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%I #24 Nov 20 2020 06:09:55
%S 40,48,54,88,108,184,250,384,424,432,448,808,864,1048,1216,1384,1528,
%T 1575,1680,1792,1864,1890,2104,2184,2457,2925,2944,3080,3120,3328,
%U 3510,3696,3712,3915,4125,4158,4288,4504,4744,4950,5224,5488,5632,5928,5940,6240
%N 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.
%C Is the sequence infinite?
%C 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, ...
%H Alois P. Heinz, <a href="/A179435/b179435.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1) = 40 = 2^3*5^1, with a = 11 and b = 13.
%e a(2) = 48 = 2^4*3^1, with a = 11 and b = 19.
%e Notice that a and b are both prime and not equal.
%p a:= proc(n) option remember; local an, bn, k, l;
%p for k from 1 +`if`(n=1, 0, a(n-1)) do
%p l:= ifactors(k)[2];
%p an:= add( i[1] * i[2], i=l);
%p bn:= add( i[1] ^ i[2], i=l);
%p if isprime(an) and isprime(bn) and an<>bn then break fi
%p od; k
%p end:
%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jan 20 2011
%t 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];
%t Array[a, 50] (* _Jean-François Alcover_, Nov 20 2020 *)
%K nonn
%O 1,1
%A _Bobby Browning_ and Rohan Hemasinha (rhemasin(AT)uwf.edu), Jan 07 2011
%E More terms from _Alois P. Heinz_, Jan 20 2011
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