A334031 - OEIS

%I #10 Apr 18 2020 11:51:18

%S 1,2,4,6,8,18,12,30,16,54,36,150,24,90,60,210,32,162,108,750,72,450,

%T 300,1470,48,270,180,1050,120,630,420,2310,64,486,324,3750,216,2250,

%U 1500,10290,144,1350,900,7350,600,4410,2940,25410,96,810,540,5250,360,3150

%N The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

%C All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.

%C Unsorted prime signature is the sequence of exponents in a number's prime factorization.

%C The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%F a(n) = A057335(A059893(n)).

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

%e        1: {}

%e        2: {1}

%e        4: {1,1}

%e        6: {1,2}

%e        8: {1,1,1}

%e       18: {1,2,2}

%e       12: {1,1,2}

%e       30: {1,2,3}

%e       16: {1,1,1,1}

%e       54: {1,2,2,2}

%e       36: {1,1,2,2}

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

%e       24: {1,1,1,2}

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

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

%e      210: {1,2,3,4}

%e       32: {1,1,1,1,1}

%e      162: {1,2,2,2,2}

%e For example, the 13th composition in standard order is (1,2,1), and the least number with prime signature (1,2,1) is 90 = 2^1 * 3^2 * 5^1, so a(13) = 90.

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Product[Prime[i]^stc[n][[-i]],{i,DigitCount[n,2,1]}],{n,0,100}]

%Y The range is A055932.

%Y The non-reversed version is A057335.

%Y Unsorted prime signature is A124010.

%Y Numbers whose prime signature is aperiodic are A329139.

%Y Normal numbers with standard compositions as prime signature are A334032.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Sum is A070939.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Aperiodic compositions are A328594.

%Y - Normal compositions are A333217.

%Y - Heinz number is A333219.

%Y Cf. A029931, A048793, A056239, A112798, A228351, A233249, A272020, A333218, A333220, A334032.

%K nonn

%O 0,2

%A _Gus Wiseman_, Apr 17 2020