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.

Showing 1-10 of 10 results.

A342927 The smallest polygonality of numbers that have exactly two different representations as polygonal numbers (A177029).

Original entry on oeis.org

3, 4, 3, 5, 4, 7, 5, 9, 4, 10, 11, 12, 7, 5, 14, 8, 15, 9, 17, 4, 10, 19, 20, 11, 21, 22, 8, 24, 25, 14, 15, 29, 10, 30, 16, 31, 5, 32, 17, 11, 34, 35, 19, 37, 39, 13, 21, 4, 42, 22, 14, 44, 23, 45, 8, 47, 25, 50, 51, 17, 54, 28, 55, 29, 57, 4, 30, 59, 60, 31
Offset: 1

Views

Author

Michel Marcus, Mar 29 2021

Keywords

Examples

			6 is A177029(1); it is a 3-gonal and 6-gonal number; so a(1) = 3.
9 is A177029(2); it is a 4-gonal and 9-gonal number; so a(2) = 4.

Links

Crossrefs

Cf. A177028, A177029, A342928.

Programs

  • PARI
    row(n) = my(v=List()); fordiv(2*n, k, if(k<2, next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v, s))); Vecrev(v); \\ A177028
lista(nn) = {for (n=3, nn, my(r = row(n)); if (#r == 2, print1(r[1], ", ")););}

A342928 The smallest polygonal index of numbers that have exactly two different representations as polygonal numbers (A177029).

Original entry on oeis.org

3, 3, 4, 3, 4, 3, 4, 3, 5, 3, 3, 3, 4, 5, 3, 4, 3, 4, 3, 7, 4, 3, 3, 4, 3, 3, 5, 3, 3, 4, 4, 3, 5, 3, 4, 3, 8, 3, 4, 5, 3, 3, 4, 3, 3, 5, 4, 11, 3, 4, 5, 3, 4, 3, 7, 3, 4, 3, 3, 5, 3, 4, 3, 4, 3, 13, 4, 3, 3, 4, 3, 3, 4, 5, 3, 3, 4, 5, 3, 4, 3, 4, 5, 7, 3, 4
Offset: 1

Views

Author

Michel Marcus, Mar 29 2021

Keywords

Comments

By definition, a(n) can never be equal to 2. Up to 10^7, no n has been found with a(n) = 6, 10 or 16.

Examples

			6 is A177029(1); it is a 3-gonal and 6-gonal number; it is the 3rd triangular number so a(1) = 3.
9 is A177029(2); it is a 4-gonal and 9-gonal number; it is the 3rd square number so a(2) = 3.

Crossrefs

Cf. A177028, A177029, A342927, A342550.

Programs

  • PARI
    row(n) = my(v=List()); fordiv(2*n, k, if(k<2, next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v, s))); Vecrev(v); \\ A177028
lista(nn) = {for (n=3, nn, my(r = row(n)); if (#r == 2, my(k); ispolygonal(n, r[1], &k); print1(k, ", ")););}

A195527 Integers n that are k-gonal for precisely 3 distinct values of k, where k >= 3.

Original entry on oeis.org

15, 21, 28, 51, 55, 64, 70, 75, 78, 91, 96, 100, 111, 112, 117, 126, 135, 136, 141, 144, 145, 148, 154, 156, 165, 175, 176, 186, 189, 195, 201, 204, 216, 232, 235, 238, 246, 255, 256, 285, 286, 288, 291, 297, 300, 306, 315, 316, 321, 322, 324, 330, 333, 336
Offset: 1

Views

Author

Ant King, Sep 21 2011

Keywords

Comments

See A177025 for number of ways a number can be represented as a polygonal number.

Examples

			21 is in the sequence because it is a triangular number (A000217), an octagonal number (A000567) and an icosihenagonal number (A051873).

Links

Crossrefs

Cf. A000217, A000567, A051873, A177025, A177029.

Programs

  • Mathematica
    data1=Reduce[1/2 n (n(k-2)+4-k)== # && k>=3 && n>0, {k,n}, Integers]&/@Range[336]; data2=If[Head[#]===And, 1, Length[#]] &/@data1; data3=DeleteCases[Table[If[data2[[k]]==3, k], {k, 1, Length[data2]}], Null]
  • Python
    A195527_list = []
for m in range(1,10**4):
    n, c = 3, 0
    while n*(n+1) <= 2*m:
        if not 2*(n*(n-2) + m) % (n*(n - 1)):
            c += 1
            if c > 2:
                break
        n += 1
    if c == 2:
        A195527_list.append(m) # Chai Wah Wu, Jul 28 2016

A195528 Integers n that are k-gonal for precisely 4 distinct values of k, where k >= 3.

Original entry on oeis.org

36, 45, 66, 81, 105, 120, 153, 171, 190, 196, 210, 261, 280, 351, 378, 396, 400, 405, 406, 456, 465, 477, 484, 496, 532, 576, 585, 606, 621, 630, 645, 666, 715, 726, 729, 736, 741, 742, 765, 780, 784, 801, 855, 876, 891, 910, 945, 960, 981, 1015, 1045, 1056
Offset: 1

Views

Author

Ant King, Sep 21 2011

Keywords

Comments

See A177025 for number of ways a number can be represented as a polygonal number.

