A230385 -id:A230385 - cp's OEIS frontend

A230386 Least sum of a set of n evil numbers (A001969) such that any two or more add to an odious number (A000069).

0, 8, 31, 64, 191, 1797, 18463

Offset: 1

Vladimir Shevelev and M. F. Hasler, Oct 17 2013

Row sums of A230385: see there for the corresponding sets. See A230387 for a "dual" version.
Is this sequence finite, or is there for any n at least one admissible set of n evil numbers, i.e., such that any sum of two or more elements add up to an odious number?
By definition, this is a subsequence of the odious numbers A000069.

			The table A230385 reads
n=1: {0} with sum = 0,
n=2: {3, 5} with sum = 8,
n=3: {5, 9, 17} or {9, 10, 12} with sum = 31,
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.
n=7: {801, 802, 804, 808, 816, 4896, 9536} with sum = 18463.

M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

(is_A69=n->bittest(hammingweight(n),0)); A1969=select(n->!is_A69(n), vector(1600,n,n)) /* no 0 here! */; A230386(n,m=9e9)={ local(v=vector(n,i,i), ve=vector(n,i,A1969[i]), t=0, s=vector(n,i,if(i>1,A230386(i-1))), S(v)=sum(j=1,#v,v[j]), ok(e)=!forstep(i=3,2^#e-1,2, is_A69( S( vecextract( e,i )))||return), inc(i)=for(j=1,n-i,v[j]=j); for(j=n-i+1,n-1, v[j]++S(ve) && /*print*/([m=S(ve),ve]); inc(n));m} /* This code is very fast up to n=5 and much too slow for n>5. */

a(6) added by M. F. Hasler, Oct 18 2013

a(7) from Donovan Johnson, Oct 27 2013

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

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

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

M. F. Hasler, Oct 17 2013

Is this sequence finite, or is there for any n at least one admissible set of n odious numbers, i.e., such that any sum of two or more elements add up to an evil number?

			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.

M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

Cf. A230386, A230385.

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]++

Row sums of A230384.

a(5)-a(6) from Charles R Greathouse IV, Oct 18 2013

a(7) from Donovan Johnson, Oct 27 2013

A230384 Table read by rows: Least set of n odious numbers (A000069) such that any two or more add to an evil number (A001969); ordered by total sum of elements, then by largest element(s).

Original entry on oeis.org

1, 1, 2, 2, 7, 8, 4, 19, 49, 67, 42, 84, 138, 174, 357, 168, 348, 372, 702, 906, 1407

Offset: 1

M. F. Hasler, Oct 17 2013

Row sums are given in A230387. See A230385 for a "dual" version.

Is this sequence finite, or is there for any n at least one admissible set of n odious numbers, i.e., such that any sum of two or more elements add up to an evil number?

			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.

M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

Cf. A230385, A230386.

a(11)-a(21) from Charles R Greathouse IV, Oct 18 2013

cp's OEIS Frontend

A230386 Least sum of a set of n evil numbers (A001969) such that any two or more add to an odious number (A000069).

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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).

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A230387 Least sum of a set of n odious numbers (A000069) such that the sum of two or more is an evil number (A001969).

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PARI

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Extensions

A230384 Table read by rows: Least set of n odious numbers (A000069) such that any two or more add to an evil number (A001969); ordered by total sum of elements, then by largest element(s).

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Author

Keywords

Comments

Examples

Links

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Extensions