This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382292 #10 Mar 21 2025 10:04:21
%S A382292 8,24,27,32,40,54,56,64,72,88,96,104,108,120,125,135,136,152,160,168,
%T A382292 184,189,192,200,224,232,243,248,250,264,270,280,288,296,297,312,320,
%U A382292 328,343,344,351,352,360,375,376,378,392,408,416,424,432,440,448,456,459,472,480,486,488,500
%N A382292 Numbers k such that A382290(k) = 1.
%C A382292 First differs from A374590 and A375432 at n = 25: A374590(25) = A375432(25) = 216 is not a term of this sequence.
%C A382292 Numbers k such that A382291(k) = 2, i.e., numbers whose number of infinitary divisors is twice the number of their unitary divisors.
%C A382292 Numbers whose prime factorization has a single exponent that is a sum of two distinct powers of 2 (A018900) and all the other exponents, if they exist, are powers of 2. Equivalently, numbers of the form p^e * m, where p is a prime, e is a term in A018900, and m is a term in A138302 that is coprime to p.
%C A382292 If k is a term then k^2 is also a term. If m is a term in A138302 that is coprime to k then k * m is also a term. The primitive terms, i.e., the terms that cannot be generated from smaller terms using these rules, are the numbers of the form p^(2^i+1), where p is prime and i >= 1.
%C A382292 Analogous to A060687, which is the sequence of numbers k with prime excess A046660(k) = 2.
%C A382292 The asymptotic density of this sequence is A271727 * Sum_{p prime} (((1 - 1/p)/f(1/p)) * Sum_{k>=1} 1/p^A018900(k)) = 0.11919967112489084407..., where f(x) = 1 - x^3 + Sum_{k>=2} (x^(2^k)-x^(2^k+1)).
%H A382292 Amiram Eldar, <a href="/A382292/b382292.txt">Table of n, a(n) for n = 1..10000</a>
%t A382292 f[p_, e_] := DigitCount[e, 2, 1] - 1; q[1] = False; q[n_] := Plus @@ f @@@ FactorInteger[n] == 1; Select[Range[500], q]
%o A382292 (PARI) isok(k) = vecsum(apply(x -> hammingweight(x) - 1, factor(k)[, 2])) == 1;
%Y A382292 Cf. A018900, A046660, A060687, A271727, A374590, A375432, A382290, A382291.
%Y A382292 Subsequences (numbers of the form): A030078 (p^3), A050997 (p^5), A030516 (p^6), A179665 (p^9), A030629 (p^10), A030631 (p^12), A065036 (p^3*q), A178740 (p^5*q), A189987 (p^6*q), A179692 (p^9*q), A143610 (p^2*q^3), A179646 (p^5*q^2), A189990 (p^2*q^6), A179702 (p^4*q^5), A179666 (p^4*q^3), A190464 (p^4*q^6), A163569 (p^3*q^2*r), A189975 (p*q*r^3), A190115 (p^2*q^3*r^4), A381315, A048109.
%K A382292 nonn,easy
%O A382292 1,1
%A A382292 _Amiram Eldar_, Mar 21 2025