This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382255 #29 Aug 25 2025 22:32:45
%S A382255 1,2,4,3,6,8,6,5,10,12,16,12,9,12,10,7,14,20,24,18,24,32,24,20,15,18,
%T A382255 24,18,15,20,14,11,22,28,40,30,36,48,36,30,40,48,64,48,36,48,40,28,21,
%U A382255 30,36,27,36,48,36,30,25,30,40,30,21,28,22,13,26,44,56,42
%N A382255 Heinz number of the partition corresponding to run lengths in the bits of n.
%C A382255 The run lengths (number of consecutive bits that are equal) in the binary numbers in [2^(L-1), 2^L-1], i.e., of bit length L, yield all possible compositions of L, i.e., the partitions with any possible order of the parts.
%C A382255 Associated to any composition (p1, ..., pK) is their Heinz number prime(p1)*...*prime(pK) which depends only on the partition, i.e., not on the order of the parts.
%C A382255 The sequence can also be read as a table with row lengths 1, 1, 2, 4, 8, 16, 32, ... (= A011782), where row L = 0, 1, 2, 3, ... lists the 2^(L-1) compositions of L through their Heinz numbers (which will appear more than once if they contain at least two distinct parts).
%H A382255 Alois P. Heinz, <a href="/A382255/b382255.txt">Table of n, a(n) for n = 0..32768</a>
%F A382255 a(2^n) = A001747(n+1).
%F A382255 a(2^n-1) = A008578(n+1).
%F A382255 a(2^n+1) = A001749(n-1) for n>=2.
%e A382255 n | binary | partition | a(n) = Heinz number
%e A382255 ---+--------+-----------+--------------------
%e A382255 0 | (0) | empty sum | 1 = empty product
%e A382255 1 | 1 | 1 | 2 = prime(1)
%e A382255 2 | 10 | 1+1 | 4 = prime(1) * prime(1)
%e A382255 3 | 11 | 2 | 3 = prime(2)
%e A382255 4 | 100 | 1+2 | 6 = prime(1) * prime(2)
%e A382255 5 | 101 | 1+1+1 | 8 = 2^3 = prime(1) * prime(1) * prime(1)
%e A382255 6 | 110 | 2+1 | 6 = prime(2) * prime(1)
%e A382255 7 | 111 | 3 | 5 = prime(3)
%e A382255 8 | 1000 | 1+3 | 10 = 2*5 = prime(1) * prime(3)
%e A382255 9 | 1001 | 1+2+1 | 12 = 2^2*3 = prime(1) * prime(2) * prime(1)
%e A382255 ...| ... | ... | ...
%e A382255 For example, n = 4 = 100[2] (in binary) has run lengths (1, 2), namely: one bit 1 followed by two bits 0. This gives a(4) = prime(1)*prime(2) = 6.
%e A382255 Next, n = 5 = 101[2] (in binary) has run lengths (1, 1, 1): one bit 1, followed by one bit 0, followed by one bit 1. This gives a(4) = prime(1)^3 = 8.
%e A382255 Then, n = 6 = 110[2] (in binary) has run lengths (2, 1): first two bits 1, then one bit 0. This is the same as for 4, just in reverse order, so it yields the same Heinz number a(6) = prime(2)*prime(1) = 6.
%e A382255 Then, n = 7 = 111[2] (in binary) has run lengths (3), namely: three bits 1. This gives a(5) = prime(3) = 5.
%e A382255 Sequence written as irregular triangle:
%e A382255 1;
%e A382255 2;
%e A382255 4, 3;
%e A382255 6, 8, 6, 5;
%e A382255 10, 12, 16, 12, 9, 12, 10, 7;
%e A382255 14, 20, 24, 18, 24, 32, 24, 20, 15, 18, 24, 18, 15, 20, 14, 11;
%e A382255 ...
%p A382255 a:= proc(n) option remember; `if`(n<2, 1+n, (p->
%p A382255 a(iquo(n, 2^p))*ithprime(p))(padic[ordp](n+(n mod 2), 2)))
%p A382255 end:
%p A382255 seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 20 2025
%o A382255 (PARI)
%o A382255 Heinz(p)=vecprod([ prime(k) | k <- p ])
%o A382255 RL(v) = if(#v, v=Vec(select(t->t,concat([1,v[^1]-v[^-1],1]),1)); v[^1]-v[^-1])
%o A382255 apply( {A382255(n) = Heinz(RL(binary(n)))}, [0..99] )
%Y A382255 Cf. A112798 and A296150 (partitions sorted by Heinz number).
%Y A382255 Cf. A185974, A334433, A334435, A334438, A334434, A129129, A334436 (partitions given as Heinz numbers, in Abramowitz-Stegun, Maple, Mathematica order).
%Y A382255 For "constructive" lists of partitions see A036036 (Abramowitz and Stegun order), A036036 (reversed), A080576 (Maple order), A080577 (Mathematica order).
%Y A382255 Row sums of triangle give A030017(n+1).
%Y A382255 Cf. A007088 (the binary numbers).
%Y A382255 Cf. A011782, A001747, A001749, A008578.
%Y A382255 Cf. A101211 (the run lengths as rows of a table).
%K A382255 nonn,look,base,changed
%O A382255 0,2
%A A382255 _M. F. Hasler_ and _Ali Sada_, Mar 19 2025