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

%I A373921 #19 Jul 06 2024 13:59:56
%S A373921 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,
%T A373921 29,19,31,32,21,27,30,6,37,38,25,32,41,27,43,44,12,37,47,31,45,50,20,
%U A373921 42,53,35,44,56,37,47,59,39,61,62,41,44,57,12,67,68,45,49,71,47,73,74,32
%N A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.
%C A373921 Inspired by A342772 and A187202.
%C A373921 The n-th row of A177028 are all integers k for which n is a k-gonal number.
%C A373921 As an example: row 10 of A177028 contain 3 and 10, because 10 is a 10-gonal number but also a triangular number.
%C A373921 -3n/2 < a(n) <= n.
%C A373921 a(n) = n if n is an odd prime (A065091), an odd composite number in A274967, or even numbers in A274968.
%C A373921 a(n) = 0: 231, tested up to 150000.
%C A373921 a(n) < 0: 441, 540, 561, 1089, 1128, 1296, 1521, 1701, 1716, 1881, 2016, 2211, 2541, 2556, 2601, ..., .
%C A373921 a(n) is negative less than 1% of the time.
%H A373921 Robert G. Wilson v, <a href="/A373921/b373921.txt">Table of n, a(n) for n = 3..150000</a>
%e A373921 a(15) = 6, because the 15th row of A177028 is {3,6,15} -> {3,9} -> {6};
%e A373921 a(36) = 6, because the 36th row of A177028 is {3,4,13,36} -{1,9,23} - {8,14} -> {6};
%e A373921 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};
%e A373921 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.
%t A373921 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]
%Y A373921 Cf. A063778, A177025, A177028, A176774, A176948, A274967, A342550, A342805.
%K A373921 easy,sign
%O A373921 3,1
%A A373921 _Robert G. Wilson v_, Jun 22 2024