A109353 -id:A109353 - cp's OEIS frontend

A199745 Numbers such that the sum of the largest and the smallest prime divisor equals the sum of the other distinct prime divisors.

2145, 2730, 4641, 4845, 5005, 5460, 5610, 6435, 7410, 8190, 8778, 9177, 10725, 10920, 11220, 11305, 11730, 13485, 13585, 13650, 13923, 14535, 14820, 16380, 16830, 17017, 17556, 19110, 19305, 20010, 20930, 21489, 21505, 21840, 22230, 22440, 23460, 23529, 23595

Offset: 1

Michel Lagneau, Nov 09 2011

nonn

The definition implies that members of the sequence have at least four distinct prime factors. An even term must have at least five distinct prime factors.

			22440 is in the sequence because the distinct prime divisors are  {2, 3, 5, 11, 17} and 17+2 = 3+5+11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A020639, A006530, A074320, A008472, A109353.

a199745 n = a199745_list !! (n-1)
a199745_list = filter (\x -> 2 * (a074320 x) == a008472 x) [1..]
-- Reinhard Zumkeller, Nov 10 2011

isA199745 := proc(n)
  local p;
  p := sort(convert(numtheory[factorset](n),list)) ;
  if nops(p) >= 3 then
    return ( op(1,p) + op(-1,p) = add(op(i,p),i=2..nops(p)-1) ) ;
  else
    false;
  end if;
end proc:
for n from 2 to 20000 do
  if isA199745(n) then
    printf("%d,",n) ;
  end if ;
end do: # R. J. Mathar, Nov 10 2011

Select[Range[25000],Plus@@(pl=First/@FactorInteger[#])/2==pl[[1]]+pl[[-1]]&] (* Ray Chandler, Nov 10 2011 *)

def isA199745(n) :
    p = factor(n)
    return len(p) > 2 and p[0][0] + p[-1][0] == add(p[i][0] for i in (1..len(p)-2))
[n for n in (2..20000) if isA199745(n)]  # Peter Luschny, Nov 10 2011

n such that A008472(n)/2 = A074320(n) = A020639(n) + A006530 (n). - Ray Chandler, Nov 10 2011

Sum_{k=2..A001221(a(n))} A027748(a(n),k) = A027748(a(n),1) + A027748(a(n), A001221(a(n))). - Reinhard Zumkeller, Nov 10 2011

A138296 Table T(k,n) read along antidiagonals: sum of the k-th powers of the distinct prime factors of A024619(n).

Original entry on oeis.org

5, 13, 7, 35, 29, 5, 97, 133, 13, 9, 275, 641, 35, 53, 8, 793, 3157, 97, 351, 34, 5, 2315, 15689, 275, 2417, 152, 13, 7, 6817, 78253, 793, 16839, 706, 35, 29, 10, 20195, 390881, 2315, 117713, 3368, 97, 133, 58, 13, 60073, 1953637, 6817, 823671, 16354, 275, 641

Offset: 1

R. J. Mathar, May 07 2008

Row k=1 is A109353. Rows k=2,3 and 4 are subsequences of A005063-A005065.

			Upper left corner of the table starting at row k=1, column n=1:
1|......5.......7.......5.......9.......8.......5.......7.
2|.....13......29......13......53......34......13......29.
3|.....35.....133......35.....351.....152......35.....133.
4|.....97.....641......97....2417.....706......97.....641.
5|....275....3157.....275...16839....3368.....275....3157.
6|....793...15689.....793..117713...16354.....793...15689.
7|...2315...78253....2315..823671...80312....2315...78253.
8|...6817..390881....6817.5765057..397186....6817..390881.

J.-M de Koninck, F. Luca, Integers divisible by sums of powers of their prime factors, J. Num. Theory vol 128 (2008) 557-563.

A024619 := proc(n)
    local a;
    if n = 1 then
        RETURN(6);
    else
        for a from A024619(n-1)+1 do
            if A001221(a) > 1 then
               RETURN(a) ;
            fi ;
        od:
    fi ;
end:
A138296 := proc(n,j)
    local f,beta ;
    beta := 0 ;
    for f in ifactors( A024619(n) )[2] do
        beta := beta+op(1,f)^j ;
    od:
    RETURN(beta) ;
end:
for d from 1 to 10 do for n from 1 to d do printf("%d,",A138296(n,d-n+1)) ; od: od: # R. J. Mathar, May 07 2008

T(k,n) = sum_{d in A000040, d| A024619(n)} d^k.

cp's OEIS Frontend

A199745 Numbers such that the sum of the largest and the smallest prime divisor equals the sum of the other distinct prime divisors.

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