A333626 Number of ways to write n as the sum of two positive integers that have the same number of divisors.

0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 3, 1, 4, 0, 4, 1, 4, 2, 4, 1, 5, 2, 3, 1, 4, 3, 6, 2, 6, 2, 6, 2, 8, 2, 3, 2, 7, 4, 7, 4, 7, 3, 6, 3, 12, 5, 7, 0, 7, 4, 10, 3, 8, 3, 5, 3, 13, 7, 6, 4, 11, 6, 11, 3, 8, 3, 11, 3, 17, 8, 7, 4, 9, 6, 12, 5, 12, 4, 9, 5

Offset

1,8

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor(n/2)} [d(i) = d(n-i)], where [] is the Iverson bracket and d is the number of divisors of n (A000005).

Example

a(14) = 3; There are 3 ways to write 14 as the sum of two numbers with the same number of divisors: 14 = 11+3 (11 and 3 both have 2 divisors), 14 = 8+6 (8 and 6 both have 4 divisors), 14 = 7+7 (7 has 2 divisors).

Mathematica

Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]], {i, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, n\2, numdiv(i) == numdiv(n-i)); \\ Michel Marcus, Mar 30 2020
(PARI) first(n) = {my(res, v, c); a2182inv = 128; res = vector(n); res[2] = 1; my(v = List(vector(a2182inv, i, List()))); for(i = 2, n, c = numdiv(i); for(i = #v + 1, c, listput(v, List()); ); listput(v[c], i) ); for(i = 1, a2182inv, for(j = 1, #v[i], for(k = j, #v[i], c = v[i][j] + v[i][k]; if(c <= n, res[c]++  ) ) ) ); res } \\ David A. Corneth, Apr 20 2020

Crossrefs

Cf. A000005, A333707.

Sequence in context: A112329 A335884 A325033 * A117448 A093321 A302015

Adjacent sequences: A333623 A333624 A333625 * A333627 A333628 A333629

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 29 2020

Status

approved