A177025 -id:A177025 - cp's OEIS frontend

A333915 Number of ways to represent n as a pyramidal number.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2

Offset: 4

Ilya Gutkovskiy, Apr 09 2020

nonn

Frequency of n in the array A261720 of pyramidal numbers.

			a(10) = 2 because 10 is the third tetrahedral (or triangular pyramidal) number and also the second 9-gonal pyramidal number.
a(30) = 3 because 30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 29-gonal pyramidal number.

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A080851, A129654, A177025, A261720, A333916.

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

Original entry on oeis.org

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

Robert G. Wilson v, Jun 22 2024

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.

			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.

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

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

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]

cp's OEIS Frontend

A333915 Number of ways to represent n as a pyramidal number.

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