A179435 For positive n with prime decomposition n = Product_{j=1..m} (p_j^k_j) define a_n = Sum_{j=1..m} (p_j*k_j) and b_n = Sum_{j=1..m} (p_j^k_j). This sequence gives those n for which a_n and b_n are both prime and unequal.

40, 48, 54, 88, 108, 184, 250, 384, 424, 432, 448, 808, 864, 1048, 1216, 1384, 1528, 1575, 1680, 1792, 1864, 1890, 2104, 2184, 2457, 2925, 2944, 3080, 3120, 3328, 3510, 3696, 3712, 3915, 4125, 4158, 4288, 4504, 4744, 4950, 5224, 5488, 5632, 5928, 5940, 6240

Offset: 1

Author

Bobby Browning and Rohan Hemasinha (rhemasin(AT)uwf.edu), Jan 07 2011

Keywords

• nonn

Comments

Is the sequence infinite?

Odd terms in the sequence are: a(18) = 1575, a(25) = 2457, a(26) = 2925, a(34) = 3915, a(35) = 4125, a(47) = 6345, a(50) = 6669, ...

Examples

			a(1) = 40 = 2^3*5^1, with a = 11 and b = 13.
a(2) = 48 = 2^4*3^1, with a = 11 and b = 19.
Notice that a and b are both prime and not equal.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Programs

Maple

a:= proc(n) option remember; local an, bn, k, l;
      for k from 1 +`if`(n=1, 0, a(n-1)) do
        l:= ifactors(k)[2];
        an:= add( i[1] * i[2], i=l);
        bn:= add( i[1] ^ i[2], i=l);
        if isprime(an) and isprime(bn) and an<>bn then break fi
      od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 20 2011

Mathematica

a[n_] := a[n] = Module[{an, bn, k, p, e}, For[k = 1 + If[n==1, 0, a[n-1]], True, k++, {p, e} = Transpose[FactorInteger[k]]; an = p.e; bn = Total[p^e]; If[PrimeQ[an] && PrimeQ[bn] && an != bn, Break[]]]; k];
Array[a, 50] (* Jean-François Alcover, Nov 20 2020 *)

Extensions

More terms from Alois P. Heinz, Jan 20 2011