A187822 Smallest k such that the partial sums of the divisors of k (taken in increasing order) contain exactly n primes.

1, 2, 4, 16, 64, 140, 440, 700, 1650, 2304, 5180, 3960, 3900, 14400, 15600, 43560, 39600, 57600, 56700, 81900, 25200, 112896, 100100, 177840, 198000, 411840, 222768, 226800, 637560, 752400, 556920, 907200, 409500, 565488, 1306800, 1984500, 1884960

Offset: 0

Author

Michel Lagneau, Dec 27 2012

Keywords

• nonn

Comments

It appears that a(n) is even for n > 0 and nonsquarefree for n > 1. We also conjecture that there is an infinite subsequence of squares 1, 4, 16, 64, 2304, 14400, 57600, 112896, ....

The corresponding triangle in which row n gives the n primes starts with:

k = 1 -> no prime

k = 2 -> 3;

k = 4 -> 3, 7;

k = 16 -> 3, 7, 31;

k = 64 -> 3, 7, 31, 127;

k = 140 -> 3, 7, 19, 29, 43;

k = 440 -> 3, 7, 41, 61, 83, 167; ...

Examples

			a(4) = 64 because the partial sums of the divisors {1, 2, 4, 8, 16, 32, 64} that generate 4 prime numbers are:
1 + 2 = 3;
1 + 2 + 4 = 7;
1 + 2 + 4 + 8 + 16  = 31;
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..126

Crossrefs

Cf. A023194, A062700, A000203.

Programs

Maple

read("transforms") :
A187822 := proc(n)
    local k,ps,pct ;
    if n = 0 then
        return 1;
    end if;
    for k from 1 do
        ps := sort(convert(numtheory[divisors](k),list)) ;
        ps := PSUM(ps) ;
        pct := 0 ;
        for p in ps do
            if isprime(p) then
                pct := pct+1 ;
            end if;
        end do:
        if pct = n then
            return k ;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 18 2013

Mathematica

a[n_] := Catch[ For[k = 1, True, k++, cnt = Count[ Accumulate[ Divisors[k]], ?PrimeQ]; If[cnt == n, Print[{n, k}]; Throw[k]]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Dec 27 2012 *)

PARI

A187822(n)={n<1||for(k=1,9e9,numdiv(k)M. F. Hasler, Dec 29 2012