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A328964 Smallest k such that omega(k) * bigomega(k) = n, where omega = A001221, bigomega = A001222.

%I A328964 #22 Feb 12 2022 20:28:52
%S A328964 1,2,4,8,6,32,12,128,24,30,48,2048,60,8192,192,120,210,131072,240,
%T A328964 524288,420,480,3072,8388608,840,2310,12288,1920,1680,536870912,3840,
%U A328964 2147483648,3360,7680,196608,9240,6720,137438953472,786432,30720,13440,2199023255552,60060,8796093022208
%N A328964 Smallest k such that omega(k) * bigomega(k) = n, where omega = A001221, bigomega = A001222.
%H A328964 Andrew Howroyd, <a href="/A328964/b328964.txt">Table of n, a(n) for n = 0..1000</a>
%F A328964 a(p) = 2^p, for p prime. - _Daniel Suteu_, Nov 03 2019
%F A328964 a(n) = min_{d|n, d<=n/d} 2^(n/d-d)*A002110(d) for n > 0. - _Andrew Howroyd_, Nov 04 2019
%e A328964 The sequence of terms together with their prime signatures begins:
%e A328964      1: ()
%e A328964      2: (1)
%e A328964      4: (2)
%e A328964      8: (3)
%e A328964      6: (1,1)
%e A328964     32: (5)
%e A328964     12: (2,1)
%e A328964    128: (7)
%e A328964     24: (3,1)
%e A328964     30: (1,1,1)
%e A328964     48: (4,1)
%e A328964   2048: (11)
%e A328964     60: (2,1,1)
%e A328964   8192: (13)
%e A328964    192: (6,1)
%e A328964    120: (3,1,1)
%e A328964    210: (1,1,1,1)
%t A328964 dat=Table[PrimeOmega[n]*PrimeNu[n],{n,1000}];
%t A328964 Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]
%o A328964 (PARI) a(n)={if(n<1, 1, my(m=oo); fordiv(n, d, if(d<=n/d, m=min(m, 2^(n/d-d)*vecprod(primes(d))))); m)} \\ _Andrew Howroyd_, Nov 04 2019
%Y A328964 Positions of first appearances in A113901.
%Y A328964 Cf. A001221, A001222, A002110, A025487, A124010, A307409, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328965.
%K A328964 nonn
%O A328964 0,2
%A A328964 _Gus Wiseman_, Nov 02 2019
%E A328964 More terms from _Daniel Suteu_, Nov 03 2019