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a(n) is the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.
2

%I #22 Feb 16 2025 08:34:04

%S 9,12880,20449,10764222,794629045,33205080888,5985,13925100

%N a(n) is the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>

%e For n = 2: 9 = 2 + 3 + 4 = 4 + 5.

%e For n = 3: 12880 = 91 + ... + 903 = 300 + ... + 990.

%e For n = 4: 20449 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.

%e For n = 5: 10764222 = 1617 + ... + 115787 = 31032 + ... + 126005.

%e From _Michael S. Branicky_, Feb 19 2023: (Start)

%e n-th term and indices of n-gonal numbers summing to it:

%e a(2) = 9: 2..4, 4..5,

%e a(3) = 12880: 13..42, 24..44,

%e a(4) = 20449: 7..39, 38..48,

%e a(5) = 10764222: 33..278, 144..290,

%e a(6) = 794629045: 1305..1505, 5321..5334,

%e a(7) = 33205080888: 616..3422, 3235..4192,

%e a(8) = 5985: 1..18, 11..19,

%e a(9) = 13925100: 103..235, 220..282. (End)

%Y Cf. A057145, A062681, A307608, A307614, A343777, A360777.

%K nonn,more,changed

%O 2,1

%A _Ilya Gutkovskiy_, Feb 19 2023

%E a(6)-a(9) from _Michael S. Branicky_, Feb 19 2023