cp's OEIS Frontend

A382256 Smallest binary number whose run lengths of bits correspond to a partition with Heinz number n.

%I A382256 #12 Mar 24 2025 18:37:39
%S A382256 0,1,3,2,7,4,15,5,12,8,31,11,63,16,24,10,127,19,255,23,48,32,511,22,
%T A382256 56,64,51,47,1023,39,2047,21,96,128,112,44,4095,256,192,46,8191,79,
%U A382256 16383,95,103,512,32767,45,240,71,384,191,65535,76,224,94,768,1024,131071,92,262143,2048,207,42,448,159
%N A382256 Smallest binary number whose run lengths of bits correspond to a partition with Heinz number n.
%C A382256 This is another way of assigning a canonical representative (natural number and composition) to any partition, and therefore to define an order on the set of all partitions.
%C A382256 Since the binary number corresponding to a partition of n is in the interval [2^(n-1), 2^n-1], it does respect the primary order criterion of all classical partition orderings (Abramowitz and Stegun or Hindenburg order: A036036, "Maple order": A080576, "Mathematica order": A080577, ...) which is the sum of the partitions. This is not the case for the Heinz number, cf. Examples.
%e A382256 Heinz number 1 corresponds to the empty product, empty sum and partition which is that of a(1) = 0.
%e A382256 Heinz number 2 = prime(1) corresponds to the partition (1), and run lengths of the binary number a(2) = 1 which has just one bit 1.
%e A382256 Heinz number 3 = prime(2) corresponds to the partition (2), which equals the run lengths of the binary number 11[2] = 3 = a(3) which has two consecutive bits 1.
%e A382256 Heinz number 4 = prime(1)^2 corresponds to the partition (1,1), which equals the run lengths of the binary number 10[2] = 2 = a(4) which has one bit 1, then one bit 0.
%e A382256 Heinz number 5 = prime(3) corresponds to the partition (3), which equals the run lengths of the binary number 111[2] = 7 = a(5) which has three consecutive bits 1.
%e A382256 Heinz number 6 = prime(1)*prime(2) corresponds to the partition (1,2), which equals the run lengths of the binary number 100[2] = 4 = a(6).
%e A382256 Heinz number 7 = prime(4) corresponds to the partition (4), which equals the run lengths of the binary number 1111[2] = 15 = a(7).
%e A382256 Heinz number 8 = prime(1)^3 corresponds to the partition (1,1,1), which equals the run lengths of the binary number 101[2] = 5 = a(8). Here, we have a partition with smaller sum than the previous one, but with a larger Heinz number. This shows that ordering the partitions by their Heinz number puts partitions with smaller sums behind partitions with larger sums. The binary numbers, however, do increase with increasing sum of the partition.
%e A382256 Similarly, Heinz number 14 = 2*7 = prime(1)*prime(4) corresponds to the partition (1,4) of 5, which equals the run lengths of the binary number 10000[2] = 16 = a(14), while the larger Heinz number 16 = 2^4 = prime(1)^4 corresponds to the partition (1,1,1,1) with smaller sum 4, and run lengths of the smaller binary number 1010[2] = 10 = a(16).
%o A382256 (PARI) apply( {A382256(n)=if(n=factor(n)~, n[1,]=apply(primepi, n[1,]); my(i=1, b, L, R=2^n[1,1]-1); n=Vec(n); while(n[i][2]-- || n=n[^i], R<<=L=n[i=b*#n+!b][1]; b && R+=2^L-1; b=1-b); R)}, [1..66])
%Y A382256 Cf. A382255,  A036036, A080576, A080577 (partitions in Abramowitz-Stegun, Maple and Mathematica order).
%K A382256 nonn
%O A382256 1,3
%A A382256 _M. F. Hasler_, Mar 19 2025