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A381535 a(n) is the least nonnegative number that can be represented as the sum of two (not necessarily distinct) generalized pentagonal numbers in exactly n ways.

%I A381535 #9 Mar 01 2025 12:18:43
%S A381535 11,0,2,27,92,352,1002,16927,2302,7827,25052,220052,13352,1487552,
%T A381535 101752,195677,85177,137532552,173577
%N A381535 a(n) is the least nonnegative number that can be represented as the sum of two (not necessarily distinct) generalized pentagonal numbers in exactly n ways.
%C A381535 a(n) is the least k >= 0 such that A093518(k) = n.
%C A381535 a(17) > 5.4 * 10^7 if it exists.
%C A381535 From _Pontus von Brömssen_, Feb 28 2025: (Start)
%C A381535 a(19) > 3*10^9 if it exists.
%C A381535 After a(19), the following are all terms below 3*10^9:
%C A381535    n |    a(n)
%C A381535   ---+-----------
%C A381535   20 |     333802
%C A381535   21 |    4891927
%C A381535   22 |  391438802
%C A381535   23 |    2543802
%C A381535   24 |     494027
%C A381535   25 |   55039427
%C A381535   27 |    3764827
%C A381535   28 |    8345052
%C A381535   30 |    4339427
%C A381535   32 |    2737177
%C A381535   35 | 1375985677
%C A381535   36 |    6422352
%C A381535   38 |  429902552
%C A381535   40 |   12350677
%C A381535   41 |   85573502
%C A381535   42 |  108485677
%C A381535   45 |   94120677
%C A381535   48 |   29014077
%C A381535   50 |  733363177
%C A381535   54 |  120983227
%C A381535   56 |  308766927
%C A381535   60 |  160558802
%C A381535   63 | 2353016927
%C A381535   64 |  101275552
%C A381535   68 | 2139337552
%C A381535   72 |  344336877
%C A381535   80 |  725351927
%C A381535   96 | 1073520852
%C A381535 (End)
%F A381535 A093518(a(n)) = n.
%e A381535 a(3) = 27 because 27 = 1 + 26 = 5 + 22 = 12 + 15 has 3 representations as the sum of two generalized pentagonal numbers, and no smaller number works.
%p A381535 GP:= [0,seq(op([m*(3*m-1)/2, m*(3*m+1)/2]),m=1..2000)]:
%p A381535 N:= GP[-1]:
%p A381535 V:= Array(0..N, datatype=integer[4]):
%p A381535 for i from 1 to nops(GP) do
%p A381535 for j from 1 to i do
%p A381535    r:= GP[i]+GP[j];
%p A381535    if r > N then break fi;
%p A381535    V[r]:= V[r]+1
%p A381535 od od:
%p A381535 W:= Array(0..16): count:= 0:
%p A381535 for i from 1 to N while count < 17 do
%p A381535   v:= V[i]; if v <= 16 and W[v] = 0 then W[v]:= i; count:= count + 1 fi
%p A381535 od:
%p A381535 W[1]:= 0:
%p A381535 convert(W,list);
%Y A381535 Cf. A001318, A093518.
%K A381535 nonn,more
%O A381535 0,1
%A A381535 _Robert Israel_, Feb 26 2025
%E A381535 a(17)-a(18) from _Pontus von Brömssen_, Feb 28 2025