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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A132768 a(n) = n*(n + 26).

Original entry on oeis.org

0, 27, 56, 87, 120, 155, 192, 231, 272, 315, 360, 407, 456, 507, 560, 615, 672, 731, 792, 855, 920, 987, 1056, 1127, 1200, 1275, 1352, 1431, 1512, 1595, 1680, 1767, 1856, 1947, 2040, 2135, 2232, 2331, 2432, 2535, 2640, 2747, 2856, 2967, 3080, 3195, 3312, 3431
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767.

Programs

Formula

a(n) = n*(n + 26).
a(n) = 2*n + a(n-1) + 25, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(26)/26 = A001008(26)/A102928(26) = 34395742267/232016584800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 18051406831/696049754400. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: x*(27 - 25*x)/(1-x)^3.
E.g.f.: x*(27 + x)*exp(x). (End)

A178242 Numerator of n*(5+n)/((n+1)*(n+4)).

Original entry on oeis.org

0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, 51, 117, 133, 75, 84, 187, 207, 114, 125, 273, 297, 161, 174, 375, 403, 216, 231, 493, 525, 279, 296, 627, 663, 350, 369, 777, 817, 429, 450, 943, 987, 516, 539, 1125, 1173, 611, 636, 1323, 1375, 714, 741, 1537, 1593
Offset: 0

Views

Author

Paul Curtz, Dec 20 2010

Keywords

Comments

Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).
Denominator: A160050(n+2).

Examples

			The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..

Links

Crossrefs

Cf. A001793, A014695, A028557, A060819, A160050, A176027.

Programs

  • Magma
    [Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011
  • Maple
    A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ;  numer(%) ;end proc:
seq(A178242(n),n=0..80) ; # R. J. Mathar, Dec 20 2010
  • Mathematica
    f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n,0,50}] (* G. C. Greubel, Sep 21 2018 *)
  • PARI
    vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ G. C. Greubel, Sep 21 2018

Formula

a(n) = numerator(A176027(n)/A001793(n+1)).
a(n) = A060819(n)*A060819(n+5).
a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.: x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).
a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - Bruno Berselli, Dec 30 2010
From Peter Bala, Aug 07 2022: (Start)
a(n) = numerator of n*(n+5)/4.
a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)
Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - Amiram Eldar, Aug 16 2022

A116268 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

81, 77394227, 89158933, 36623663376237623663376335, 37633762366336633762366235, 62366237633663366237633761, 63376336623762376336623661, 86194223018927804587702128, 88063202723646452838040443, 35574229497606875609044578088011, 35693849662968953146129859753682, 42317841210726174031503123524229
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			89158933 * 89158938 = 79493157//79493154, where // denotes concatenation.

Links

Crossrefs

Cf. A028557, A116099, A116261, A116267, A116269, A116276.

Programs

  • Maple
    f:= proc(d) local k, K;
      K:= map(t -> rhs(op(t)), [msolve(k^2+5*k+3=0,10^d+1)]);
      op(sort(select(k -> k^2 + 5*k + 3 >= (10^d+1)*10^(d-1), K)));
end proc:
map(f, [$1..62]); # Robert Israel, Jul 10 2025

Extensions

More terms from Robert Israel, Jul 10 2025

A116289 Numbers k such that k*(k+5) gives the concatenation of a number m with itself.

Original entry on oeis.org

6, 96, 385, 429, 567, 611, 814, 996, 4521, 5475, 9996, 90910, 99996, 316832, 683164, 999996, 3636364, 6363632, 9999996, 82352942, 99999996, 331668332, 368421053, 395604391, 442767754, 461538462, 488721800, 511278196, 538461534, 557232242, 604395605, 631578943, 668331664, 700089385, 727272728
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Apr 09 2025: (Start)
Numbers k such that k * (k + 5) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d-4 for all d >= 1. (End)

Links

Crossrefs

Cf. A028557, A020338.
Cf. A116158, A116281, A116288, A116290, A116303.

Programs

  • Maple
    q:= proc(d,m) local R,t,a,b,x,q;
   t:= 10^d+1;
   R:= NULL;
   for a in numtheory:-divisors(t) do
     b:= t/a;
     if igcd(a,b) > 1 then next fi;
     for x from chrem([0,-m],[a,b]) by t do
       q:= x*(x+m)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort(convert({R},list));
end proc:
seq(op(q(d,5)),d=1..10); # Robert Israel, Apr 09 2025

Extensions

More terms from Robert Israel, Apr 09 2025

A116324 Numbers k such that k * (k+5) gives the concatenation of two numbers m and m+5.

Original entry on oeis.org

31, 65, 42754, 57242, 75424, 425073, 574923, 979529, 4301394, 5698602, 7028667, 4925000748, 5074999248, 7748266575, 8511881485, 8814851185, 7059602159673, 7106167933829, 7439286611622, 7485852385778, 46791112884926
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Links

Crossrefs

Cf. A028557, A032610.
Cf. A116193, A116310, A116323, A116325, A116331.

Programs

  • Maple
    f:= proc(d) local S,R,r,s,m,n;
  r:= 10^d+1;
  S:= map(t -> rhs(op(t)), [msolve(n*(n+5)=5,r)]);
  S:= select(proc(s) local t; t:= (s*(s+5)-5)/r; t+5 >= (r-1)/10 and t+5 < r-1 end proc, S);
  op(sort(S));
end proc:
map(f, [$1..20]); # Robert Israel, Jun 21 2024

A132769 a(n) = n*(n + 27).

Original entry on oeis.org

0, 28, 58, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 520, 574, 630, 688, 748, 810, 874, 940, 1008, 1078, 1150, 1224, 1300, 1378, 1458, 1540, 1624, 1710, 1798, 1888, 1980, 2074, 2170, 2268, 2368, 2470, 2574, 2680, 2788, 2898, 3010, 3124, 3240, 3358, 3478
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759.
Cf. A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768.
Cf. A132756.

Programs

Formula

a(n) = 2*n + a(n-1) + 26, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=28, a(2)=58; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 14 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(27)/27 = A001008(27)/A102928(27) = 312536252003/2168462696400, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/27 - 57128792093/2168462696400. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 2*x*(14 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(28 + x).
a(n) = 2*A132756(n). (End)

A116261 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

7, 97, 766, 857, 909, 997, 3284, 6712, 9997, 45451, 54545, 99997, 990099, 999997, 8181818, 9999997, 70588232, 99999997, 343130553, 362637362, 363636360, 420053630, 421052631, 497975708, 502024288, 578947365, 579946366
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A028557, A116130, A116255, A116260, A116262, A116268.

A116276 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

70, 79822843, 69852478553064869297984899963805, 77473062193002372448027740546437, 77747359197583788609974143907617, 84341826458653210947638195982113, 85367942837521291760016984490249
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			79822843 * 79822848 = 63716866//63716864, where // denotes concatenation.

Crossrefs

Cf. A028557, A116107, A116268, A116275, A116277, A116281.

A116281 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

39, 57, 32262231, 67737765, 79321055, 3341093417798787499092, 3861488851737861033960, 4747922651210186579786, 5252077348789813420210, 6138511148262138966036, 6658906582201212500904, 7232275368591793618230
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			79321055 * 79321060 = 62918301//62918300, where // denotes concatenation.

Crossrefs

Cf. A028557, A116112, A116276, A116280, A116282, A116289.

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
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