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A380574 For an integer k with prime factorization p_1*p_2*p_3* ... *p_m let k* = (p_1+1)*(p_2+1)*(p_3+1)* ... *(p_m+1); sequence gives k* such that k* is divisible by k.

%I A380574 #22 Apr 22 2025 06:32:20
%S A380574 1,12,36,144,432,1296,1728,5184,15552,20736,46656,62208,186624,248832,
%T A380574 559872,746496,1679616,2239488,2985984,6718464,8957952,20155392,
%U A380574 26873856,35831808,60466176,80621568,107495424,241864704,322486272,429981696,725594112,967458816
%N A380574 For an integer k with prime factorization p_1*p_2*p_3* ... *p_m let k* = (p_1+1)*(p_2+1)*(p_3+1)* ... *(p_m+1); sequence gives k* such that k* is divisible by k.
%C A380574 Terms of A064518 in increasing order.
%C A380574 Numbers of the form 3^i*4^j with j <= i <= 2j.
%C A380574 Subsequence of A064476.
%H A380574 Amiram Eldar, <a href="/A380574/b380574.txt">Table of n, a(n) for n = 1..10000</a>
%F A380574 Sum_{n>=1} 1/a(n) = 432/385. - _Amiram Eldar_, Mar 29 2025
%t A380574 With[{max = 10^9}, Select[Flatten[Table[3^i*4^j, {j, 0, Log[12, max]}, {i, j, 2*j}]] // Sort, # <= max &]] (* _Amiram Eldar_, Mar 29 2025 *)
%o A380574 (Python)
%o A380574 from sympy import integer_log
%o A380574 def A380574(n):
%o A380574     def bisection(f,kmin=0,kmax=1):
%o A380574         while f(kmax) > kmax: kmax <<= 1
%o A380574         kmin = kmax >> 1
%o A380574         while kmax-kmin > 1:
%o A380574             kmid = kmax+kmin>>1
%o A380574             if f(kmid) <= kmid:
%o A380574                 kmax = kmid
%o A380574             else:
%o A380574                 kmin = kmid
%o A380574         return kmax
%o A380574     def f(x): return n+x-sum(max(0,min((j:=i<<1),integer_log(x>>j,3)[0])-i+1) for i in range(x.bit_length()+1>>1))
%o A380574     return bisection(f,n,n)
%Y A380574 Cf. A064518, A064514, A064515, A064476.
%K A380574 nonn,easy
%O A380574 1,2
%A A380574 _Chai Wah Wu_, Mar 26 2025