A085379 Greatest prime as sum of distinct divisors of n.

3, 3, 7, 5, 11, 7, 13, 13, 17, 11, 23, 13, 23, 23, 31, 17, 37, 19, 41, 31, 23, 23, 59, 31, 41, 37, 53, 29, 71, 31, 61, 47, 53, 47, 89, 37, 59, 53, 89, 41, 89, 43, 83, 73, 71, 47, 113, 7, 83, 71, 97, 53, 113, 71, 113, 79, 89, 59, 167, 61, 31, 103, 127, 83, 139, 67

Offset: 2

Author

Reinhard Zumkeller, Jun 26 2003

Keywords

• nonn

Examples

			The divisors of n = 50 are {1,2,5,10,25,50}, the sums of distinct divisors that are prime: 2, 3 = 2+1, 5, 7 = 5+2, 11 = 10+1, 13 = 10+2+1, 17 = 10+5+2, 31 = 25+5+1, 37 = 25+10+2, 41 = 25+10+5+1, 43 = 25+10+5+2+1, 53 = 50+2+1, 61 = 50+10+1, 67 = 50+10+5+2 and 83 = 50+25+5+2+1. Therefore a(50) = 83 < 89 = A070801(50) and A085381(3) = 50.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10001
• Eric Weisstein's World of Mathematics, Divisor Function.

Crossrefs

Cf. A000203, A023194, A062700, A070801, A085380, A085381.

Programs

Mathematica

a[n_] := Module[{d = Divisors[n], c, x}, c = Rest@CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, Total[d]}], x]; Max[Select[Position[c, ?(# > 0 &)] // Flatten, PrimeQ]]]; Array[a, 100, 2] (* _Amiram Eldar, Mar 01 2024 *)

Formula

a(n) <= A070801(n) <= A000203(n).

a(A085380(n)) = A070801(A085380(n)).

a(A085381(n)) < A070801(A085381(n)).

a(A023194(n)) = A000203(A023194(n)) = A062700(n).