A174889 -id:A174889 - cp's OEIS frontend

A174892 a(n) = A074206(n) - A174889(n).

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 2, 7, 1, 7, 1, 7, 2, 2, 1, 19, 1, 2, 3, 7, 1, 13, 1, 15, 2, 2, 2, 25, 1, 2, 2, 19, 1, 13, 1, 7, 7, 2, 1, 47, 1, 7, 2, 7, 1, 19, 2, 19, 2, 2, 1, 43, 1, 2, 7, 31, 2, 13, 1, 7, 2, 13, 1, 75, 1, 2, 7, 7, 2

Offset: 1

Mats Granvik, Apr 01 2010

nonn

Cf. A074206, A174888, A174889, A174891.

A174888 Triangle read by rows. Row sums = Mobius function A008683, row counts of nonzero elements = A074206.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0

Offset: 1

Mats Granvik, Apr 01 2010

This table cannot be completed as a triangle beyond row 95 because the ordered factorization value for 96 is 112, which is greater than the row length 96. Values in the first column are either zero or one, and are chosen so that the number of nonzero elements in each row equals A074206 while rows sums equals A008683. The row indices for nonzero elements in the 1st column are A174891.

			Table begins:
n\k|....1....2....3....4....5....6....7....8....9...10
---|--------------------------------------------------
1..|....1
2..|....0...-1
3..|....0...-1....0
4..|....1...-1....0....0
5..|....0...-1....0....0....0
6..|....1...-1....1....0....0....0
7..|....0...-1....0....0....0....0....0
8..|....1...-1....1...-1....0....0....0....0
9..|....1...-1....0....0....0....0....0....0....0
10.|....1...-1....1....0....0....0....0....0....0....0
Notice that the 30th row begins with a zero:
0,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1

Cf. A008683, A074206, A174889, A174891, A174892.

First column = A174889.

Beginning from the second column: a(n)=if k<=A074206 then ((-1)^(k+1)) else 0.

A174891 Row indices for nonzero elements in first column of A174888.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 68, 69, 72, 74, 75, 76, 77

Offset: 1

Mats Granvik, Apr 01 2010

nonn

Compare with nonprimes A018252. 30 is not a member of this sequence. Neither is 42.

Cf. A174889, A174888, A018252.

A261447 The number of distinct prime factors of the number of proper divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1

Offset: 2

R. J. Mathar, Aug 19 2015

nonn

Starts to differ from A174889 at n=64.
The first 2 occurs at n = 64, the first 3 at n = 1081080.
The first 4 occurs at n = 75424789440 and the first 5 occurs at 2474044411193114760000. - Charles R Greathouse IV, Aug 20 2015

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
L. Troupe, On the number of prime factors of values of the sum-of-proper-divisors function, J. Number Theory 150 (2015)120-135

Cf. A001221, A032741, A007416, A002182.

A261447 := proc(n)
        A001221(A032741(n)) ;
end proc:

PrimeNu[DivisorSigma[0,#]-1&/@Range[2,82]] (* Ivan N. Ianakiev, Aug 20 2015 *)

a(n)=omega(numdiv(n)-1) \\ Charles R Greathouse IV, Aug 20 2015

a(n) = A001221(A032741(n)).

cp's OEIS Frontend

A174892 a(n) = A074206(n) - A174889(n).

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A174888 Triangle read by rows. Row sums = Mobius function A008683, row counts of nonzero elements = A074206.

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A174891 Row indices for nonzero elements in first column of A174888.

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A261447 The number of distinct prime factors of the number of proper divisors of n.

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