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A007956
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Product of the proper divisors of n.
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41
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1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, 1936, 2025, 46, 1, 5308416, 7, 2500, 51, 2704, 1, 157464, 55, 175616, 57, 58, 1, 777600000, 1, 62, 3969, 32768, 65
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OFFSET
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1,4
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COMMENTS
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a(n) = 1 iff n = 1 or n is prime.
a(n) = n when n > 1 iff n has exactly four divisors, equally, iff n is either the cube of a prime or the product of two different primes, so iff n belongs to A030513 (very nice proof in Sierpiński).
a(p^3) = 1 * p * p^2 = p^3; a(p*q) = 1 * p * q = p*q.
As a(1) = 1, {1} Union A030513 = A007422, fixed points of this sequence. (End)
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REFERENCES
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József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.
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LINKS
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FORMULA
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a(n) = A007955(n)/n = n^(A000005(n)/2-1) = sqrt(n^(number of factors of n other than 1 and n)).
Sum_{k=1..n} 1/a(k) ~ pi(n) + log(log(n))^2 + c_1*log(log(n)) + c_2 + O(log(log(n))/log(n)), where pi(n) = A000720(n) and c_1 and c_2 are constants (Weiyi, 2004; Sandor and Crstici, 2004). - Amiram Eldar, Oct 29 2022
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EXAMPLE
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MAPLE
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A007956 := n -> mul(i, i=op(numtheory[divisors](n) minus {1, n}));
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MATHEMATICA
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Table[Times@@Most[Divisors[n]], {n, 65}] (* Alonso del Arte, Apr 18 2011 *)
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PROG
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(Haskell)
a007956 = product . a027751_row
(Python)
from math import isqrt
from sympy import divisor_count
def A007956(n): return isqrt(n)**(d-2) if (d:=divisor_count(n))&1 else n**((d>>1)-1) # Chai Wah Wu, Jun 18 2023
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CROSSREFS
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Cf. A001065 (sums of proper divisors).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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R. Muller
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EXTENSIONS
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More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)
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STATUS
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approved
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