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A187822 Smallest k such that the partial sums of the divisors of k (taken in increasing order) contain exactly n primes. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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; ...

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, Dec 27 2012 *)

PROG

(PARI) A187822(n)={n<1||for(k=1, 9e9, numdiv(k)<n&next; my(t=divisors(k), s=1, c); for(i=2, #t, isprime(s+=t[i])&c++==n&return(k)))} \\ M. F. Hasler, Dec 29 2012

CROSSREFS

Cf. A023194, A062700, A000203.

Sequence in context: A138871 A001901 A127588 * A092585 A106186 A155543

Adjacent sequences:  A187819 A187820 A187821 * A187823 A187824 A187825

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 27 2012

STATUS

approved

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Last modified October 26 18:30 EDT 2021. Contains 348268 sequences. (Running on oeis4.)