A092517 -id:A092517 - cp's OEIS frontend

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

1, 2, 2, 2, 5, 2, 3, 4, 4, 3, 2, 8, 6, 8, 2, 4, 4, 6, 6, 4, 4, 2, 10, 4, 15, 4, 10, 2, 4, 4, 12, 6, 6, 12, 4, 4, 3, 11, 4, 16, 8, 16, 4, 11, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 10, 10, 22, 4, 30, 4, 22, 10, 10, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - N. J. A. Sloane, Feb 02 2013

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
[ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
[ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
[ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
[ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
[ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
[ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
[ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
[10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
[12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
.
Displayed as a triangular array:
1,
2,  2,
2,  5,  2,
3,  4,  4,  3,
2,  8,  6,  8, 2,
4,  4,  6,  6, 4,  4,
2, 10,  4, 15, 4, 10, 2,
4,  4, 12,  6, 6, 12, 4,  4,
3, 11,  4, 16, 8, 16, 4, 11, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, 2012. - From _N. J. A. Sloane_, Feb 02 2013

Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
Main diagonal is A060724.
Rows give A000005, A060710, A054584, A221951, A221852.

with(numtheory):
T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # Alois P. Heinz, Sep 12 2012
T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # N. J. A. Sloane, Feb 02 2013

t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 21 2013 *)

def A216624(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(gcd, cp))
for n in (1..12): [A216624(n,k) for k in (1..12)]

T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A113935(n) = prime(n)+3.

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008

Offset: 1

Labos Elemer, May 21 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).

Cf. A083481, A083539, A092517.

a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012

a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022

Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

Original entry on oeis.org

3, 12, 28, 42, 72, 96, 120, 195, 234, 216, 336, 392, 336, 576, 744, 558, 702, 780, 840, 1344, 1152, 864, 1440, 1860, 1302, 1680, 2240, 1680, 2160, 2304, 2016, 3024, 2592, 2592, 4368, 3458, 2280, 3360, 5040, 3780, 4032, 4224, 3696, 6552, 5616, 3456, 5952

Offset: 1

Labos Elemer, May 21 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A002378, A060781, A060780, A083538, A330322 (partial sums).

Cf. A083481, A083542, A092517.

f[x_] := DivisorSigma[1, x]; t=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[DivisorSigma[1,Range[50]],2,1] (* Harvey P. Dale, May 21 2014 *)

a(n)=sigma(n)*sigma(n+1) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A000203(A002378(n)). - Amiram Eldar, Jul 10 2024

A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 4, 4, 3, 2, 6, 5, 6, 2, 4, 4, 6, 6, 4, 4, 2, 8, 4, 10, 4, 8, 2, 4, 4, 10, 6, 6, 10, 4, 4, 3, 8, 4, 12, 7, 12, 4, 8, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 8, 8, 14, 4, 20, 4, 14, 8, 8, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6, 2, 12, 4, 12, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A060648(n) = Sum_{d|n} Dedekind_Psi(d).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A062011(n) = 2*tau(n).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A052147(n) = prime(n)+2.

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8,  4, 12
[ 3] 2,  4,  5,  6,  4, 10,  4,  8,  8,  8,  4, 15
[ 4] 3,  6,  6, 10,  6, 12,  6, 14,  9, 12,  6, 20
[ 5] 2,  4,  4,  6,  7,  8,  4,  8,  6, 14,  4, 12
[ 6] 4,  8, 10, 12,  8, 20,  8, 16, 16, 16,  8, 30
[ 7] 2,  4,  4,  6,  4,  8,  9,  8,  6,  8,  4, 12
[ 8] 4,  8,  8, 14,  8, 16,  8, 22, 12, 16,  8, 28
[ 9] 3,  6,  8,  9,  6, 16,  6, 12, 17, 12,  6, 24
[10] 4,  8,  8, 12, 14, 16,  8, 16, 12, 28,  8, 24
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13, 12
[12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50
.
Displayed as a triangular array:
   1,
   2, 2,
   2, 4,  2,
   3, 4,  4,  3,
   2, 6,  5,  6, 2,
   4, 4,  6,  6, 4,  4,
   2, 8,  4, 10, 4,  8, 2,
   4, 4, 10,  6, 6, 10, 4, 4,
   3, 8,  4, 12, 7, 12, 4, 8, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(igcd(c,d)), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Sep 12 2012

t[n_, k_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

def A216620(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(gcd, cp)))
for n in (1..12): [A216620(n,k) for k in (1..12)]

A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7

Offset: 1

Alois P. Heinz, Aug 05 2019

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020

			a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14).
a(7) = 1: T(7) = T(28)-T(27).
a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35).
a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44).
a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120
16, 135, 136. - _N. J. A. Sloane_, Mar 31 2020

Alois P. Heinz, Table of n, a(n) for n = 1..20000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332.
See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.
Cf. A092517, A063440.

with(numtheory): seq(tau(n*(n+1))-tau(n*(n+1)/2)-1, n=1..80); # Ridouane Oudra, Dec 08 2023

