A349494 -id:A349494 - cp's OEIS frontend

A365451 Odd composite numbers k such that A349494(k) = A000120(k).

15, 27, 51, 63, 85, 95, 111, 119, 123, 125, 187, 219, 221, 255, 335, 365, 411, 447, 485, 511, 629, 655, 685, 697, 771, 831, 879, 959, 965, 1011, 1139, 1241, 1285, 1405, 1535, 1563, 1649, 1731, 1779, 1791, 1799, 1923, 1983, 2005, 2019, 2031, 2043, 2045, 2227, 2605, 2735, 2815, 2827, 2885, 3099

Offset: 1

Robert Israel, Sep 03 2023

Odd composite numbers k such that for all divisors d of k, A000120(d) * A000120(k/d) = A000120(k).

			a(4) = 63 is a term because 63 = 3 * 21 = 7 * 9 with A000120(63) = 6, A000120(3) * A000120(21) = 2 * 3 = 6 and A000120(7) * A000120(9) = 3 * 2 = 6.

Robert Israel, Table of n, a(n) for n = 1..7500

Cf. A000120, A349494.
Includes x^3 for x in A019434.
Includes all members of A235040 except 1.

g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
filter:= proc(n) local d, t;
  if isprime(n) then return false fi;
  t:= g(n);
  andmap(d -> g(d) * g(n/d) = t, select(d -> d^2 <= n, numtheory:-divisors(n)))
end proc:
select(filter, [seq(i,i=3..10000,2)]);

q[n_] := CompositeQ[n] && Ordering[(d = DigitCount[Divisors[n], 2, 1])*Reverse[d], -1][[1]] == Length[d]; Select[Range[3, 3100, 2], q] (* Amiram Eldar, Sep 04 2023 *)

is(n) = if(n%2 != 1 || isprime(n), return(0)); my(h=hammingweight(n), d=divisors(n), i); for(i=2,(#d+1)\2, if(hammingweight(d[i]) * hammingweight(d[#d+1-i]) > h, return(0))); n > 1 \\ David A. Corneth, Sep 04 2023

A365476 a(n) is the minimum of A000120(k)*A000120(A071904(n)/k) for divisors k of the n-th odd composite number A071904(n) other than 1 and A071904(n).

Original entry on oeis.org

4, 4, 6, 4, 4, 6, 6, 6, 4, 9, 4, 6, 6, 6, 6, 8, 6, 9, 4, 4, 8, 9, 10, 6, 4, 6, 6, 8, 6, 6, 9, 6, 6, 8, 9, 8, 10, 9, 8, 6, 4, 10, 8, 12, 4, 9, 6, 6, 10, 10, 6, 6, 6, 4, 6, 12, 6, 6, 9, 8, 8, 15, 6, 6, 6, 6, 10, 10, 6, 6, 9, 8, 12, 8, 9, 8, 8, 8, 9, 9, 10, 8, 9, 4, 6, 10, 4, 12, 12, 8, 10, 10, 6

Offset: 1

Robert Israel, Sep 04 2023

a(n) = 4 iff A071904(n) is the product of two (not necessarily distinct) members of A000051.
a(n) >= A000120(A071904(n)) because A000120(x) * A000120(y) >= A000120(x*y).
a(n) <= A349494(A071904(n)).

			a(9) = 4 because A071904(9) = 45 = 3 * 15 = 5 * 9 with A000120(3) * A000120(15) = 2 * 4 = 8 and A000120(5) * A000120(9) = 2 * 2 = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000051, A000120, A071904, A349494.

g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
f:= proc(n) local t, r;
      min(seq(g(t)*g(n/t), t = numtheory:-divisors(n) minus {1,n}))
    end proc:
map(f, remove(isprime, [seq(i,i=3..1000,2)]));

from sympy import primepi, divisors
def A365476(n):
    if n == 1: return 4
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return min(int(d).bit_count()*int(m//d).bit_count() for d in divisors(m,generator=True) if 1Chai Wah Wu, Aug 02 2024

cp's OEIS Frontend

A365451 Odd composite numbers k such that A349494(k) = A000120(k).

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