Examples

			36 is in the sequence because it is a triangular number (A000217), a square number (A000290), a tridecagonal number (A051865), and a 36-gonal number.

Links

Crossrefs

Cf. A000217, A000290, A051865, A177025, A177029.

Programs

  • Mathematica
    data1=Reduce[1/2 n (n(k-2)+4-k)==# && k>=3 && n>0, {k,n}, Integers]&/@Range[1056]; data2=If[Head[#]===And, 1, Length[#]] &/@data1; data3=DeleteCases[Table[If[data2[[k]]==4, k], {k, 1, Length[data2]}], Null]
  • Python
    A195528_list = []
for m in range(1,10**4):
    n, c = 3, 0
    while n*(n+1) <= 2*m:
        if not 2*(n*(n-2) + m) % (n*(n - 1)):
            c += 1
            if c > 3:
                break
        n += 1
    if c == 3:
        A195528_list.append(m) # Chai Wah Wu, Jul 28 2016

A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

1559439365121, 2468046593376, 7760419091425
Offset: 1

Views

Author

Hugh Erling, Oct 24 2018

Keywords

Examples

			a(1): 1559439365121 has representations P(n,k) = P(3, 519813121708)=P(6, 103962624343)=P(9, 43317760144)=P(11, 28353443004)=P(18, 10192414153)=P(27, 4442847196)=P(33, 2953483648)=P(57, 977092336)=P(66, 727011361)=P(69, 664722664)=P(81, 481308448)=P(86, 426659199)=P(129, 188885584)=P(131, 183140268)=P(171, 107288572)=P(209, 71744544)=P(237, 55761976)=P(414, 18240979)=P(473, 13969968)=P(513, 11874388)=P(711, 6178324)=P(729, 5876784)=P(1881, 881968)=P(3537, 249376)=P(16899, 10924)=P(720981, 8).
a(2): 2468046593376 has representations P(n,k) = P(3, 822682197793)=P(6, 164536439560)=P(12, 37394645356)=P(18, 16131023488)=P(24, 8942197804)=P(26, 7593989520)=P(39, 3330697159)=P(42, 2866488496)=P(56, 1602627660)=P(72, 965589436)=P(84, 707988124)=P(96, 541238290)=P(116, 370021980)=P(126, 313402744)=P(392, 32204796)=P(416, 28591830)=P(576, 14903665)=P(647, 11809911)=P(783, 8061483)=P(936, 5640220)=P(1827, 1479601)=P(2912, 582306)=P(4302, 266776)=P(5823, 145603)=P(7056, 99160)=P(145551, 235).
a(3): 7760419091425 has representations P(n,k) = P(5, 776041909144)=P(7, 369543766260)=P(10, 172453757589)=P(13, 99492552456)=P(19, 45382567788)=P(25, 25868063640)=P(35, 13042721164)=P(37, 11652280920)=P(49, 6598995828)=P(55, 5225871444)=P(65, 3730970719)=P(82, 2336771785)=P(143, 764347396)=P(145, 743335164)=P(154, 658723293)=P(205, 371134344)=P(290, 185190769)=P(325, 147396376)=P(475, 68935548)=P(1225, 10351368)=P(1378, 8179601)=P(1729, 5194893)=P(2755, 2045644)=P(7585, 269814)=P(1969825, 6)=P(3939649, 3).

Links

Crossrefs

Cf. A057145, A063778, A129654, A139601, A177029, A195527, A195528.
A subsequence of A090466.

Programs

A321156 Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

561, 1485, 1701, 2016, 2556, 2601, 2850, 3025, 3060, 3256, 3321, 4186, 4761, 4851, 5226, 5320, 5565, 5841, 6175, 6216, 6336, 6525, 6670, 7425, 7821, 7840, 8001, 8100, 8625, 8646, 9730, 9856, 9945, 9976, 10116, 10296, 10450, 10585, 11025, 11305, 11340, 12025, 12090
Offset: 1

Views

Author

Hugh Erling, Oct 28 2018

Keywords

Comments

n | 2*m where m is a term in this sequence. - David A. Corneth, Oct 29 2018

Examples

			561 has representations P(3, 188)=P(6, 39)=P(11, 12)=P(17, 6)=P(33, 3).
1485 has representations P(3, 496)=P(5, 150)=P(9, 43)=P(15, 16)=P(54, 3).
1701 has representations P(3, 568)=P(6, 115)=P(9, 49)=P(18, 13)=P(21, 10).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321157, A321158, A321159, A321160, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 5; \\ Michel Marcus, Oct 29 2018
  • PARI
    is(n) = my(d=divisors(n<<1)); sum(i=2, #d, k=2*(d[i]^2 - 2 * d[i] + n) / (d[i] - 1) / d[i]; k == k\1 && min(d[i], k) >=3) == 5 \\ David A. Corneth, Oct 29 2018