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #),
      (TNn + TNn^2 - # + #^2)/(2 #)} & /@
    Complement[Divisors[TNn (TNn + 1)], {TNn}],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]
(* Bradley Klee, Mar 01 2020 *)

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020
a(n) = A092517(n) - A063440(n) - 1. - Ridouane Oudra, Dec 08 2023

A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 6, 6, 4, 6, 6, 4, 8, 10, 5, 6, 6, 6, 12, 8, 4, 8, 12, 6, 8, 12, 6, 8, 8, 6, 12, 8, 8, 18, 9, 4, 8, 16, 8, 8, 8, 6, 18, 12, 4, 10, 15, 9, 12, 12, 6, 8, 16, 16, 16, 8, 4, 12, 12, 4, 12, 21, 14, 16, 8, 6, 12, 16, 8, 12, 12, 4, 12, 18, 12, 16, 8, 10, 25, 10, 4, 12, 24, 8, 8

Offset: 1

Vladeta Jovovic, Aug 08 2001

nonn

Unordered solutions to the equation 1/n = 1/r+1/s+1/(r*s) are r=d+n, s=n*(n+1)/d+n, where d is factor of n*(n+1) not greater than n.

Number of divisors of n-th oblong number not greater than n. - Chandler

			a(2)=2 because 1/2=1/3+1/8+1/24=1/4+1/5+1/20.

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A063520.

a[n_]:=DivisorSigma[0,n]DivisorSigma[0,(n+1)]/2; Array[a,86] (* Stefano Spezia, Aug 11 2025 *)

a(n) = numdiv(n)*numdiv(n+1)/2 \\ Michel Marcus, Jun 17 2013

a(n) = tau(n)*tau(n+1)/2 = A092517(n)/2.

A133947 a(n) = the number of "non-isolated divisors" of n(n+1). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 4, 6, 7, 4, 6, 6, 4, 8, 8, 4, 5, 5, 6, 11, 7, 4, 6, 8, 4, 5, 8, 4, 7, 7, 4, 8, 5, 4, 15, 6, 4, 5, 10, 6, 7, 7, 4, 12, 9, 4, 6, 9, 4, 7, 8, 4, 5, 10, 10, 9, 5, 4, 8, 8, 4, 7, 10, 6, 9, 5, 4, 6, 10, 4, 8, 8, 4, 7, 10, 4, 11, 5, 6, 13, 5, 4, 8, 15, 4, 5, 8, 4, 9, 13, 6, 6, 5, 4, 12, 6, 4, 9

Offset: 1

Leroy Quet, Sep 30 2007

nonn

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A133948, A133949, A092517.

Table[Length[Select[Divisors[n*(n + 1)],If[ # > 1, Mod[n*(n + 1), #*(# - 1)] == 0] || Mod[n*(n + 1), #*(# + 1)] == 0 &]], {n, 1, 80}] (* Stefan Steinerberger, Nov 01 2007 *)

a(n) = A092517(n) - A133948(n) = A132747(A002378(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A133948 a(n) = the number of "isolated divisors" of n(n+1). A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 6, 5, 4, 6, 6, 4, 8, 12, 6, 7, 7, 6, 13, 9, 4, 10, 16, 8, 11, 16, 8, 9, 9, 8, 16, 11, 12, 21, 12, 4, 11, 22, 10, 9, 9, 8, 24, 15, 4, 14, 21, 14, 17, 16, 8, 11, 22, 22, 23, 11, 4, 16, 16, 4, 17, 32, 22, 23, 11, 8, 18, 22, 12, 16, 16, 4, 17, 26, 20, 21, 11, 14, 37, 15, 4, 16

Offset: 1

Leroy Quet, Sep 30 2007

nonn

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A133947, A133950, A092517.

Table[Length[Divisors[n*(n + 1)]] - Length[Select[Divisors[n*(n + 1)], If[ # > 1, Mod[n*(n + 1), #*(# - 1)] == 0] || Mod[n*(n + 1), #*(# + 1)] == 0 &]], {n, 1, 80}] (* Stefan Steinerberger, Nov 01 2007 *)

a(n) = A092517(n) - A133947(n) = A132881(A002378(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A330320 a(n) = Sum_{i=1..n} tau(i)*tau(i+1), where tau(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 54, 66, 74, 86, 98, 106, 122, 142, 152, 164, 176, 188, 212, 228, 236, 252, 276, 288, 304, 328, 340, 356, 372, 384, 408, 424, 440, 476, 494, 502, 518, 550, 566, 582, 598, 610, 646, 670, 678, 698, 728, 746, 770, 794, 806, 822, 854, 886, 918, 934, 942, 966, 990, 998, 1022, 1064, 1092, 1124, 1140

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

For background references see A330570.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 61.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. 1, No. 3 (1927), pp. 202-208.
Nikolay V. Kuznetsov, Convolution of the Fourier coefficients of the Eisenstein-Maass series, Journal of Soviet Mathematics, Vol. 29, No. 2 (1985), pp. 1131-1159.

Cf. A000005, A059956, A063123.

Partial sums of A092517.

