A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.

3, 4, 5, 3, 7, 8, 5, 7, 11, 7, 13, 14, 6, 12, 17, 11, 19, 20, 8, 17, 23, 15, 21, 26, 17, 19, 29, 19, 31, 32, 21, 27, 30, 6, 37, 38, 25, 32, 41, 27, 43, 44, 12, 37, 47, 31, 45, 50, 20, 42, 53, 35, 44, 56, 37, 47, 59, 39, 61, 62, 41, 44, 57, 12, 67, 68, 45, 49, 71, 47, 73, 74, 32

Offset: 3

Author

Robert G. Wilson v, Jun 22 2024

Keywords

• easy
• sign

Comments

Inspired by A342772 and A187202.

The n-th row of A177028 are all integers k for which n is a k-gonal number.

As an example: row 10 of A177028 contain 3 and 10, because 10 is a 10-gonal number but also a triangular number.

-3n/2 < a(n) <= n.

a(n) = n if n is an odd prime (A065091), an odd composite number in A274967, or even numbers in A274968.

a(n) = 0: 231, tested up to 150000.

a(n) < 0: 441, 540, 561, 1089, 1128, 1296, 1521, 1701, 1716, 1881, 2016, 2211, 2541, 2556, 2601, ..., .

a(n) is negative less than 1% of the time.

Examples

			a(15) = 6, because the 15th row of A177028 is {3,6,15} -> {3,9} -> {6};
a(36) = 6, because the 36th row of A177028 is {3,4,13,36} -{1,9,23} - {8,14} -> {6};
a(225) = 37, because the 225th row of A177028 is {4,8,24,76,225} -> {4,16,52,149} -> {12,36,97} -> {24,61} -> {37};
a(561) = -82, because the 561st row of A177028 is {3,6,12,39,188,561} -> {3,6,27,149,373} -> {3,21,122,224} -> {18,101,102}, {83,1} -> {-82}; etc.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 3..150000

Crossrefs

Cf. A063778, A177025, A177028, A176774, A176948, A274967, A342550, A342805.

Programs

Mathematica

planeFigurateQ[n_, r_] := IntegerQ[((r -4) + Sqrt[(r -4)^2 + 8n (r -2)])/(2 (r -2))]; a[n_] := Block[{pg = Select[ Range[3, n], planeFigurateQ[n, #] &]}, Differences[pg, Length@ pg - 1][[1]]]; Array[a, 73, 3]