A063647 -id:A063647 - cp's OEIS frontend

A146746 Duplicate of A073541.

5, 14, 54, 55, 56, 58, 70, 87, 88, 89, 114, 118, 123, 125, 178, 201, 204, 206, 220, 246, 354, 358, 417, 419, 459, 478, 500, 534, 549, 594, 604, 620, 646, 654, 663, 715, 718, 753, 834, 835, 838, 894, 957

Offset: 1

Ctibor O. Zizka, Nov 01 2008

dead

A000203(n+1)*A000203(n)/(A000203(n+1)+A000203(n)) = c, c an integer.

Same as A073541. - Georg Fischer, Nov 02 2018

Cf. A083795, A063647, A073541, A000203.

sQ[n_]:=Module[{sn=DivisorSigma[1,n],sn1=DivisorSigma[1,n+1]},Divisible[sn sn1,sn+sn1]]; Select[Range[1000],sQ]  (* Harvey P. Dale, Apr 21 2011 *)

Corrected by Harvey P. Dale, Apr 21 2011

A174225 Array: row n consists of the positive integers x for which there are exactly n positive integers y such that x+y divides x*y.

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 7, 25, 27, 6, 11, 49, 125, 10, 32, 13, 121, 343, 14, 243, 64, 17, 169, 1331, 15, 3125, 729, 12, 19, 289, 2197, 16, 16807, 15625, 18, 256, 23, 361, 4913, 21, 161051, 117649, 20, 6561, 512, 29, 529, 6859, 22

Offset: 1

Clark Kimberling, Mar 12 2010

Many properties of the array follow easily from Comments at A063647:
Every positive integer except 1 occurs, exactly once.
Row 1 consists of the primes.
Row k includes p^k for all primes p, for k>=1.
Row 4 includes all products of two distinct primes.
Column 1 consists of even numbers.

			Corner of the array:
2....3....5....7....11....13....
4....9....25...49...121...169...
8....27...125..343..1331..2197..
6....10...14...15...16....21....
6 is in row 4 because there are 4 numbers y for which
6+y divides 6*y; they are 3,6,12,30.

Cf. A063647.

A284167 a(n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.

Original entry on oeis.org

2, 5, 7, 10, 8, 15, 8, 18, 16, 18, 10, 29, 8, 19, 25, 28, 10, 33, 10, 35, 26, 20, 12, 50, 18, 20, 31, 36, 12, 51, 10, 42, 27, 23, 33, 62, 8, 22, 30, 60, 12, 53, 10, 40, 52, 22, 14, 78, 20, 41, 28, 38, 12, 63, 36, 63, 30, 24, 16, 95, 8, 23, 59, 60, 32, 54, 10

Offset: 1

Ctibor O. Zizka, Mar 21 2017

nonn

Let S(n,n) be the number of solutions of the equation n/x + n/y = c where n, c, x, and y are positive integers. Then S(n,n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.
For c = 1 , S(n,n) = A000005(n).
Let S(n,m) be the number of solutions of the equation n/x + m/y = c where n, m, c, x, and y are positive integers, n not equal to m. Let k(i) be the i-th divisor of n, and k(j) the j-th divisor of m. Let d(t) be the number of divisors of t. Let R = d(k(i) + k(j)). Then S(n,m) = Sum_{i=1..A000005(n)} Sum_{j=1..A000005(m)} [R*1 if gcd(k(i),k(j)) = 1 , R*0 else].
For c = 1 , S(n,m) = A000005(n) * A000005(m) - P, where P is the number of divisor pairs such that gcd(k(i),k(j)) >= 2.

			For n = 4, divisors of 4 are 1, 2, 4; thus a(4) = d(4+1) + d(4+2) + d(4+4) = d(5) + d(6) + d(8) = 2 + 4 + 4 = 10.

R. L. Graham, Paul Erdos and Egyptian Fractions, Bolyai Society Mathematical Studies 25, pp 289-309, 2013.

Cf. A000005, A018892, A063647, A089233, A275387.

a[n_] := Sum[DivisorSigma[0, d + n], {d, Divisors@n}]; Array[a, 67] (* Giovanni Resta, Mar 21 2017 *)

for(n=1, 101, print1(sumdiv(n, d, numdiv(d + n)),", ")) \\ Indranil Ghosh, Mar 22 2017

from sympy import divisor_count, divisors
def a(n):
    return sum(divisor_count(n + d) for d in divisors(n)) # Indranil Ghosh, Mar 22 2017

a(21)-a(67) from Giovanni Resta, Mar 21 2017

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A146746 Duplicate of A073541.

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A174225 Array: row n consists of the positive integers x for which there are exactly n positive integers y such that x+y divides x*y.

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A284167 a(n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.

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