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

1, 4, 6, 16, 12, 64, 24, 256, 48, 60, 96, 4096, 120, 16384, 384, 240, 420, 262144, 480, 1048576, 840, 960, 6144, 16777216, 1680, 4620, 24576, 3840, 3360, 1073741824, 7680, 4294967296, 6720, 15360, 393216, 18480, 13440, 274877906944, 1572864, 61440, 26880, 4398046511104

Offset: 0

Author

Gus Wiseman, Nov 02 2019

Keywords

• nonn

Comments

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

Examples

			The sequence of terms together with their prime signatures begins:
      1: ()
      4: (2)
      6: (1,1)
     16: (4)
     12: (2,1)
     64: (6)
     24: (3,1)
    256: (8)
     48: (4,1)
     60: (2,1,1)
     96: (5,1)
   4096: (12)
    120: (3,1,1)
  16384: (14)
    384: (7,1)
    240: (4,1,1)
    420: (2,1,1,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Positions of first appearances in A307409.

Cf. A001221, A001222, A002110, A113901, A124010, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328964.

Programs

Mathematica

dat=Table[(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]

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

Formula

From Andrew Howroyd, Nov 03 2019: (Start)

a(p) = 2^(p + 1) for odd prime p.

a(n) = min_{d|n, d<=n/d+1} 2^(n/d-d+1)*A002110(d) for n > 0. (End)

Extensions

Terms a(23) and beyond from Andrew Howroyd, Nov 03 2019