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A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.

%I A328965 #21 Nov 04 2019 15:19:12
%S A328965 1,4,6,16,12,64,24,256,48,60,96,4096,120,16384,384,240,420,262144,480,
%T A328965 1048576,840,960,6144,16777216,1680,4620,24576,3840,3360,1073741824,
%U A328965 7680,4294967296,6720,15360,393216,18480,13440,274877906944,1572864,61440,26880,4398046511104
%N A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.
%C A328965 For n > 0, a(n) is of the form 2^k*primorial(d) where d is a divisor of n and k = n / d - d + 1.  a(n) is never 0 since A307409(2^(n+1)) = n. - _Andrew Howroyd_, Nov 04 2019
%H A328965 Andrew Howroyd, <a href="/A328965/b328965.txt">Table of n, a(n) for n = 0..1000</a>
%F A328965 From _Andrew Howroyd_, Nov 03 2019: (Start)
%F A328965 a(p) = 2^(p + 1) for odd prime p.
%F A328965 a(n) = min_{d|n, d<=n/d+1} 2^(n/d-d+1)*A002110(d) for n > 0. (End)
%e A328965 The sequence of terms together with their prime signatures begins:
%e A328965       1: ()
%e A328965       4: (2)
%e A328965       6: (1,1)
%e A328965      16: (4)
%e A328965      12: (2,1)
%e A328965      64: (6)
%e A328965      24: (3,1)
%e A328965     256: (8)
%e A328965      48: (4,1)
%e A328965      60: (2,1,1)
%e A328965      96: (5,1)
%e A328965    4096: (12)
%e A328965     120: (3,1,1)
%e A328965   16384: (14)
%e A328965     384: (7,1)
%e A328965     240: (4,1,1)
%e A328965     420: (2,1,1,1)
%t A328965 dat=Table[(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
%t A328965 Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]
%o A328965 (PARI) a(n)={if(n<1, 1, my(m=oo); fordiv(n, d, if(d<=n/d+1, m=min(m, 2^(n/d-d+1)*vecprod(primes(d))))); m)} \\ _Andrew Howroyd_, Nov 04 2019
%Y A328965 Positions of first appearances in A307409.
%Y A328965 Cf. A001221, A001222, A002110, A113901, A124010, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328964.
%K A328965 nonn
%O A328965 0,2
%A A328965 _Gus Wiseman_, Nov 02 2019
%E A328965 Terms a(23) and beyond from _Andrew Howroyd_, Nov 03 2019