A051838 Numbers k such that sum of first k primes divides product of first k primes.

1, 3, 8, 13, 23, 38, 39, 41, 43, 48, 50, 53, 56, 57, 58, 66, 68, 70, 73, 77, 84, 90, 94, 98, 126, 128, 134, 140, 143, 145, 149, 151, 153, 157, 160, 164, 167, 168, 172, 174, 176, 182, 191, 194, 196, 200, 210, 212, 215, 217, 218, 219, 222, 225, 228, 229

Offset: 1

Author

G. L. Honaker, Jr., Dec 12 1999

Keywords

• nonn

Examples

			Sum of first 8 primes is 77 and product of first 8 primes is 9699690. 77 divides 9699690 therefore a(3)=8.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A007504, A002110, A159639, A196415.

A116536 gives the quotients, A140763 the divisors and A159578 the dividends.

Programs

GAP

P:=Filtered([1..2000],IsPrime);;
Filtered([1..Length(P)],n->Product([1..n],i->P[i]) mod Sum([1..n],i->P[i])=0); # Muniru A Asiru, Dec 20 2018

Haskell

import Data.List (elemIndices)
a051838 n = a051838_list !! (n-1)
a051838_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod a002110_list a007504_list
-- Reinhard Zumkeller, Oct 03 2011

Mathematica

p = Prime@ Range@ 250; Flatten@ Position[ Mod[ First@#, Last@#] & /@ Partition[ Riffle[ Rest[ FoldList[ Times, 1, p]], Accumulate@ p], 2], 0] (* Harvey P. Dale, Dec 19 2010 *)

PARI

for(n=1,100,P=prod(i=1,n,prime(i));S=sum(i=1,n,prime(i));if(!(P%S),print1(n,", "))) \\ Derek Orr, Jul 19 2015

PARI

isok(n) = my(p = primes(n)); (vecprod(p) % vecsum(p)) == 0; \\ Michel Marcus, Dec 20 2018

Formula

From Reinhard Zumkeller, Oct 03 2011: (Start)

A002110(a(n)) mod A007504(a(n)) = 0.

A116536(n) = A002110(a(n)) / A007504(a(n)). (End)