cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A279830 a(n) = the least integer that is centered polygonal in exactly n ways.

Original entry on oeis.org

4, 7, 37, 31, 91, 181, 211, 421, 631, 1891, 1261, 2521, 6931, 18481, 20791, 13861, 27721, 41581, 83161, 138601, 245701, 235621, 180181, 556921, 360361, 540541, 1670761, 1081081, 1413721, 2702701, 2162161, 6486481, 3063061, 8288281, 13430341, 6846841, 10270261, 6126121
Offset: 1

Views

Author

Daniel Sterman, Dec 20 2016

Keywords

Comments

a(n) has exactly n representations as a centered r-gonal number P(r,m) = 1 + r*m*(m+1)/2, with m > 1, r > 0.
a(n) appears n+1 times in A101321, due to the second column containing every positive integer.
a(n)-1 is the first appearance of n+1 in A007862.

Examples

			a(4)=31, because 31 is a centered triangular number (A005448), a centered pentagonal number (A005891), a centered decagonal number (A062786), and a central polygonal number (A002061). No number less than 31 has 4 representations.

Links

Crossrefs

Cf. A007862 (see alternative definition: the number of ways to represent n+1 as a centered polygonal number).
Cf. A063778 (the equivalent for polygonal numbers).
Subset of A275340 (the list of nontrivial centered polygonal numbers).
Subset of A101321 (centered polygonal numbers read by antidiagonals).

Programs

  • Mathematica
    f[n_] := Length@Select[Divisors[2 n - 2], IntegerQ@Sqrt[1 + 4 #] &] - 1;
Do[If[IntegerQ[A279830[f[i]]], , A279830[f[i]] = i], {i, 10000}];
A279830 /@ Range[13]
(* Davin Park, Dec 28 2016 *)

Extensions

Corrected and extended by Davin Park, Dec 27 2016

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

Views

Author

Robert G. Wilson v, Jun 22 2024

Keywords

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

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]

A322660 Numbers k > 1 for which the number of representations as an m-gonal number P(m,r) = r*((m-2)*r-(m-4))/2, with m>1, r>1, equals the number of divisors of k.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 111, 113, 121, 127, 131, 137, 139, 141, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 201
Offset: 1

Views

Author

Daniel Suteu, Dec 22 2018

Keywords

Comments

Numbers k > 1 such that A129654(k) = A000005(k).
Each prime number greater than 2 is a term of this sequence.
The first 20 composite terms are: 9, 15, 21, 25, 49, 51, 55, 81, 91, 111, 121, 141, 145, 169, 201, 235, 289, 291, 321, 325.

Examples

			15 is a term of this sequence, as it has 4 divisors and it can be represented in 4 different ways as an m-gonal number P(m,r) = r*((m-2)*r-(m-4))/2, with m>1, r>1, as following: 15 = P(15,2) = P(6,3) = P(3,5) = P(2,15).

Links

Crossrefs

Cf. A000005, A063778, A129654.

Programs

  • PARI
    isok(k) = (k>1) && (sigma(k,0) == sumdiv(2*k, d, (d>1) && (2*k/d + 2*d - 4) % (d-1) == 0));

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0
Offset: 1

Views

Author

Robert G. Wilson v, Apr 27 2020

Keywords

Comments

\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
r\
_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998
_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956
_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477
_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217
_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290
_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326
_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384
_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566
_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567
_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A001106
10 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A001107
11 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A051682
12 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A051624
13 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A051865
14 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A051866
15 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

Links

Crossrefs

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Programs

  • Mathematica
    Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

Formula

P(r, c) = (r - 2)(c(c-1)/2) + c.

A333932 a(n) is the least integer that is 4-dimensional pyramidal in exactly n ways.

Original entry on oeis.org

5, 15, 35, 140, 1820, 11375, 820820, 19019000, 10790015600, 1568726956160, 7278234628665, 7271181889157550
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 10 2020

Keywords

Comments

a(n) has exactly n representations as an 4-dimensional pyramidal number P(m, k) = binomial(k + 2, 3)*(k*(m - 2) - m + 6) / 4, with m > 2, k > 1.

Examples

			a(3) = 35 because 35 is the least integer which is 4-dimensional pyramidal in 3 ways (35 = P(3, 4) = P(7, 3) = P(33, 2)).

Links

Crossrefs

Cf. A063778, A080852, A261721, A333916.

Extensions

a(9) from Giovanni Resta, Apr 11 2020
a(9) corrected and a(10)-a(12) from Bert Dobbelaere, Apr 14 2020
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