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 1-5 of 5 results.

A082259 Triangle read by rows in which the n-th row contains n multiples of n with digit sum n.

Original entry on oeis.org

1, 2, 20, 3, 12, 21, 4, 40, 112, 220, 5, 50, 140, 230, 320, 6, 24, 42, 60, 114, 132, 7, 70, 133, 322, 511, 700, 1015, 8, 80, 152, 224, 440, 512, 800, 1016, 9, 18, 27, 36, 45, 54, 63, 72, 81, 190, 280, 370, 460, 550, 640, 730, 820, 910, 1090, 209, 308, 407, 506, 605, 704, 803, 902, 2090, 3080, 4070
Offset: 1

Views

Author

Amarnath Murthy, Apr 12 2003

Keywords

Examples

			Triangle begins:
1
2, 20;
3, 12,  21;
4, 40, 112, 220;
5, 50, 140, 230, 320;
6, 24,  42,  60, 114, 132;
7, 70, 133, 322, 511, 700, 1015;
8, 80, 152, 224, 440, 512,  800, 1016;
9, 18,  27,  36,  45,  54,   63,   72,  81;

Links

Crossrefs

Cf. A082260, A082261, A082262, A082263, A002998.

Programs

  • Mathematica
    Table[Take[Select[n Range[10000],Total[IntegerDigits[#]]==n&],n],{n,20}] // Flatten (* Harvey P. Dale, Sep 16 2019 *)

Extensions

Corrected by Anne Donovan, May 29 2003
Corrected and extended by David Wasserman, Aug 26 2004

A082260 a(n) = n-th multiple of n with digit sum n.

Original entry on oeis.org

1, 20, 21, 220, 320, 132, 1015, 1016, 81, 1090, 4070, 516, 2353, 2534, 1185, 3760, 3842, 846, 5662, 14960, 3738, 7546, 12857, 7296, 28825, 25298, 6885, 39088, 43877, 78960, 61969, 78368, 64977, 98872, 258965, 69984, 187849, 367688, 199758
Offset: 1

Views

Author

Amarnath Murthy, Apr 12 2003

Keywords

Links

Crossrefs

Main diagonal of A082259 and of A245062.
Row sums give A082261.
Cf. A082262, A082263, A002998.

Extensions

Corrected and extended by Ray Chandler, Oct 08 2005

A082261 Row sums in A082259.

Original entry on oeis.org

1, 22, 36, 376, 745, 378, 2758, 3232, 405, 6040, 13684, 3528, 16198, 19474, 10575, 34384, 35479, 10674, 65854, 178600, 48321, 108394, 161575, 117000, 445225, 392158, 136404, 695800, 778882, 2011950, 1332349, 1806496, 1849320, 2563600
Offset: 1

Views

Author

Amarnath Murthy, Apr 12 2003

Keywords

Crossrefs

Cf. A082259, A082260, A082262, A082263, A002998.

Programs

  • Mathematica
    Do[s = t = 0; i = 1; While[t < n, If[Plus @@ IntegerDigits[i*n] == n, t++; s += i*n]; i++ ]; Print[s], {n, 1, 50}] (* Ryan Propper, Jul 16 2005 *)

Extensions

Corrected and extended by Ryan Propper, Jul 16 2005

A082263 Triangle A082259 with row n divided by n.

Original entry on oeis.org

1, 1, 10, 1, 4, 7, 1, 10, 28, 55, 1, 10, 28, 46, 64, 1, 4, 7, 10, 19, 22, 1, 10, 19, 46, 73, 100, 145, 1, 10, 19, 28, 55, 64, 100, 127, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 19, 28, 37, 46, 55, 64, 73, 82, 190, 280, 370, 4, 7, 13, 16, 19, 22, 28, 31, 34, 37
Offset: 1

Views

Author

Amarnath Murthy, Apr 12 2003

Keywords

Examples

			Triangle begins:
1
1 10
1 4 7
1 10 28 55
1 10 28 46 64
1 4 7 10 19 22

Crossrefs

Cf. A082259, A082260, A082261, A082262, A002998.

