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A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

%I A277905 #33 Jul 23 2025 19:55:26
%S A277905 1,2,3,4,6,8,5,9,12,16,10,18,24,32,15,20,27,36,48,64,30,40,54,72,96,
%T A277905 128,7,25,45,60,80,81,108,144,192,256,14,50,90,120,160,162,216,288,
%U A277905 384,512,21,28,75,100,135,180,240,243,320,324,432,576,768,1024,42,56,150,200,270,360,480,486,640,648,864,1152,1536,2048,35,63,84,112,125,225,300,400
%N A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.
%C A277905 Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.
%C A277905 Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.
%C A277905 Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - _Gus Wiseman_, May 25 2024
%C A277905 Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - _Flávio V. Fernandes_, Jul 19 2025
%H A277905 Antti Karttunen, <a href="/A277905/b277905.txt">Table of n, a(n) for n = 1..166</a>
%H A277905 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A277905 a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]
%F A277905 Other identities. For all n >= 1:
%F A277905 A048675(a(n)) = A277903(n).
%e A277905 The irregular table begins as:
%e A277905   row terms
%e A277905    0   1;
%e A277905    1   2;
%e A277905    2   3,  4;
%e A277905    3   6,  8;
%e A277905    4   5,  9,  12,  16;
%e A277905    5  10, 18,  24,  32;
%e A277905    6  15, 20,  27,  36,  48,  64;
%e A277905    7  30, 40,  54,  72,  96, 128;
%e A277905    8   7, 25,  45,  60,  80,  81, 108, 144, 192, 256;
%e A277905    9  14, 50,  90, 120, 160, 162, 216, 288, 384, 512;
%e A277905   10  21, 28,  75, 100, 135, 180, 240, 243, 320, 324, 432,  576,  768, 1024;
%e A277905   11  42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;
%e A277905 ...
%t A277905 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A277905 Table[Select[Range[0,2^k],Total[2^(prix[#]-1)]==k&],{k,0,10}] (* _Gus Wiseman_, May 25 2024 *)
%o A277905 (Scheme)
%o A277905 (definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))
%o A277905 (define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.
%o A277905 ;; Implementation based on a naive recurrence:
%o A277905 (definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))
%Y A277905 Cf. A019565 (the left edge, the only terms that are squarefree).
%Y A277905 Cf. A000079 (the trailing edge).
%Y A277905 Cf. A048675, A260443, A277886, A277896, A277903, A277904.
%Y A277905 Row lengths are A018819 (number of partitions of binary rank n).
%Y A277905 A000009 counts strict partitions, ranks A005117.
%Y A277905 A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.
%Y A277905 A048675 gives binary rank of prime indices, distinct A087207.
%Y A277905 A048793 lists binary indices, product A096111, reverse A272020.
%Y A277905 A061395 gives greatest prime index, least A055396.
%Y A277905 A112798 lists prime indices, cf. A001222, A003963, A056239, A296150.
%Y A277905 A372890 adds up binary ranks of partitions, strict A372888.
%Y A277905 Cf. A000040, A000041, A001511, A005940, A118462, A215366, A225620, A246868.
%Y A277905 Cf. A097248, A048675.
%K A277905 nonn,tabf
%O A277905 1,2
%A A277905 _Antti Karttunen_, Nov 14 2016