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%I #28 Jun 15 2022 19:38:45
%S 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,
%T 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,
%U 3,8,3,11,3,17,8,7,4,9,6,12,5,12,4,9,5
%N Number of ways to write n as the sum of two positive integers that have the same number of divisors.
%H David A. Corneth, <a href="/A333626/b333626.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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).
%e 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).
%t Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]], {i, Floor[n/2]}], {n, 100}]
%o (PARI) a(n) = sum(i=1, n\2, numdiv(i) == numdiv(n-i)); \\ _Michel Marcus_, Mar 30 2020
%o (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
%Y Cf. A000005, A333707.
%K nonn,easy
%O 1,8
%A _Wesley Ivan Hurt_, Mar 29 2020
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