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 21 results. Next

A109735 Partial sums of A109890.

Original entry on oeis.org

1, 3, 6, 12, 16, 24, 36, 45, 50, 60, 75, 100, 120, 144, 160, 192, 240, 270, 288, 324, 351, 364, 371, 424, 530, 795, 954, 1272, 1484, 1498, 1605, 1926, 2140, 2568, 3210, 3745, 3780, 3801, 3982, 3993, 4026, 4048, 4071, 4130, 4200, 4228, 4379, 4408, 4427, 4660, 5126
Offset: 1

Views

Author

N. J. A. Sloane and Nadia Heninger, Aug 10 2005

Keywords

Examples

			See A109890.

Links

Crossrefs

Cf. A109890.

Programs

  • Haskell
    a109735 n = a109735_list !! (n-1)
a109735_list = scanl1 (+) a109890_list
-- Reinhard Zumkeller, Jan 02 2015
  • Mathematica
    Accumulate[a[1]=1;a[2]=2;a[n_]:=Block[{t=Table[a[i], {i,n-1}]}, s=Plus @@ t; d=Divisors[s]; l=Complement[d,t]; If[l!={},k = First[l], k=s; While[Position[t,k]=={},k+=s];k]]; Table[a[n],{n,51}] ] (* James C. McMahon, Apr 03 2024 *)

A111238 Indices k where A109890(k) is prime.

Original entry on oeis.org

2, 3, 9, 22, 23, 24, 31, 39, 40, 43, 44, 47, 48, 49, 50, 58, 59, 66, 67, 70, 76, 81, 84, 88, 89, 97, 98, 100, 119, 122, 123, 130, 131, 138, 139, 144, 145, 150, 151, 152, 163, 168, 174, 178, 179, 185, 197, 200, 204, 207, 217, 218, 221, 225, 226, 227
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2005

Keywords

Links

Crossrefs

Cf. A109890, A111238, A111239, A111240.

Programs

  • Mathematica
    nn = 240; c[_] := False; a[1] = 1; a[2] = 2; s = 3; c[1] = c[2] = True;
{2}~Join~Reap[Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True; s += k;
If[PrimeQ[k], Sow[n]], {n, 3, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Apr 27 2024 *)

A111239 Primes in the order in which they appear in A109890.

Original entry on oeis.org

2, 3, 5, 13, 7, 53, 107, 181, 11, 23, 59, 151, 29, 19, 233, 31, 61, 197, 17, 199, 41, 193, 97, 109, 37, 281, 47, 71, 131, 79, 149, 103, 241, 137, 191, 239, 113, 163, 43, 653, 617, 853, 673, 89, 937, 67, 571, 599, 751, 83, 101, 1103, 829, 457, 499, 229
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2005

Keywords

Comments

Smallest missing prime in A109890 for n <= 10^5 is prime(1821) = 15619. - Michael De Vlieger, Apr 27 2024

Links

Crossrefs

Cf. A109890, A111238, A111240.

Programs

  • Mathematica
    nn = 2^14; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
s = a[1] + a[2]; v = NextPrime[a[2]];
t = Join[{{2, 2}},
  Reap[Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True; s += k;
  If[PrimeQ[k], Sow[{k, n}];
    If[k == v, While[c[v], v = NextPrime[v]]]], {n, 3, nn}], n] ][[-1, 1]] ];
TakeWhile[t, First[#] <= v &][[All, 1]] (* Michael De Vlieger, Apr 27 2024 *)

A111240 Index at which n-th prime appears in A109890.

Original entry on oeis.org

2, 3, 9, 23, 40, 22, 67, 49, 43, 48, 58, 89, 76, 151, 98, 24, 44, 59, 185, 100, 271, 122, 207, 178, 84, 217, 130, 31, 88, 145, 357, 119, 138, 309, 123, 47, 590, 150, 334, 684, 245, 39, 139, 81, 66, 70, 253, 642, 737, 227, 50, 144, 131, 422, 496, 479, 516, 389, 715
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2005

Keywords

Examples

			The 4th prime, 7, is A109890(23), so a(4) = 23.

Links

Crossrefs

Cf. A000040, A000720, A109890.

