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.

11, 0, 2, 27, 92, 352, 1002, 16927, 2302, 7827, 25052, 220052, 13352, 1487552, 101752, 195677, 85177, 137532552, 173577

Offset: 0

Author

Robert Israel, Feb 26 2025

Keywords

• nonn
• more

Comments

a(n) is the least k >= 0 such that A093518(k) = n.

a(17) > 5.4 * 10^7 if it exists.

From Pontus von Brömssen, Feb 28 2025: (Start)

a(19) > 3*10^9 if it exists.

After a(19), the following are all terms below 3*10^9:

n | a(n)

---+-----------

20 | 333802

21 | 4891927

22 | 391438802

23 | 2543802

24 | 494027

25 | 55039427

27 | 3764827

28 | 8345052

30 | 4339427

32 | 2737177

35 | 1375985677

36 | 6422352

38 | 429902552

40 | 12350677

41 | 85573502

42 | 108485677

45 | 94120677

48 | 29014077

50 | 733363177

54 | 120983227

56 | 308766927

60 | 160558802

63 | 2353016927

64 | 101275552

68 | 2139337552

72 | 344336877

80 | 725351927

96 | 1073520852

(End)

Examples

			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.
		

Crossrefs

Cf. A001318, A093518.

Programs

Maple

GP:= [0,seq(op([m*(3*m-1)/2, m*(3*m+1)/2]),m=1..2000)]:
N:= GP[-1]:
V:= Array(0..N, datatype=integer[4]):
for i from 1 to nops(GP) do
for j from 1 to i do
   r:= GP[i]+GP[j];
   if r > N then break fi;
   V[r]:= V[r]+1
od od:
W:= Array(0..16): count:= 0:
for i from 1 to N while count < 17 do
  v:= V[i]; if v <= 16 and W[v] = 0 then W[v]:= i; count:= count + 1 fi
od:
W[1]:= 0:
convert(W,list);

Formula

A093518(a(n)) = n.

Extensions

a(17)-a(18) from Pontus von Brömssen, Feb 28 2025