Extensions

Corrected and extended by Ray Chandler, Oct 08 2005

A281008 Least positive integer k with exactly n odd divisors greater than sqrt(2*k).

Original entry on oeis.org

1, 3, 21, 75, 105, 315, 495, 945, 1575, 2835, 3465, 4095, 11025, 17955, 10395, 23205, 17325, 24255, 31185, 36855, 51975, 61425, 45045, 108675, 143325, 121275, 184275, 155925, 135135, 176715, 239085, 315315, 294525, 225225, 606375, 626535, 405405, 700245, 1531530, 1351350, 2072070, 1289925, 855855
Offset: 0

Views

Author

Omar E. Pol, Feb 16 2017

Keywords

Comments

Conjecture: a(n) is also the smallest number k having n pairs of equidistant subparts in the symmetric representation of sigma(k).
For more information about the "subparts" see A279387.
Observations about the known terms:
Observation 1: terms a(1)-a(51) are divisible by 3.
Observation 2: terms a(3)-a(51) are divisible by 5.

Examples

			a(3) = 75 because the divisors of 75 are [1, 3, 5, 15, 25, 75], and 75 has three odd divisors greater than the square root of 2*75 = 12.2..., and it is the smallest number with that property.
Other examples (conjectured):
2) The 75th row of A237593 is [38, 13, 7, 4, 3, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 3, 4, 7, 13, 38], and the 74th row of the same triangle is [38, 13, 6, 5, 3, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 3, 5, 6, 13, 38], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are three pairs of equidistant subparts: [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75. (The diagram of the symmetric representation of sigma(75) is too large to include).
3) The 75th row of A196020 is [149, 73, 47, 0, 25, 19, 0, 0, 0, 5, 0], hence the 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0]. There are three pairs of equidistant subparts [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75.
4) The 75th row of A237048 is [1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0]. The sum of the even-indexed terms is equal to 3. That is the first row with that property, so a(3) = 75.
5) The 75th row of A261699 is [1, 75, 3, 0, 5, 25, 0, 0, 0, 15, 0]. There are three even-indexed terms that are positive integers: [75, 25, 15]. That is the first row with that property, so a(3) = 75.

Links

Crossrefs

Row 1 of A280849.
Cf. A000203, A001227, A008585, A067742, A082262, A131576, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A261699, A262626, A279387, A280850, A280940, A281005.

Programs

  • Mathematica
    cnt[k_] := cnt[k] = DivisorSum[k, Boole[OddQ[#] && #>Sqrt[2k]]&]; a[n_] := a[n] = For[k = 1, True, k++, If[cnt[k]==n, Return[k]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)
  • PARI
    a(n,{s=0},{q=1},{k=2},{w=1})={if(n<1,return(1));my(z,ii,F,d,L:list,V,p,ans:list);ans=List();if(q<1,q=1);if(k<2,k=2);while(k++,p=sqrt(2*k);F=factor(k);ii=vecsum(F[1,]);F=F[,1]~;L=List([1]);for(i=1,ii,forvec(y=vector(i,t,[1,#F]),d=prod(u=1,#y,F[y[u]]);if((d<=k)&&!(k%d),listput(L,d)),1));V=Set(Vec(L));if(n==sum(u=1,#V,(V[u]>p)&&(V[u]%2==!!w)),if(s,print1(V","));listput(ans,k);if(z++==q,if(#ans==1,return(k),return(Vec(ans))),n++)))} \\ with n>=1, "s" set to 1 also prints the divisors (of "w" version: 1 odd, 0 even) for the first "q" terms from the n-th, resuming their search with k>=2. - R. J. Cano, Feb 20 2017
  • PARI
    a(n)=my(k,s); while(k++, s=sqrtint(2*k); if(sumdiv(k>>valuation(k,2), d, d>s)==n, return(k))) \\ Charles R Greathouse IV, Feb 20 2017

Extensions

a(10)-a(30) from Jean-François Alcover, Feb 16 2017
a(31)-a(43) from Michael De Vlieger, Feb 18 2017
Showing 1-5 of 5 results.

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