A302753 - OEIS

%I #13 Aug 23 2018 17:28:19

%S 30,42,60,70,84,90,102,132,140,150,170,174,180,186,228,252,270,290,

%T 294,300,306,318,350,364,378,396,442,540,588,618,650,730,750,774,804,

%U 882,894,900,906,915,980,986,1015,1116,1182,1314,1364,1804,1892,1935,2058

%N Prime-additive numbers: numbers n with at least 2 distinct prime factors, that can be represented as n = Sum_{p|n} p^e_p, with e_p > 0.

%H Giovanni Resta, <a href="/A302753/b302753.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős and Norbert Hegyvári, <a href="http://real-j.mtak.hu/5469/1/StudScientMath_27.pdf">On prime-additive numbers</a>, Studia Scientiarum Mathematicarum Hungarica, Vol. 27, No. 1-2 (1992), pp. 207-212. <a href="https://www.emis.de/classics/Erdos/cit/68810043.htm">Review</a>.

%e 30 = 2 * 3 * 5 = 2 + 3 + 5^2.

%e 42 = 2 * 3 * 7 = 2^3 + 3^3 + 7 = 2^5 + 3 + 7.

%t primes[n_] := First[Transpose[FactorInteger[n]]]; maxPower[p_,n_] := Module[{k=0,nn=n},While[nn>1, nn/=p;k++];k-1]; a[n_] := Module[ {ps=primes[n]},np=Length[ps]; pws=Table[maxPower[ps[[k]],n], {k,1,np}]; npws=Length[pws]; Coefficient[Product[Sum[x^(ps[[k]]^j), {j,1,pws[[k]]}],{k,1,np}],x,n]]; s={}; Do[b=a[n]; If[b>0,AppendTo[s,n]], {n,1,2100}]; s

%Y Cf. A296717, A302755.

%K nonn

%O 1,1

%A _Amiram Eldar_, Apr 12 2018