A054519 -id:A054519 - cp's OEIS frontend

A058189 Number of increasing geometric progressions ending in n (in the positive integers), including those of length 1 or 2.

1, 2, 3, 5, 5, 6, 7, 10, 11, 10, 11, 13, 13, 14, 15, 21, 17, 20, 19, 21, 21, 22, 23, 26, 29, 26, 31, 29, 29, 30, 31, 38, 33, 34, 35, 41, 37, 38, 39, 42, 41, 42, 43, 45, 47, 46, 47, 53, 55, 54, 51, 53, 53, 58, 55, 58, 57, 58, 59, 61, 61, 62, 65, 77, 65, 66, 67, 69, 69, 70, 71

Offset: 1

Henry Bottomley, Nov 22 2000

nonn

			a(4) = 5 since the possibilities are (4), (1,4), (2,4), (3,4) and (1,2,4).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A054519 for arithmetic progressions.

Cf. A058190.

ends_max_progression_of_length(n,ratio) = { my(k=1); while(1,if(denominator(n)>1,return(k)); n *= ratio; k++;) };
A058190(n) = sum(d=1,(n-1),max(0,ends_max_progression_of_length(d,d/n)-2));
A058189(n) = (A058190(n)+n); \\ Antti Karttunen, Nov 19 2017

a(n) = A058190(n) + n.

A326494 Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

The only allowed sets are the empty set, any singleton, any initial interval of positive integers and {2,4}. This can be shown by induction. - Andrew Howroyd, Aug 25 2019

			The a(0) = 1 through a(6) = 13 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {2}    {2}      {2}        {2}          {2}
           {1,2}  {3}      {3}        {3}          {3}
                  {1,2}    {4}        {4}          {4}
                  {1,2,3}  {1,2}      {5}          {5}
                           {2,4}      {1,2}        {6}
                           {1,2,3}    {2,4}        {1,2}
                           {1,2,3,4}  {1,2,3}      {2,4}
                                      {1,2,3,4}    {1,2,3}
                                      {1,2,3,4,5}  {1,2,3,4}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,5,6}

Subsets with difference are A054519.
Subsets with quotients are A326023.
Subsets with quotients > 1 are A326079.
Subsets without differences or quotients are A326490.
Cf. A005408, A007865, A051026, A325849, A326076, A326491.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Union[Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&],Subtract@@@Select[Tuples[#,2],Greater@@#&]]]&]],{n,0,10}]

a(n) = 2*n + 1 = A005408(n) for n > 3. - Andrew Howroyd, Aug 25 2019

Terms a(20) and beyond from Andrew Howroyd, Aug 25 2019

A326440 a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 5, 3, 7, 4, 8, 6, 12, 10, 14, 10, 15, 13, 19, 17, 23, 19, 23, 21, 29, 26, 30, 26, 32, 30, 38, 36, 42, 38, 42, 38, 47, 45, 49, 45, 53, 51, 59, 57, 63, 57, 61, 59, 69, 66, 72, 68, 74, 72, 80, 76, 84, 80, 84, 82, 94, 92, 96, 90, 97, 93, 101, 99

Offset: 0

Gus Wiseman, Jul 06 2019

nonn

Is this sequence nonnegative?

As tau(n) is odd when n is a square, there are alternating strings of even and odd integers with change of parity for each n square. Indeed, between m^2 and (m+1)^2-1, there is a string of 2m+1 even terms if m is odd, or a string of 2m+1 odd terms if m is even. - Bernard Schott, Jul 10 2019

			The first 6 terms of A000005 are 1, 2, 2, 3, 2, 4, so a(6) = 1 - 1 + 2 - 2 + 3 - 2 + 4 = 5.

Michel Marcus, Table of n, a(n) for n = 0..5000

Cf. A000005, A001222, A008683, A054519, A071321, A195017, A268387, A307704, A316523, A316524, A319273.

[1] cat [1+(&+[(-1)^(k)*#Divisors(k):k in [1..n]]):n in [1..70]]; // Marius A. Burtea, Jul 10 2019

Accumulate[Table[If[k==0,1,(-1)^k*DivisorSigma[0,k]],{k,0,30}]]

a(n) = 1 - sum(k=1, n, (-1)^(k+1)*numdiv(k)); \\ Michel Marcus, Jul 09 2019

a(n) = 1 + Sum_{k=1..n} (-1)^k A000005(k).
For n > 0, a(n) = 1 + A307704(n).
If p prime, a(p) = a(p-1) - 2. - Bernard Schott, Jul 10 2019

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A058189 Number of increasing geometric progressions ending in n (in the positive integers), including those of length 1 or 2.

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A326494 Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.

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A326440 a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.

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