A307119 - OEIS

%I #35 Aug 16 2019 16:18:26

%S 1,2,4,4,6,5,7,6,8,6,9,7,9,7,10,8,10,7,11,9,11,7,11,10,12,8,11,9,13,9,

%T 13,9,11,9,14,12,12,7,13,11,15,9,13,11,13,9,13,12,16,10,13,9,13,11,17,

%U 13,13,7,15,13,15,9,14,14,16,11,13,9,15,11,19,13,15,9,13,13,15,11,17,14,16,8

%N a(1) = 1, for n>1, a(n) = dp(n-1) + dp(n) + dp(n+1), where dp(n) is the number of divisors of n less than n (A032741).

%C Proper divisibility of n's 1-area. This is the first step to examine the divisibility of n's k-area. n's k-area is the set of m for which |n-m| is less than or equal to k (n, k, m are natural numbers). 1's 1-area is {1,2}, 5's 1-area {4,5,6}, 3's 2-area {1,2,3,4,5}. We could call this natural area, and still talk about nonnegative or integer areas etc.

%t {1}~Join~MapAt[# + 1 &, Total /@ Partition[DivisorSigma[0, Range@82] - 1, 3, 1], 1] (* _Michael De Vlieger_, Jun 06 2019 *)

%o (PARI) dp(n) = if (n < 1, 0, numdiv(n) - 1);

%o a(n) = dp(n-1) + dp(n) + dp(n+1); \\ _Michel Marcus_, Jun 11 2019

%Y Cf. A032741, A307118, A307120.

%K nonn

%O 1,2

%A _Todor Szimeonov_, Mar 25 2019