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 1-7 of 7 results.

A163101 Partial sums of A163100.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 22, 24, 25, 26, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 44, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 62, 65, 66, 67, 69, 73, 75, 76, 77, 78, 79, 81, 84, 87, 89, 90, 91, 92, 93, 95, 99, 103, 105, 106, 107, 110, 113, 114
Offset: 1

Views

Author

Omar E. Pol, Jul 20 2009

Keywords

Crossrefs

Cf. A147861, A163100.

Extensions

More terms from R. J. Mathar, Oct 09 2010

A056737 Minimum nonnegative integer m such that n = k*(k+m) for some positive integer k.

Original entry on oeis.org

0, 1, 2, 0, 4, 1, 6, 2, 0, 3, 10, 1, 12, 5, 2, 0, 16, 3, 18, 1, 4, 9, 22, 2, 0, 11, 6, 3, 28, 1, 30, 4, 8, 15, 2, 0, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 2, 0, 5, 14, 9, 52, 3, 6, 1, 16, 27, 58, 4, 60, 29, 2, 0, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 10, 15, 4
Offset: 1

Views

Author

Leroy Quet, Aug 26 2000

Keywords

Comments

a(n) is difference between the least divisor of n that is >= square root(n) and the greatest divisor of n that is <= square root(n).
From Omar E. Pol, Aug 12 2009: (Start)
a(n) = 0 iff n is a square.
a(n) = n-1 is a new record iff n is a prime number. (End)
For odd n = 2k-1, a(n) = 2*A219695(k) is even. - M. F. Hasler, Nov 25 2012
Conjecture: There exists some constant, k, approximately equal to 1.7, such that a(n) is of average order k*n/log(n). See Tooth Link for evidence. - Clive Tooth, Mar 18 2025

Examples

			a(8) = 2 because 8 = 2*(2+2) and 8 = k*(k+1) or 8 = k^2 have no solutions for k = a positive integer.

Links

Crossrefs

Cf. A033676, A033677.
Cf. A000040, A000290, A147861, A163100, A163280.

Programs

  • Mathematica
    A033676[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2]], Sqrt[n]]; A033677[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2+1]], Sqrt[n]]; Table[A033677[n] - A033676[n], {n, 1, 77}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 27 2004 *)
Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 0, d[[1 + len/2]] - d[[len/2]]], {n, 100}] (* T. D. Noe, Jun 04 2012 *)
  • PARI
    A056737(n)={n=divisors(n);n[(2+#n)\2]-n[(1+#n)\2]}  \\ M. F. Hasler, Nov 25 2012

Formula

a(n) = Min_{t - d | 0 < d <= t <= n and d*t=n}. - Reinhard Zumkeller, Feb 25 2002
a(n) = A033677(n)-A033676(n). - Omar E. Pol, Jun 21 2009
a(2n-1) = 2*A219695(n). - M. F. Hasler, Nov 25 2012

A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 10, 15, 20, 25, 11, 14, 18, 24, 30, 36, 13, 22, 21, 28, 35, 42, 49, 17, 26, 27, 32, 40, 48, 56, 64, 19, 34, 33, 44, 45, 54, 63, 72, 81, 23, 38, 39, 52, 50, 60, 70, 80, 90, 100, 29, 46, 51, 68, 55, 66, 77, 88, 99, 110, 121, 31, 58, 57, 76, 65, 78, 84, 96, 108, 120, 132, 144
Offset: 1

Views

Author

Omar E. Pol, Aug 07 2009

Keywords

Comments

This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.
Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - Franklin T. Adams-Watters, Aug 12 2009

Examples

			Array begins:
   1,  4,  9,  16,  25,  36,  49,  64,  81, 100, 121, 144, ...
   2,  6, 12,  20,  30,  42,  56,  72,  90, 110, 132, 156, ...
   3,  8, 15,  24,  35,  48,  63,  80,  99, 120, 143, 168, ...
   5, 10, 18,  28,  40,  54,  70,  88, 108, 130, 154, 180, ...
   7, 14, 21,  32,  45,  60,  77,  96, 117, 140, 165, 192, ...
  11, 22, 27,  44,  50,  66,  84, 104, 126, 150, 176, 204, ...
  13, 26, 33,  52,  55,  78,  91, 112, 135, 160, 187, 216, ...
  17, 34, 39,  68,  65, 102,  98, 128, 153, 170, 198, 228, ...
  19, 38, 51,  76,  75, 114, 105, 136, 162, 190, 209, 264, ...
  23, 46, 57,  92,  85, 138, 119, 152, 171, 200, 220, 276, ...
  29, 58, 69, 116,  95, 174, 133, 184, 189, 230, 231, 348, ...
  31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
  ...

Links

Crossrefs

Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Another version: A163990.
Cf. A000027, A000040, A033676, A147861, A163100, A164000.

Programs

  • Maple
    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: # R. J. Mathar, Aug 09 2009
  • Mathematica
    nmax = 12;
pm = Prime[nmax];
sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2019 *)

Formula

Column k lists the numbers j such that A033676(j)=k.

