product of n, but not fewer than n, evil numbers (A001969).
Original entry on oeis.org
25, 575, 51175, 4554575, 405357175
Offset: 2
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
A230385
Table read by rows: Least set of n evil numbers (A001969) such that any two or more add up to an odious number (A000069); ordered by total sum of the elements, then by the size of the largest element(s).
Original entry on oeis.org
0, 3, 5, 9, 10, 12, 5, 9, 17, 33, 33, 34, 36, 40, 48, 257, 264, 278, 288, 326, 384
Offset: 1
The table reads
n=1: {0} with sum = 0,
n=2: {3,5} with sum = 8,
n=3: {9, 10, 12} with sum = 31 (the set {5, 9, 17} having the same sum but a larger maximum),
n=4: {5, 9, 17, 33} with sum = 64,
n=5: {33, 34, 36, 40, 48 } with sum = 191.
n=6: {257, 264, 278, 288, 326, 384} with sum = 1797.
For example, for n=4, all 11 numbers 5+9=14,5+17=22,5+33=38,9+17=26, 9+33=42, 17+33=50, 5+9+17=31, 5+9+33=47, 5+17+33=55, 9+17+33=59, 5+9+17+33=64 are odious.
A230387
Least sum of a set of n odious numbers (A000069) such that the sum of two or more is an evil number (A001969).
Original entry on oeis.org
1, 3, 17, 139, 795, 3903, 28575
Offset: 1
For n=1 to 4, we have the sets
n=1: {1} with sum = 1,
n=2: {1, 2} with sum = 3
n=3: {2, 7, 8} with sum = 17,
n=4: {4, 19, 49, 67} with sum = 139.
E.g., for n=3, the numbers 2, 7 and 8 have an odd bit sum, but 2+7, 2+8, 7+8 and 2+7+8 all have an odd bit sum.
For n=4, we also have the admissible set {14, 31, 44, 61} which has a smaller maximal element, but a larger total sum.
n=5: {42, 84, 138, 174, 357} with sum = 795.
n=6: {168, 348, 372, 702, 906, 1407} with sum = 3903.
n=7: {2273, 2274, 2276, 2280, 2288, 3296, 13888} with sum = 28575.
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A69=select(is_A69=n->bittest(hammingweight(n),0),vector(700,n,n)); A230387(n,m=9e9)={ local(v=vector(n,i,i), ve=vector(n,i,A69[i]), t=0, s=vector(n,i,if(i>1,A230387(i-1))), ok(e)=!forstep(i=3,2^#e-1,2, is_A69( sum( j=1,#t=vecextract(e,i),t[j] )) && return), inc(i)=for(j=1,n-i,v[j]=j); for(j=n-i+1,n-1, v[j]++
Showing 1-3 of 3 results.
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