Programs

  • Maple
    A109890 := proc(nmin) local a,i,k,apsum; a := [1] ; apsum := 1 ; while nops(a) < nmin do k := 1; while k in a or not ( apsum mod k = 0 or k mod apsum = 0 ) do k := k+1 ; od ; a := [op(a),k] ; apsum := apsum+k ; od; RETURN(a) ; end: A111240 := proc(nmin) local a,a109890,n,i; a := [] ; a109890 := A109890(nmin) ; n := 1; while member( ithprime(n),a109890,'i') do a := [op(a),i] ; n := n+1 ; od; RETURN(a) ; end: A111240(560) ; # R. J. Mathar, Aug 20 2007
  • Mathematica
    nn = 1000; c[] := False; p[] := 0;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
r = 0; s = a[1] + a[2]; p[2] = 2;
Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True;
  Map[(If[p[#] == 0, Set[p[#], n]]; If[# > r, r = #]) &,
    FactorInteger[k][[All, 1]]];
  s += k, {n, 3, nn}], n];
s = 1; Reap[While[Set[k, p[Prime[s]]] > 0, Sow[k]; s++] ][[-1, 1]] (* Michael De Vlieger, Apr 26 2024 *)

Extensions

More terms from R. J. Mathar, Aug 20 2007
More terms from David Wasserman, Jan 07 2009

A111315 Positions where A109890(n) = Sum_{i = 1..n-1} A109890(i).

Original entry on oeis.org

3, 4, 222, 232, 279, 568, 634, 844, 1307, 1704, 2016, 2050, 2164, 2325, 2701, 3130, 3225, 3322, 3524, 4673, 6007, 6692, 7141, 7221, 7954, 8689, 11051, 11898, 13959, 14166, 16192, 17078, 17220, 17521, 18071, 22324, 22414, 24757, 28290, 28979
Offset: 1

Views

Author

N. J. A. Sloane, Nov 04 2005

Keywords

Comments

These are the positions where A109735(n) = 2*A109735(n-1), or equally, where A111241(n) = 1.

Crossrefs

Cf. A109890, A109735, A111316, A111241.

Programs

  • PARI
    s = 3; S = Set([1, 2]); i = 3; while (1, d = divisors(s); j = 1; while (setsearch(S, d[j]), j++); n = d[j]; if (n == s, print(i)); s += n; S = setunion(S, Set(n)); i++); \\ David Wasserman, Jan 09 2009

Extensions

More terms from David Wasserman, Jan 09 2009

A111316 a(n) = A109890(A111315(n)).

Original entry on oeis.org

3, 6, 113573, 230801, 941071, 5166859, 30956561, 123081011, 3050325741, 14086296281, 60060345973, 120331687901, 316465918571, 634257678809, 8042099198501, 34761370800833, 134702376451061, 269979973606237
Offset: 1

Views

Author

N. J. A. Sloane, Nov 04 2005

Keywords

Crossrefs

Cf. A109890, A111315.

Programs

  • PARI
    s = 3; S = Set([1, 2]); i = 3; while (1, d = divisors(s); j = 1; while (setsearch(S, d[j]), j++); n = d[j]; if (n == s, print1(s, ", ")); s += n; S = setunion(S, Set(n)); i++); \\ David Wasserman, Jan 09 2009

Extensions

More terms from David Wasserman, Jan 09 2009

A253443 Smallest missing number within the first n terms in A109890.

Original entry on oeis.org

4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 34, 37, 37, 37, 37, 37
Offset: 4

Views

Author

Reinhard Zumkeller, Jan 01 2015

Keywords

Comments

A253584(n) occurs exactly A253444(n) times.

Links

Crossrefs

Cf. A095258, A095259, A253444 (run lengths), A253584 (range), A253415.

Programs

  • Haskell
    import Data.List (insert)
a253443 n = a253443_list !! (n-4)
a253443_list = f (4, []) 6 where
   f (m,ys) z = g $ dropWhile (< m) $ a027750_row' z where
     g (d:ds) | elem d ys = g ds
              | otherwise = m : f (ins [m, m+1 ..] (insert d ys)) (z + d)
     ins (u:us) vs'@(v:vs) = if u < v then (u, vs') else ins us vs
-- Reinhard Zumkeller, Jan 03 2015

A111241 a(n) = A109735(n)/A109890(n+1).

