A318736 -id:A318736 - cp's OEIS frontend

A318734 a(n) = Sum_{k=1..n} (-1)^(k + 1) * d(2*k - 1), where d(k) is the number of divisors of k (A000005).

1, -1, 1, -1, 2, 0, 2, -2, 0, -2, 2, 0, 3, -1, 1, -1, 3, -1, 1, -3, -1, -3, 3, 1, 4, 0, 2, -2, 2, 0, 2, -4, 0, -2, 2, 0, 2, -4, 0, -2, 3, 1, 5, 1, 3, -1, 3, -1, 1, -5, -3, -5, 3, 1, 3, -1, 1, -3, 3, -1, 2, -2, 2, 0, 4, 2, 6, -2, 0, -2, 2, -2

Offset: 1

Hugo Pfoertner, Sep 05 2018

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Records of A318734, showing negative and positive records.

Cf. A000005, A099774, A006218, A222068.

Records and their positions: A318735, A318736, A318737, A318738.

a[n_] := Sum[(-1)^(k + 1) DivisorSigma[0, 2 k - 1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Sep 17 2018 *)

s=0;j=-1;forstep(k=1,141,2,j=-j;s=s+j*numdiv(k);print1(s,", "))

a(n) = sum(k=1, n, (-1)^(k+1)*numdiv(2*k-1)); \\ Michel Marcus, Sep 08 2018

A318735 Positive records in A318734.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 21, 23, 25, 30, 31, 32, 34, 39, 45, 47, 48, 51, 52, 56, 60, 62, 71, 76, 78, 83, 84, 88, 91, 103, 108, 119, 127, 129, 132, 142, 143, 151, 166, 168, 171, 178, 181, 183, 189, 197, 215, 237, 241, 244, 266, 270, 274

Offset: 1

Hugo Pfoertner, Sep 05 2018

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..268

Cf. A000005, A099774, A318734, A318736, A318737, A318738.

s=0;smax=0;j=-1;forstep(k=1,20000000,2,j=-j;s=s+j*numdiv(k);if(s>smax,smax=s;print1(s,", ")))

A318737 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 9, 25, 49, 85, 133, 169, 225, 445, 845, 973, 1125, 2205, 2209, 2469, 2829, 7929, 9429, 9945, 23569, 24073, 24645, 26145, 40425, 68153, 71289, 72413, 89517, 112233, 112245, 128973, 162405, 162409, 162429, 297073, 477489, 477493, 502713, 561253

Offset: 1

Hugo Pfoertner, Sep 05 2018

nonn

			a(2) = 9, because s = d(1)-d(3)+d(5)-d(7)+d(9) = 1-2+2-2+3 = 2 exceeds d(1)=1, d(1)-d(3)=-1, d(1)-d(3)+d(5)=1, d(1)-d(3)+d(5)-d(7)=-1.

Hugo Pfoertner, Table of n, a(n) for n = 1..268

Cf. A000005, A099774, A318734, A318735, A318736, A318738.

s=0;smax=0;j=-1;forstep(k=1,600000,2,j=-j;s=s+j*numdiv(k);if(s>smax,smax=s;print1(k,", ")))

A318738 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new negative record, with d(n) = number of divisors of n (A000005).

Original entry on oeis.org

3, 15, 39, 63, 99, 259, 319, 403, 675, 679, 943, 1615, 1779, 2919, 4899, 5775, 7399, 7407, 13475, 13479, 25635, 29835, 29839, 44955, 78463, 78475, 108927, 108931, 126819, 136959, 136975, 136983, 244875, 244879, 256355, 276675, 276687, 457275, 530139

Offset: 1

Hugo Pfoertner, Sep 08 2018

nonn

			a(1) = 3, because s = d(1)-d(3) = 1-2 = -1 is the first negative record.
a(2) = 15, because s = d(1)-d(3)+d(5)-d(7)+d(9)-d(11)+d(13)-d(15) =
1-2+2-2+3-2+2-4 = -2 is the first sum less than -1.

Hugo Pfoertner, Table of n, a(n) for n = 1..282

Cf. A000005, A099774, A318734, A318735, A318736, A318737.

s=0;j=-1;smin=0;forstep(k=1,600000,2,j=-j;s=s+j*numdiv(k);if(s

cp's OEIS Frontend

A318734 a(n) = Sum_{k=1..n} (-1)^(k + 1) * d(2*k - 1), where d(k) is the number of divisors of k (A000005).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A318735 Positive records in A318734.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A318737 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A318738 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new negative record, with d(n) = number of divisors of n (A000005).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A318734 a(n) = Sum_{k=1..n} (-1)^(k + 1) * d(2*k - 1), where d(k) is the number of divisors of k (A000005).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A318735 Positive records in A318734.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A318737 Numbers n=2*k-1 where Sum_{j=1..k} (-1)^(j+1) * d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A318738 Numbers n=2*k-1 where Sum_{j=1..k} (-1)^(j+1) * d(2*j-1) achieves a new negative record, with d(n) = number of divisors of n (A000005).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A318737 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).

A318738 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new negative record, with d(n) = number of divisors of n (A000005).