A365473 Odd semiprimes p*q such that A000120(p)*A000120(q) = A000120(p*q).

15, 51, 85, 95, 111, 119, 123, 187, 219, 221, 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, 1799, 1923, 1983, 2005, 2019, 2031, 2045, 2227, 2605, 2735, 2815, 2827, 2885, 3099, 3183, 3279, 3281, 3291, 3327

Offset

1,1

Comments

If p is an odd prime < 2^m and A365475(m) exists, then p * A365475(m) is a term. Thus, if A365475 is infinite, this sequence contains infinitely many multiples of every odd prime.

Being the intersection of A001358 and A235040 (as stated by the original author), these are odd semiprimes p*q such that p*q = A048720(p,q), where A048720 is the carryless binary multiplication. - Antti Karttunen, Dec 07 2025

Links

Robert Israel, Table of n, a(n) for n = 1..7230

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to carryless arithmetic.

Example

a(3) = 85 is a term because 85 = 5 * 17 is an odd semiprime with A000120(5) * A000120(17) = 2 * 2 = 4 = A000120(85).

Maple

g:= proc(n) convert(convert(n, base, 2), `+`) end proc:
N:= 10^4: # for terms <= N
S:= NULL: p:= 2:
while 3*p <= N do
  p:= nextprime(p);
  t:= g(p);
  q:= 2:
  do
    q:= nextprime(q);
    if q = p or q*p > N then break fi;
    if g(q)*t = g(p*q) then S:= S, p*q fi;
od od:
sort([S]);

Prog

(PARI) is_A365473(n) = if(!(n%2) || 2!=bigomega(n), 0, my(f=factor(n), a = f[1, 1], b = f[#f~, 1]); (hammingweight(n)==hammingweight(a)*hammingweight(b))); \\ Antti Karttunen, Dec 07 2025
(PARI)
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
is_A365473(n) = if(!(n%2) || 2!=bigomega(n), 0, my(f=factor(n), a = f[1, 1], b = f[#f~, 1]); (n==A048720(a, b))); \\ Antti Karttunen, Dec 07 2025

Crossrefs

Cf. A000120, A048720, A365475.

Intersection of A001358 and A235040, and intersection of A001358 and A365451.

Subsequence of A391253.

Cf. also A391256, A391332 (analogous sequence for the carryless base-3 multiplication).

Sequence in context: A392922 A339880 A235040 * A075933 A075929 A337295

Adjacent sequences: A365470 A365471 A365472 * A365474 A365475 A365476

Keyword

nonn,base

Author

Robert Israel, Sep 04 2023

Status

approved