Original entry on oeis.org

1, 1, 3, 2, 2, 4, 9, 5, 4, 3, 5, 5, 9, 5, 4, 8, 15, 8, 12, 27, 52, 7, 4, 2, 5, 3, 6, 106, 14, 5, 9, 5, 4, 6, 107, 180, 21, 362, 121, 183, 176, 69, 59, 150, 28, 151, 232, 19, 10, 2, 11, 9, 233, 360, 247, 304, 155, 244, 195, 98, 231, 174, 196, 50, 591, 296, 198, 51, 199
Offset: 2

Views

Author

N. J. A. Sloane, Oct 30 2005

Keywords

Comments

This is always an integer for n>=2.
a(n) = 1 for n in A111315. When this happens A109890(n+1) makes a large jump. The corresponding values of A109890(n+1) are in A111316 (cf. A111242).

Examples

			A109735(4)=12, A109890(5)=4, so a(4) = 12/4 = 3.

Links

Crossrefs

Cf. A109735, A109890, A094340, A111315, A111316, A111242.
Different from A094339.

Programs

  • Mathematica
    nn = 71; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[k = SelectFirst[Divisors[s], ! c[#] &];
    c[k] = True; Sow[s/k];
s += k, {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Apr 26 2024 *)

Formula

a(n) = A094340(n) for all n > 1. - David Wasserman, Jan 06 2009

A371909 Number of divisors of the partial sums of A109890.

Original entry on oeis.org

1, 2, 4, 6, 5, 8, 9, 6, 6, 12, 6, 9, 16, 15, 12, 14, 20, 16, 18, 15, 8, 12, 4, 8, 8, 8, 12, 16, 12, 8, 8, 12, 12, 16, 16, 8, 48, 8, 8, 8, 16, 20, 8, 16, 48, 12, 4, 16, 4, 12, 8, 8, 18, 16, 60, 12, 20, 8, 24, 24, 15, 36, 40, 36, 12, 8, 20, 32, 8, 12, 36, 16, 24
Offset: 1

Views

Author

Michael De Vlieger, Apr 26 2024

Keywords

Links

Crossrefs

Cf. A000005, A109735, A109890, A371910.

Programs

  • Mathematica
    nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{1}~Join~Reap[Do[k = SelectFirst[Divisors[s], ! c[#] &];
     c[k] = True; Sow[DivisorSigma[0, s]];
     s += k, {n, 3, nn}] ][[-1, 1]]

Formula

a(n) = A000005(A109735(n)).

A371910 Position of A109890(n) among the sorted set of divisors of A109735(n-1).

Original entry on oeis.org

2, 4, 4, 4, 7, 6, 3, 4, 9, 5, 6, 12, 9, 9, 11, 14, 9, 13, 9, 4, 4, 3, 6, 7, 6, 10, 12, 5, 5, 6, 8, 9, 13, 12, 4, 15, 5, 3, 4, 6, 8, 4, 9, 17, 7, 2, 5, 3, 8, 7, 6, 13, 8, 17, 6, 7, 4, 9, 10, 8, 13, 17, 15, 7, 3, 7, 13, 5, 6, 16, 8, 11, 8, 5, 4, 13, 12, 17, 5, 6
Offset: 3

Views

Author

Michael De Vlieger, Apr 26 2024

Keywords

Comments

A109890(n) is the a(n)-th smallest divisor of A109735(n-1).

Examples

			Table relating sequences b = A109890, s = A109735, c = A371909. a(n) = c(n) implies both A111315(i) = n and A111316(i) = b(n) = s(n-1).
    n     b(n)   s(n-1)  a(n)  c(n)    i
   --------------------------------------
    3       3   =    3     2     2     1
    4       6   =    6     4     4     2
    5       4       12     4     6
    6       8       16     4     5
    7      12       24     7     8
    8       9       36     6     9
    9       5       45     3     6
   10      10       50     4     6
   11      15       60     9    12
   12      25       75     5     6
  ...
  222  113573 = 113573     4     4     3
  ...
  232  230801 = 230801     4     4     4
  ...
  279  941071 = 941071     4     4     5
  ...

Links

Crossrefs

Cf. A000005, A027750, A109735, A109890, A111315, A111316, A371909.

Programs

  • Mathematica
    nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[d = Divisors[s]; k = SelectFirst[d, ! c[#] &];
    c[k] = True; Sow[FirstPosition[d, k][[1]]];
    s += k, {n, 3, nn}] ][[-1, 1]]

Formula

1 < a(n) <= A371909(n), where A371909(n) = A000005(A109735(n-1)), corollary of Sloane's theorem in the comments in A109890.
A109890(n) = T(j, k), where T = A027750, j = A109735(n-1), and k = a(n).
A371909(n) = A371910(n) if and only if A109890(n) = A109735(n-1).
Showing 1-10 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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