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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A164006 Zero together with row 6 of the array in A163280.

Original entry on oeis.org

0, 11, 22, 27, 44, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164005, A164007.
Cf. A028557 for n > 4. - R. J. Mathar, Aug 09 2009

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: A164006 := proc(n) if n = 0 then 0; else A163280(6,n) ; fi; end: seq(A164006(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0,11,22,27}, Table[n*(n + 5), {n, 4, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    concat(0, Vec(x*(8*x^6-21*x^5+23*x^4-18*x^3+6*x^2+11*x-11)/(x-1)^3 + O(x^100))) \\ Colin Barker, Nov 24 2014

Formula

From Colin Barker, Nov 24 2014: (Start)
a(n) = n*(n+5) for n > 4.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x*(8*x^6 - 21*x^5 + 23*x^4 - 18*x^3 + 6*x^2 + 11*x - 11) / (x-1)^3. (End)
E.g.f.: (x/2)*(10 + 8*x + x^2 + 2*(x + 6)*exp(x)). - G. C. Greubel, Aug 28 2017

Extensions

Extended beyond a(12) by R. J. Mathar, Aug 09 2009

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

A162787 a(n) = A162527(n)/7.

Original entry on oeis.org

7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 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
Offset: 1

Views

Author

Omar E. Pol, Jul 13 2009

Keywords

Links

Crossrefs

Cf. A161344, A162527, A162786, A162788.

A163925 Table, row n is nonprime numbers k such that the largest divisor of n*k <= sqrt(n*k) is n.

Original entry on oeis.org

1, 4, 4, 6, 9, 4, 6, 8, 6, 8, 9, 10, 15, 25, 6, 8, 9, 10, 8, 9, 10, 12, 14, 15, 21, 25, 35, 49, 8, 9, 10, 12, 14, 16, 9, 10, 12, 14, 15, 18, 21, 27, 10, 12, 14, 15, 16, 20, 25, 12, 14, 15, 16, 18, 20, 21, 22, 25, 27, 33, 35, 49, 55, 77, 121, 12, 14, 15, 16, 18, 22, 14, 15, 16, 18, 20
Offset: 1

Views

Author

Franklin T. Adams-Watters, Aug 06 2009

Keywords

Comments

Every prime > n also has this property.
If a*b is a composite number > n^2, with a <= b, then a*n and b are both > n, and one of them must be <= sqrt(n*a*b); thus n^2 is an upper bound for the numbers in row n.

Examples

			The table starts:
1: 1
2: 4
3: 4,6,9
4: 4,6,8
5: 6,8,9,10,15,25
6: 6,8,9,10

Links

Crossrefs

Cf. A163926 (row lengths), A161344, A033676.
Cf. A018252, A027750.

Programs

  • Haskell
    a163925 n k = a163925_tabf !! (n-1) !! (k-1)
a163925_tabf = map a163925_row [1..]
a163925_row n = [k | k <- takeWhile (<= n ^ 2) a018252_list,
                     let k' = k * n, let divs = a027750_row k',
                     last (takeWhile ((<= k') . (^ 2)) divs) == n]
-- Reinhard Zumkeller, Mar 15 2014
  • PARI
    arow(n)=local(v,d);v=[];for(k=n,n^2,if(!isprime(k),d=divisors(n*k);if(n==d[(#d+1)\2],v=concat(v,[k]))));v

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

A231167 a(1) = a(2) = 0, for n>=3: (sum of non-divisors of n) modulo (number of non-divisors of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 8, 6, 0, 0, 0, 0, 12, 1, 0, 0, 8, 1, 16, 20, 0, 19, 0, 23, 20, 1, 24, 8, 0, 1, 24, 26, 0, 25, 0, 32, 21, 1, 0, 26, 18, 38, 32, 38, 0, 31, 40, 36, 36, 1, 0, 30, 0, 1, 31, 15, 48, 37, 0, 50, 44, 47, 0, 33, 0, 1, 35, 56
Offset: 1

Views

Author

Jaroslav Krizek, Nov 07 2013

Keywords

Comments

a(n) = 0 for n = 1, 2 and numbers from A140826.
a(n) = 1 for numbers of form 2*p (p=prime) from A100484 and other numbers, e.g. 8 and 13456 are only numbers n < 10^5 which are not of form 2*p with a(n) = 1.

Examples

			For n=6, a(6) = A024816(6) mod A049820(6) = 9 mod 2 = 1.

Links

Crossrefs

Cf. A054025 (sigma(n) mod tau(n)), A024816, A049820, A024816, A049820, A065091, A230605, A161344.

Programs

  • Mathematica
    ndn[n_]:=Module[{nd=Complement[Range[n],Divisors[n]]},Mod[Total[ nd],Length[ nd]]]; Join[{0,0},Array[ndn,80,3]] (* Harvey P. Dale, Apr 11 2022 *)

Formula

a(n) = A024816(n) mod A049820(n).

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
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