A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

11935, 12376, 21736, 24220, 41041, 45441, 51360, 52326, 53361, 54145, 54405, 58311, 58696, 73360, 82720, 89425, 90321, 96580, 101025, 102025, 108801, 113050, 117216, 118405, 122265, 122500, 122760, 123201, 123256, 127281, 128961, 135201, 144585, 152076, 165376, 166635, 169456, 174097
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			11935 has representations P(n,k) = P(5, 1195) = P(7, 570) = P(10, 267) = P(14, 133) = P(35, 22) = P(55, 10) = P(154, 3).
12376 has representations P(n,k) = P(4, 2064) = P(7, 591) = P(16, 105) = P(26, 40) = P(34, 24) = P(56, 10) = P(91, 5).
21736 has representations P(n,k) = P(4, 3624) = P(8, 778) = P(11, 397) = P(16, 183) = P(19, 129) = P(22, 96) = P(208, 3).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321158, A321159, A321160, A320943.

A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m*((k-2)*m - (k-4))/2, k and m >= 3.

Original entry on oeis.org

11781, 61776, 75141, 133056, 152361, 156520, 176176, 179740, 188650, 210925, 241605, 266085, 292825, 298936, 338625, 342585, 354025, 358281, 360801, 365365, 371925, 391392, 395200, 400960, 417340, 419805, 424270, 438516
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 11781 has representations P(m,k) = P(3, 3928)=P(6, 787)=P(9,329)=P(11, 216)=P(21, 58)=P(63, 8)=P(77, 6)=P(153, 3).
a(2) 61776 has representations P(m,k) = P(3, 20593)=P(6, 4120)=P(8,2208)=P(11, 1125)=P(26, 192)=P(36, 100)=P(176, 6)=P(351, 3).
a(3) 75141 has representations P(m,k) = P(3, 25048)=P(6, 5011)=P(9,2089)=P(11, 1368)=P(18, 493)=P(27, 216)=P(66, 37)=P(69, 34).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321159, A321160, A320943.

Programs

  • Mathematica
    r[n_] := Module[{k}, Sum[Boole[d >= 3 && (k = 2(d^2 - 2d + n)/(d^2 - d); IntegerQ[k] && k >= 3)], {d, Divisors[2n]}]];
Select[Range[500000], r[#] == 8&] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)
  • PARI
    r(n)={sumdiv(2*n, d, if(d>=3, my(k=2*(d^2 - 2*d + n)/(d^2 - d)); !frac(k) && k>=3))}
for(n=1, 5*10^5, if(r(n)==8, print1(n, ", "))) \\ Andrew Howroyd, Nov 26 2018
  • Python
    # See link.

A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

27405, 126225, 194481, 201825, 273105, 478401, 538461, 615681, 718641, 859600, 862785, 1056160, 1187145, 1257201, 1328481, 1413126, 1439361, 1532601, 1540540, 1619541, 1625625, 1708785, 1842400, 1849926, 1890945
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 27405 has representations P(n,k) = P(3, 9136)=P(5, 2742)=P(9, 763)=P(14, 303)=P(18, 181)=P(27, 80)=P(35, 48)=P(63, 16)=P(105, 7).
a(2) 126225 has representations P(n,k) = P(3, 42076)=P(5, 12624)=P(9, 3508)=P(15, 1204)=P(17, 930)=P(33, 241)=P(50, 105)=P(99, 28)=P(225, 7).
a(3) 194481 has representations P(n,k) = P(3, 64828)=P(6, 12967)=P(9, 5404)=P(14, 2139)=P(18, 1273)=P(21, 928)=P(27, 556)=P(81, 62)=P(441, 4).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321160, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 9; \\ Michel Marcus, Nov 02 2018
  • Python
    # See Erling link.

A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

220780, 519156, 1079001, 1154440, 1324576, 1447551, 2429505, 2454705, 2491776, 2603601, 2665125, 2700621, 2772225, 2953665, 3000025, 3086721, 3316600, 3665376, 4488561, 4741660, 5142501, 5388201, 5785101, 6076225
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 220780 has representations P(n,k) = P(4, 36798) = P(7, 10515) = P(10, 4908) = P(14, 2428) = P(19, 1293) = P(28, 586) = P(35, 373) = P(38, 316) = P(40, 285) = P(664, 3).
a(2) 519156 has representations P(n,k) = P(3, 173053) = P(6, 34612) = P(8, 18543) = P(11, 9441) = P(27, 1481) = P(36, 826) = P(66, 244) = P(92, 126) = P(99, 109) = P(456, 7).
a(3) 1079001 has representations P(n,k) = P(3, 359668) = P(6, 71935) = P(9, 29974) = P(11, 19620) = P(14, 11859) = P(21, 5140) = P(27, 3076) = P(66, 505) = P(81, 335) = P(126, 139).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321159, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 10; \\ Michel Marcus, Nov 02 2018
  • Python
    # See links.
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