cp's OEIS Frontend

A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

%I A334031 #10 Apr 18 2020 11:51:18
%S A334031 1,2,4,6,8,18,12,30,16,54,36,150,24,90,60,210,32,162,108,750,72,450,
%T A334031 300,1470,48,270,180,1050,120,630,420,2310,64,486,324,3750,216,2250,
%U A334031 1500,10290,144,1350,900,7350,600,4410,2940,25410,96,810,540,5250,360,3150
%N A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.
%C A334031 All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.
%C A334031 Unsorted prime signature is the sequence of exponents in a number's prime factorization.
%C A334031 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 A334031 a(n) = A057335(A059893(n)).
%e A334031 The sequence of terms together with their prime indices begins:
%e A334031        1: {}
%e A334031        2: {1}
%e A334031        4: {1,1}
%e A334031        6: {1,2}
%e A334031        8: {1,1,1}
%e A334031       18: {1,2,2}
%e A334031       12: {1,1,2}
%e A334031       30: {1,2,3}
%e A334031       16: {1,1,1,1}
%e A334031       54: {1,2,2,2}
%e A334031       36: {1,1,2,2}
%e A334031      150: {1,2,3,3}
%e A334031       24: {1,1,1,2}
%e A334031       90: {1,2,2,3}
%e A334031       60: {1,1,2,3}
%e A334031      210: {1,2,3,4}
%e A334031       32: {1,1,1,1,1}
%e A334031      162: {1,2,2,2,2}
%e A334031 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 A334031 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334031 Table[Product[Prime[i]^stc[n][[-i]],{i,DigitCount[n,2,1]}],{n,0,100}]
%Y A334031 The range is A055932.
%Y A334031 The non-reversed version is A057335.
%Y A334031 Unsorted prime signature is A124010.
%Y A334031 Numbers whose prime signature is aperiodic are A329139.
%Y A334031 Normal numbers with standard compositions as prime signature are A334032.
%Y A334031 All of the following pertain to compositions in standard order (A066099):
%Y A334031 - Length is A000120.
%Y A334031 - Sum is A070939.
%Y A334031 - Strict compositions are A233564.
%Y A334031 - Constant compositions are A272919.
%Y A334031 - Aperiodic compositions are A328594.
%Y A334031 - Normal compositions are A333217.
%Y A334031 - Heinz number is A333219.
%Y A334031 Cf. A029931, A048793, A056239, A112798, A228351, A233249, A272020, A333218, A333220, A334032.
%K A334031 nonn
%O A334031 0,2
%A A334031 _Gus Wiseman_, Apr 17 2020