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A328965
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Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.
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10
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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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)
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MATHEMATICA
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dat=Table[(PrimeOmega[n]-1)*PrimeNu[n], {n, 1000}];
Table[Position[dat, i][[1, 1]], {i, First[Split[Union[dat], #2==#1+1&]]}]
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PROG
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(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
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CROSSREFS
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Positions of first appearances in A307409.
Cf. A001221, A001222, A002110, A113901, A124010, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328964.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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