Extensions

Edited by R. J. Mathar, Aug 01 2010
Example edited by Jean-François Alcover, Dec 15 2019

A212119 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the number of divisors d of n with min(d, n/d) = k.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Jul 02 2012

Keywords

Comments

Column k lists the numbers A040000: 1, 2, 2, 2, 2... interleaved with k-1 zeros, starting in row k^2.
The sum of row n gives A000005(n), the number of divisors of n.
T(n,k) is also the number of divisors of n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212120.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253. - Omar E. Pol, Dec 03 2013

Examples

			Row 10 gives 2, 2, 0 therefore the sums of row 10 is 2+2+0 = 4, the same as A000005(10), the number of divisors of 10.
Written as an irregular triangle the sequence begins:
1;
2;
2;
2, 1;
2, 0;
2, 2;
2, 0;
2, 2;
2, 0, 1;
2, 2, 0;
2, 0, 0;
2, 2, 2;
2, 0, 0;
2, 2, 0;
2, 0, 2;
2, 2, 0, 1;
2, 0, 0, 0;
2, 2, 2, 0;
2, 0, 0, 0;
2, 2, 0, 2;
2, 0, 2, 0;
2, 2, 0, 0;
2, 0, 0, 0;
2, 2, 2, 2;
2, 0, 0, 0, 1;

Links

Crossrefs

Row sums give A000005. Column 1 is A040000. Column 2 gives the absolute values of A176742.
Cf. A006218, A027750, A010766, A147861, A163100, A196020, A210959, A212120, A211343, A221645, A228812-A228814.

Extensions

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 10, 7, 36, 30, 24, 18, 14, 11, 49, 42, 35, 28, 21, 22, 13, 64, 56, 48, 40, 32, 27, 26, 17, 81, 72, 63, 54, 45, 44, 33, 34, 19, 100, 90, 80, 70, 60, 50, 52, 39, 38, 23, 121, 110, 99, 88, 77, 66, 55, 68, 51, 46, 29, 144, 132, 120, 108
Offset: 1

Views

Author

Omar E. Pol, Aug 11 2009

Keywords

Comments

This sequence is a permutation of the natural numbers.
Note that the first row is formed by 1 together the prime numbers and the first column are the squares of the natural numbers.
For more information see A163280, the main entry for this sequence. (See also A161344).

Examples

			Array begins:
1, 2, 3, 5, 7, 11,
4, 6, 8, 10, 14,
9, 12, 15, 18,
16, 20, 24,
25, 30,
36,
See also the array in A163280.

Links

Crossrefs

Cf. A000027, A000040, A033676, A147861, A163100, A163280, A163281, A163991, A164000.
Cf. Rows: 1=A008578, 2=A161344, 3=A161345, 4=A161424, 5=A161835, 6=A162526, 7=A162527, 8=A162528, 9=A162529, 10=A162530, 11=A162531, 12=A162532.
Cf. Columns: 1=A000290, 2=A002378, 4=A164004, 6=A164006, 7=A164007, 8=A164008, 9=A164009, 10=A164010, 11=A164011, 12=A164012.

Formula

Row n lists the numbers k such that A033676(k)=n.

A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

Original entry on oeis.org

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1
Offset: 1

Views

Author

Omar E. Pol, Jul 02 2012

Keywords

Comments

Column k lists the odd numbers repeated k times starting in row k^2.
1 together with the first differences of the row sums give the divisor function A000005.
T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

Examples

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;

Links

Crossrefs

Row sums give A006218, n >= 1.
Columns (1-5): A005408, A109613, A130823, A129756, A130497.
Cf. A000005, A027750, A010766, A147861, A163100, A212119.

Formula

T(n,k) = Sum_{j=1..n} A212119(j,k).

Extensions

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A163984 First differences of A056737.

Original entry on oeis.org

1, 1, -2, 4, -3, 5, -4, -2, 3, 7, -9, 11, -7, -3, -2, 16, -13, 15, -17, 3, 5, 13, -20, -2, 11, -5, -3, 25, -27, 29, -26, 4, 7, -13, -2, 36, -19, -7, -7, 37, -39, 41, -35, -3, 17, 25, -44, -2, 5, 9, -5, 43, -49, 3, -5, 15, 11, 31, -54, 56, -31, -27, -2, 8, -3, 61, -53, 7, -17, 67, -69, 71, -37, -25, 5, -11, 3, 71, -76, -2, 39, 43, -77, 7, 29, -15
Offset: 1

Views

Author

Omar E. Pol, Aug 12 2009

Keywords

Links

Crossrefs

Cf. A033676, A033677, A056737, A147861, A163100, A163280, A163990.

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:
A033677 := proc(n) local a, d; a := n ; for d in numtheory[divisors](n) do if d^2 >= n then a := min(a, d) ; fi; od: a; end proc:
A056737 := proc(n) A033677(n)-A033676(n) ; end proc:
[seq(A056737(n),n=1..120)] ;
DIFF(%) ;
Showing 1-7 of 7 results.

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