A084113 -id:A084113 - cp's OEIS frontend

A084115 A084113(n) minus A084114(n).

0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 3, 4, 1, 3, 1, 3, 3

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A084113(n) - A084114(n) = 2*A084113(n) - A032741(n) = A032741(n) - 2*A084114(n);

a(A084116(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A066423, A084116.

Cf. A084110, A084113, A084114.

a084115 n = a084113 n - a084114 n -- Reinhard Zumkeller, Jul 31 2014

Definition fixed by Reinhard Zumkeller, Jul 31 2014

A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

Original entry on oeis.org

1, 2, 3, 8, 5, 1, 7, 1, 27, 1, 11, 48, 13, 1, 1, 16, 17, 162, 19, 80, 1, 1, 23, 16, 125, 1, 1, 112, 29, 25, 31, 512, 1, 1, 1, 1944, 37, 1, 1, 25, 41, 49, 43, 176, 405, 1, 47, 48, 343, 1250, 1, 208, 53, 324, 1, 49, 1, 1, 59, 9, 61, 1, 567, 8, 1, 121, 67, 272, 1, 49, 71, 9, 73, 1

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = r(n,tau(n)), where r is defined as follows:
let d(n,j) = j-th divisor of n, 1 <= j <= tau(n) = A000005(n), r(n,1)=d(n,1), r(n,j) = if d(n,j) divides r(n,j-1) then r(n,j-1)/d(n,j) else r(n,j-1)*d(n,j), 1 < j <= tau(n);
p prime: a(p)=p, a(p^2)=p^3, a(p^3)=1, a(p^k)=p^A008344(k+1);
a(m)=1 iff m multiplicatively perfect: a(A007422(k))=1.
a(A084111(n)) = A084111(n). - Reinhard Zumkeller, Jul 31 2014

			Divisors of 48 = {1,2,3,4,6,8,12,16,24,48}: 1*2*3 = 6 -> 6*4 = 24 -> 24/6 = 4 -> 4*8 = 32 -> 32*12 = 384 -> 384/16 = 24 -> 24/24 = 1 -> 1*48 = a(48);
divisors of 49 = {1,7,49}: 1*7 = 7 -> 7*49 = 343 = a(49);
divisors of 50 = {1,2,5,10,25,50}: 1*2*5 = 10 -> 10/10 = 1 -> 1*25 = 25 -> 25*50 = 1250 = a(50).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A027750, A084111 (fixed points), A084113, A084114.

a084110 = foldl (/*) 1 . a027750_row where
   x /* y = if m == 0 then x' else x*y where (x',m) = divMod x y
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

a[n_] := Module[{d = Divisors[n], e}, e[i_] := e[i] = If[i == 1, 1, If[Divisible[e[i-1], d[[i]]], e[i-1]/d[[i]], e[i-1] d[[i]]]]; e[Length[d]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 10 2021 *)

Corrected and extended by David Wasserman, Dec 14 2004

A084116 Numbers m such that A084115(m) = 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

A084113(a(n)) = A084114(a(n)) + 1.
Union of primes and multiplicatively perfect numbers (A000040, A007422).
A084115(a(n)) = 1; A066729(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A084110, A066729, A084113, A084114, A084115, A066423 (complement).

a084116 n = a084116_list !! (n-1)
a084116_list = filter ((== 1) . a084115) [1..]
-- Reinhard Zumkeller, Jul 31 2014

Select[Range[2, 200], PrimeQ[DivisorSigma[0, #]^DivisorSigma[0, #] + 1] &] (* Carl Najafi, Oct 19 2011 *)

is(n)=isprime(n) || numdiv(n) == 4 \\ Charles R Greathouse IV, Oct 19 2015

It appears that a(n) = n such that A000005(n)^A000005(n)+1 is prime. - Carl Najafi, Oct 19 2011

Corrected and edited by Carl Najafi, Oct 19 2011

Revised by Reinhard Zumkeller, Jul 31 2014

A084114 Number of divisions when calculating A084110(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 1, 1, 1, 2, 0, 3, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 1, 1, 1, 1, 4, 0, 1, 1, 2, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n) - 1 - A084113(n) = A032741(n) - A084113(n) = (A032741(n)-A084115(n))/2;

a(n) = 0 iff n is prime or a square of prime (A000430).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A080257.

Cf. A027750, A084110, A084113.

a084114 = g 0 1 . tail . a027750_row where
   g c _ []     = c
   g c x (d:ds) = if r > 0 then g c (x * d) ds else g (c + 1) x' ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

cp's OEIS Frontend

A084115 A084113(n) minus A084114(n).

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A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

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A084116 Numbers m such that A084115(m) = 1.

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A084114 Number of divisions when calculating A084110(n).

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