A325198 - OEIS

%I #14 Feb 28 2020 22:53:50

%S 10,20,21,30,40,50,55,60,63,80,90,91,100,105,120,147,150,160,180,187,

%T 189,200,240,247,250,270,275,300,315,320,360,385,391,400,441,450,480,

%U 500,525,540,551,567,600,605,637,640,713,720,735,750,800,810,900,945

%N Positive numbers whose maximum prime index minus minimum prime index is 2.

%C Also Heinz numbers of integer partitions whose maximum minus minimum part is 2 (counted by A008805). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%H Robert Israel, <a href="/A325198/b325198.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of terms together with their prime indices begins:

%e    10: {1,3}

%e    20: {1,1,3}

%e    21: {2,4}

%e    30: {1,2,3}

%e    40: {1,1,1,3}

%e    50: {1,3,3}

%e    55: {3,5}

%e    60: {1,1,2,3}

%e    63: {2,2,4}

%e    80: {1,1,1,1,3}

%e    90: {1,2,2,3}

%e    91: {4,6}

%e   100: {1,1,3,3}

%e   105: {2,3,4}

%e   120: {1,1,1,2,3}

%e   147: {2,4,4}

%e   150: {1,2,3,3}

%e   160: {1,1,1,1,1,3}

%e   180: {1,1,2,2,3}

%e   187: {5,7}

%p N:= 1000: # for terms <= N

%p q:= 2: r:= 3:

%p Res:= NULL:

%p do

%p   p:= q; q:= r; r:= nextprime(r);

%p   if p*r > N then break fi;

%p   for i from 1 do

%p     pi:= p^i;

%p     if pi*r > N then break fi;

%p     for j from 0 do

%p       piqj:= pi*q^j;

%p       if piqj*r > N then break fi;

%p       Res:= Res, seq(piqj*r^k,k=1 .. floor(log[r](N/piqj)))

%p     od

%p   od

%p od:

%p sort([Res]); # _Robert Israel_, Apr 12 2019

%t Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]==2&]

%Y Positions of 2's in A243055.

%Y A061395(a(n)) - A055396((n)) = 2.

%Y Cf. A000961, A008805, A046660, A056239, A093641, A112798, A118914, A174090, A195086, A256617, A325180, A325197.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 11 2019