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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A164007 Zero together with row 7 of the array in A163280.

Original entry on oeis.org

0, 13, 26, 33, 52, 55, 78, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392, 2491, 2592, 2695
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164006, A164008.
Cf. A028560 for n > 6.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164007 := proc(n) if n = 0 then 0; else A163280(7,n) ; fi; end: seq(A164007(n),n=0..80) ;  # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 13, 26, 33, 52, 55, 78}, Table[n*(n + 6), {n, 7, 50}]] (* G. C. Greubel, Aug 28 2017 *)
LinearRecurrence[{3,-3,1},{0,13,26,33,52,55,78,91,112,135},50] (* Harvey P. Dale, Jul 03 2020 *)
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(x*(13 - 13*x - 6*x^2 + 18*x^3 - 28*x^4 + 36*x^5 - 30*x^6 + 18*x^7 - 6*x^8)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

From G. C. Greubel, Aug 28 2017: (Start)
a(n) = n*(n+6), n >= 7.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 7.
G.f.: x*(13 - 13*x - 6*x^2 + 18*x^3 - 28*x^4 + 36*x^5 - 30*x^6 + 18*x^7 - 6*x^8)/(1 - x)^3.
E.g.f.: (7*x + x^2)*exp(x) + 6*x +5*x^2 + x^3 + x^4/2 + x^6/120. (End)

Extensions

Extended by R. J. Mathar, Aug 09 2009

A164008 Zero together with row 8 of the array in A163280.

Original entry on oeis.org

0, 17, 34, 39, 68, 65, 102, 98, 128, 153, 170, 198, 228, 260, 294, 330, 368, 408, 450, 494, 540, 588, 638, 690, 744, 800, 858, 918, 980, 1044, 1110, 1178, 1248, 1320, 1394, 1470, 1548, 1628, 1710, 1794, 1880, 1968, 2058, 2150, 2244, 2340, 2438, 2538, 2640
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164007, A164009.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: printf("0,") ; for n from 1 to 70 do printf("%d,",A163280(8,n)) ; end do ; # R. J. Mathar, Feb 05 2010
  • Mathematica
    LinearRecurrence[{3,-3,1},{0,17,34,39,68,65,102,98,128,153,170,198,228},50] (* Harvey P. Dale, Dec 25 2022 *)

Formula

Conjecture: a(n) = A028563(n), n > 9. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond 228 from R. J. Mathar, Feb 05 2010

A164009 Zero together with row 9 of the array in A163280.

Original entry on oeis.org

0, 19, 38, 51, 76, 75, 114, 105, 136, 162, 190, 209, 264, 273, 308, 345, 384, 425, 468, 513, 560, 609, 660, 713, 768, 825, 884, 945, 1008, 1073, 1140, 1209, 1280, 1353, 1428, 1505, 1584, 1665, 1748, 1833, 1920, 2009, 2100, 2193, 2288, 2385, 2484, 2585, 2688
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164008, A164010.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end proc:
A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then return T ; end if; end if; end do: end proc:
printf("0, ") ; for n from 1 to 90 do printf("%d, ", A163280(9, n)) ; end do ; # R. J. Mathar, Jul 31 2010

Formula

Conjecture: a(n) = A028566(n), n > 12. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Jul 31 2010

A164010 Zero together with row 10 of the array in A163280.

Original entry on oeis.org

0, 23, 46, 57, 92, 85, 138, 119, 152, 171, 200, 220, 276, 286, 322, 375, 416, 442, 486, 532, 580, 630, 682, 736, 792, 850, 910, 972, 1036, 1102, 1170, 1240, 1312, 1386, 1462, 1540, 1620, 1702, 1786, 1872, 1960, 2050, 2142, 2236, 2332, 2430, 2530, 2632, 2736
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164009, A164011.

Formula

Conjecture: a(n) = A028569(n), n > 16. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Feb 06 2010

A164011 Zero together with row 11 of the array in A163280.

Original entry on oeis.org

0, 29, 58, 69, 116, 95, 174, 133, 184, 189, 230, 231, 348, 299, 350, 390, 448, 459, 522, 551, 620, 651, 704, 759, 816, 875, 936, 999, 1064, 1131, 1200, 1271, 1344, 1419, 1496, 1575, 1656, 1739, 1824, 1911, 2000, 2091, 2184, 2279, 2376, 2475, 2576, 2679, 2784
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164010, A164012.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: printf("0,") ; for n from 1 to 70 do printf("%d,",A163280(11,n)) ; end do ; # R. J. Mathar, Feb 05 2010

Formula

Conjecture: a(n) = A098603(n), n > 20. [R. J. Mathar, Jul 31 2010]

Extensions

Extended by R. J. Mathar, Feb 05 2010

A161425 a(n) = A161424(n)/2.

Original entry on oeis.org

8, 10, 12, 14, 16, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161428.

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009

A164005 Zero together with row 5 of the array in A163280.

Original entry on oeis.org

0, 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164004, A164006.
Cf. A028347.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164005 := proc(n) if n = 0 then 0; else A163280(5,n) ; fi; end: seq(A164005(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 7, 14}, Table[n*(n + 4), {n, 3, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

Conjecture: a(n) = A100451(n+2). (See A163280.)
From G. C. Greubel, Aug 28 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = n*(n+4), n >= 3.
G.f.: x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: x*(x+5)*exp(x) + 2*x + x^2. (End)

Extensions

Extended by R. J. Mathar, Aug 09 2009

A164012 Zero together with row 12 of the array in A163280.

Original entry on oeis.org

0, 31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, 312, 364, 405, 464, 476, 558, 570, 640, 693, 726, 782, 888, 925, 962, 1026, 1092, 1160, 1230, 1302, 1376, 1452, 1530, 1610, 1692, 1776, 1862, 1950, 2040, 2132, 2226, 2322, 2420, 2520, 2622, 2726
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164011.

Formula

Conjecture: a(n) = A119412(n), n > 36. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Jul 31 2010

A365081 Numbers k with the property that the symmetric representation of sigma(k) has four parts and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the diagram.

Original entry on oeis.org

21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381
Offset: 1

Views

Author

Omar E. Pol, Aug 20 2023

Keywords

Comments

Also the row numbers of the triangle A364639 where the rows are [0, 0, 1, 0, -1, 1] or where the rows start with [0, 0, 1, 0, -1, 1] and the remaining terms are zeros.
Observation: the first 29 terms coincide with the first 29 terms of A161345 that are >= 21.
Apparently a(n)=A127329(n) for n>2. - R. J. Mathar, Sep 05 2023

Examples

			The symmetric representation of sigma(21) in the first quadrant looks like this:
   _ _ _ _ _ _ _ _ _ _ _
  |_ _ _ _ _ _ _ _ _ _ _|
                        |
                        |
                        |_ _ _
                        |_ _  |_
                            |_ _|_
                                | |_
                                |_  |
                                  | |
                                  |_|_ _ _ _
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          |_|
.
There are four parts (or polygons) and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the structure so 21 is in the sequence.

Crossrefs

Subsequence of A016945, A264102, A280107 and A364414.
Cf. A033676, A161345, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A240062, A245092, A249351 (widths), A262626, A364639.

A161428 a(n) = A161424(n)/4.

Original entry on oeis.org

4, 5, 6, 7, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161425.

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009
Previous Showing 21-30 of 38 results. Next

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