A230213 -id:A230213 - 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

A230226 Odd odious numbers (A000069) which can be written as a product of two odious numbers > 1.

Original entry on oeis.org

49, 91, 121, 133, 143, 217, 247, 259, 273, 341, 361, 385, 403, 451, 475, 481, 511, 517, 539, 589, 611, 625, 637, 651, 665, 671, 721, 737, 749, 767, 775, 779, 793, 803, 805, 847, 861, 871, 875, 889, 925, 949, 959, 961, 1001, 1015, 1027, 1029, 1053, 1057, 1067

Offset: 1

Vladimir Shevelev, Oct 12 2013

			From _Bernard Schott_, Sep 23 2019: (Start)
49 = 7 * 7 with 7 = 111_2 and 49 = 110001_2 hence 49 is a term.
91 = 7 * 13 with 7 = 111_2, 13 = 1101_2 and 91 = 1011011_2, hence 91 is a term. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A230213.

odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]]; fQ[n_] := Module[{f, i}, If[PrimeQ[n], False, f = Select[Divisors[n], # > 1 && # <= Sqrt[n] &]; i = 1; While[i <= Length[f] && ! (odiousQ[f[[i]]] && odiousQ[n/f[[i]]]), i++]; i <= Length[f]]]; Select[Range[1, 1000, 2], odiousQ[#] && fQ[#] &] (* T. D. Noe, Oct 16 2013 *)

Extended by T. D. Noe, Oct 16 2013

A227932 Evil numbers (A001969) which can be written as a product of two odious numbers (A000069).

Original entry on oeis.org

77, 147, 154, 169, 175, 209, 231, 245, 275, 287, 294, 308, 325, 329, 338, 343, 350, 399, 407, 413, 418, 427, 441, 455, 462, 469, 483, 490, 525, 533, 550, 553, 567, 574, 588, 605, 609, 616, 649, 650, 658, 676, 679, 686, 700, 703, 715, 735, 759, 763, 777, 798

Offset: 1

Vladimir Shevelev, Oct 15 2013

			Evil number 275 = 25*11. Since 25 and 11 are odious, then 275 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969, A000069, A230213, A230226.

evilQ[n_] := EvenQ[DigitCount[n, 2][[1]]]; odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]]; fQ[n_] := Module[{f, i}, If[PrimeQ[n], False, f = Select[Divisors[n], # > 1 && # <= Sqrt[n] &]; i = 1; While[i <= Length[f] && ! (odiousQ[f[[i]]] && odiousQ[n/f[[i]]]), i++]; i <= Length[f]]]; Select[Range[1000], evilQ[#] && fQ[#] &] (* T. D. Noe, Oct 16 2013 *)

Extended by T. D. Noe, Oct 16 2013

A230306 Evil numbers (A001969) which can be written as a product of two evil numbers > 1.

Original entry on oeis.org

9, 15, 18, 27, 30, 36, 45, 51, 54, 60, 72, 75, 85, 90, 99, 102, 108, 120, 129, 135, 144, 150, 153, 159, 165, 170, 180, 189, 195, 198, 204, 207, 215, 216, 225, 231, 240, 243, 249, 255, 258, 267, 270, 288, 297, 300, 303, 306, 315, 318, 325, 330, 340, 360, 378

Offset: 1

Vladimir Shevelev, Oct 15 2013

			Evil number 75 = 3*5*5. Since 3 and 5 are evil numbers, than 75 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969, A227932, A230213, A230226.

evilQ[n_] := EvenQ[DigitCount[n, 2][[1]]]; fQ[n_] := Module[{f, i}, If[PrimeQ[n], False, f = Select[Divisors[n], # > 1 && # <= Sqrt[n] &]; i = 1; While[i <= Length[f] && ! (evilQ[f[[i]]] && evilQ[n/f[[i]]]), i++]; i <= Length[f]]]; Select[Range[1000], evilQ[#] && fQ[#] &] (* T. D. Noe, Oct 16 2013 *)

Extended by T. D. Noe, Oct 16 2013

a(1)=0 removed by Amiram Eldar, Sep 23 2019

A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers pq, pr, qr, and pq*r are odious (A000069).

Original entry on oeis.org

575, 1775, 2075, 2225, 2825, 3475, 6575, 8381, 8675, 8825, 8975, 8993, 10235, 11225, 11675, 11975, 12035, 12167, 12905, 13075, 14275, 14825, 18745, 19925, 21575, 22881, 23943, 24389, 25325, 25775, 26765, 27575, 30189, 30925, 30981, 31433, 32223, 32675, 32975

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 16 2013

These numbers are products of 3 evil numbers (A001969) but not represented as products of two evil numbers (A230213).

			For triple of evil primes {3,29,263} numbers 3*29 = 87, 3*263 = 789, 29*263 = 7627 and 3*29*263 = 22881. Thus 22881 is in the sequence.

Charles R Greathouse IV and Peter J. C. Moses, Table of n, a(n) for n = 1..10000 (first 500 from Moses)

Cf. A000069, A001969, A027699, A230213.

od(n)=hammingweight(n)%2
list(lim)=my(v=List(),pq); forprime(p=23,lim\25, if(od(p), next); forprime(q=5,min(lim\(3*p),p), if(od(q) || !od(pq=p*q), next); forprime(r=3,min(lim\pq,q), if(!od(r) && od(q*r) && od(p*r) && od(pq*r), listput(v, pq*r))))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 01 2013

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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A230226 Odd odious numbers (A000069) which can be written as a product of two odious numbers > 1.

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A227932 Evil numbers (A001969) which can be written as a product of two odious numbers (A000069).

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A230306 Evil numbers (A001969) which can be written as a product of two evil numbers > 1.

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A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers p*q, p*r, q*r, and p*q*r are odious (A000069).

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A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers pq, pr, qr, and pq*r are odious (A000069).