A058074 -id:A058074 - cp's OEIS frontend

A060778 a(n) = gcd(tau(n+1), tau(n)), where tau = A000005.

1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 1, 2, 2, 2, 2, 4, 2, 2, 1, 1, 4, 2, 2, 2, 2, 2, 2, 4, 4, 1, 1, 2, 4, 4, 2, 2, 2, 2, 6, 2, 2, 2, 1, 3, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 4, 2, 2, 5, 1, 2, 2, 4, 4, 4, 4, 2, 2, 4, 2, 2, 4, 4, 4, 2, 2, 6, 3, 1, 2, 2, 2, 8, 4

Offset: 1

Labos Elemer, Apr 26 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A000005, A057921, A058074.

GCD@@@Partition[DivisorSigma[0,Range[110]],2,1] (* Harvey P. Dale, May 27 2014 *)

a(n) = gcd(numdiv(n), numdiv(n+1)); \\ Michel Marcus, Jan 12 2018

from math import gcd
from sympy import divisor_count
def A060778(n): return gcd(divisor_count(n+1),divisor_count(n)) # Chai Wah Wu, Aug 12 2023

a(n) = gcd(A000005(n+1), A000005(n)).

A058075 Numbers k such that gcd(sigma(k), sigma(k+1)) = 1, where sigma(k) is sum of positive divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 15, 16, 18, 24, 25, 31, 35, 36, 48, 63, 64, 72, 80, 81, 97, 99, 100, 120, 121, 127, 128, 143, 144, 162, 200, 224, 225, 241, 255, 256, 288, 289, 323, 337, 399, 400, 483, 511, 512, 528, 529, 575, 576, 577, 578, 624, 625, 675, 721, 722, 728, 729

Offset: 1

Leroy Quet, Nov 11 2000

nonn

			8 is included because sigma(8) = 15 is relatively prime to sigma(9) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Ivan Neretin)

Cf. A058074.

Select[Range@800, GCD @@ DivisorSigma[1, {#, # + 1}] == 1 &] (* Ivan Neretin, Jan 27 2017 *)
Drop[Position[Partition[DivisorSigma[1,Range[800]],2,1],?(GCD@@#== 1&)]// Flatten,2] (* _Harvey P. Dale, Jul 31 2019 *)

lista(nn) = {for(n=1, nn, if (gcd(sigma(n), sigma(n+1)) == 1, print1(n, ", ")));} \\ Michel Marcus, May 19 2014

A130194 Let M = lower triangular matrix with 1's on and below the main diagonal, with columns multiplied by +1, +1, -1, -1, repeated; form M^2; read across rows of resulting triangle.

Original entry on oeis.org

1, 2, 1, -1, 2, 1, -4, -1, 2, 1, 1, -4, -1, 2, 1, 6, 1, -4, -1, 2, 1, -1, 6, 1, -4, -1, 2, 1, -8, -1, 6, 1, -4, -1, 2, 1, 1, -8, -1, 6, 1, -4, -1, 2, 1, 10, 1, -8, -1, 6, 1, -4, -1, 2, 1, -1, 10, 1, -8, -1, 6, 1, -4, -1, 2, 1, -12, -1, 10, 1, -8, -1, 6, 1, -4, -1, 2, 1

Offset: 1

Gary W. Adamson, May 16 2007

A000012(signed)^2.
Left border = A009531: (1, 2, -1, -4, 1, 6, -1, ...).
Row sums = A130195: (1, 3, 2, -2, -1, 5, 4, ...).
Row sums of the unsigned triangle = A058074: (1, 3, 4, 8, 9, 15, ...).
A009531 in every column: (1, 2, -1, -4, 1, 6, -1, ...).

			First few rows of the triangle:
   1;
   2,  1;
  -1,  2,  1;
  -4, -1,  2,  1;
   1, -4, -1,  2,  1;
   6,  1, -4, -1,  2,  1;
  ...

Cf. A000012, A058074, A009531.

trg(nn) = {vgf = x*(1+x)^2/(1+x^2)^2 + O(x^(nn+1)); m = matrix(nn, nn, i, j, if (i >= j, polcoeff(vgf, i-j+1))); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Oct 03 2014

More terms from Michel Marcus, Oct 03 2014

A364890 Least number k such that A060778(k) = n.

Original entry on oeis.org

1, 2, 49, 14, 80, 44, 529983, 104, 16640, 2511, 8212890624, 735, 1019423412224, 29888, 600624, 2295, 54020648488730624, 6075, 3018417549254328320, 5264, 123200, 24151040, 3264402128528250685620224, 5984, 1753599375, 689278976, 2310399, 156735, 27965083137654166225393025024, 180224, 11404289746101879774056466612224, 21735, 170853262335, 2035980763136, 207593229375, 223244

Offset: 1

Seiichi Manyama, Aug 12 2023

nonn

Cf. A000005, A058074, A060778, A084307, A364903.

a(n) = my(x=1, nx=1, ny=2); while(gcd(nx, ny) != n, x++; nx=ny; ny=numdiv(x+1)); x;

a(11),a(13),a(17),a(19),a(23)-a(36) from Max Alekseyev, Feb 18 2024

cp's OEIS Frontend

A060778 a(n) = gcd(tau(n+1), tau(n)), where tau = A000005.

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A058075 Numbers k such that gcd(sigma(k), sigma(k+1)) = 1, where sigma(k) is sum of positive divisors of k.

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A130194 Let M = lower triangular matrix with 1's on and below the main diagonal, with columns multiplied by +1, +1, -1, -1, repeated; form M^2; read across rows of resulting triangle.

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A364890 Least number k such that A060778(k) = n.

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