Accumulate[a[n_]:=DivisorSum[n+1, DivisorSigma[0, n]&]; Array[a, 66]] (* Vincenzo Librandi, Jan 10 2020 *)
Accumulate[Times@@@Partition[DivisorSigma[0,Range[70]],2,1]] (* Harvey P. Dale, Nov 02 2023 *)

a(n) = sum(i=1, n, numdiv(i*(i+1))); \\ Michel Marcus, Jan 11 2020

a(n) ~ (1/zeta(2)) * n * log(n)^2. - Amiram Eldar, Mar 05 2020

A336013 Three-column table read by rows giving triples of integers with x > 0, y > 1 and z > 0 such that y^2 - y - x*z = 0, sorted by y then by x.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 1, 3, 6, 2, 3, 3, 3, 3, 2, 6, 3, 1, 1, 4, 12, 2, 4, 6, 3, 4, 4, 4, 4, 3, 6, 4, 2, 12, 4, 1, 1, 5, 20, 2, 5, 10, 4, 5, 5, 5, 5, 4, 10, 5, 2, 20, 5, 1, 1, 6, 30, 2, 6, 15, 3, 6, 10, 5, 6, 6, 6, 6, 5, 10, 6, 3, 15, 6, 2, 30, 6, 1

Offset: 1

David Lovler, Jul 07 2020

When [x, y, z] is a row, f(a,b) = x*a*b + y*(a+b) + z is associative.
For each triple, the corresponding f(a,b) has an identity element (id), meaning f(a,id) = f(id,a) = a for all a. Id = -z/y. f(a,b) also has a zero element (call it theta), meaning f(a,theta) = f(theta,a) = theta for all a. Theta = -y/x.
I and θ are both integers only for rows with x = 1. - David Lovler, Feb 12 2022
f(a,b), defined by each row, also has a distributive rule when the generalized zero is taken into account. This means that if we define a "partition" of b by b = b1 + b2 - theta, then f(a,b) = f(a,b1 + b2 - theta) = f(a,b1) + f(a,b2) - theta for all a and b, and all "partitions" of b. Notice that when theta = 0, we have the usual distributive rule.
Another way to write f(a,b) for a row is to first compute id and theta from x, y and z. Then f(a,b) = (a*b - theta*(a+b) + id*theta)/(id - theta); or equivalently f(a,b) = (a*b - theta*(a+b) + theta^2)/(id - theta) + theta. Notice that id cannot equal theta because of id - theta in the denominator. Also notice that when id = 1 and theta = 0, f(a,b) = a*b, but multiplication is not represented in the table since the corresponding row would be [1, 0, 0], which is not allowed.
If (i) two rows are [x1, y1, z1] and [x2, y2, z2],
   (ii) id_1 = -z1/y1, id_2 = -z2/y2, theta_1 = -y1/x1, theta_2 = -y2/x2, and
   (iii) id_1/theta_2 + id_2/theta_1 = 2,
  then [x1+x2, y1+y2, z1+z2] is a row. Consequently, if
   (i) f1(a,b) = x1*a*b + y1*(a+b) + z1 is associative and
       f2(a,b) = x2*a*b + y2*(a+b) + z2 is associative,
   (ii) id_1 = -z1/y1, id_2 = -z2/y2, theta_1 = -y1/x1, theta_2 = -y2/x2, and
   (iii) id_1/theta_2 + id_2/theta_1 = 2,
  then f1(a,b) + f2(a,b) = (x1+x2)*a*b + (y1+y2)*(a+b) + z1+z2 is associative. Proof: Given [x1, y1, z1] and [x2, y2, z2] are rows and id_1/theta_2 + id_2/theta_1 = 2, the following algebraic manipulations show that [x1+x2, y1+y2, z1+z2] is a row.
(-z1/y1)/(-y2/x2) + (-z2/y2)/(-y1/x1) = 2
(x2*z1)/(y1*y2) + (x1*z2)/(y1*y2) = 2   [Multiply by y1*y2 and move to the right.]
0 = 2*y1*y2 - x1*z2 - x2*z1
[Add to the right side y1^2 - y1 - x1*z1 and y2^2 - y2 - x2*z2 which are both 0.]
0 = 2*y1*y2 - x1*z2 - x2*z1 + y1^2 - y1 - x1*z1 + y2^2 - y2 - x2*z2 [Rearrange.]
0 = (y1^2 + 2*y1*y2 + y2^2) - y1 - y2 - x1*z1 - x1*z2 - x2*z1 - x2*z2
0 = (y1+y2)^2 - (y1+y2) - (x1+x2)*(z1+z2). QED.
The idea of summing rows to get another row can be extended. If (i) three rows are [x1, y1, z1], [x2, y2, z2], and [x3, y3, z3], (ii) id and theta are defined as above, and (iii) (id_1/theta_2 + id_2/theta_1 - 2)/y3 + (id_1/theta_3 + id_3/theta_1 - 2)/y2 + (id_2/theta_3 + id_3/theta_2 - 2)/y1 = 0, then [x1+x2+x3, y1+y2+y3, z1+z2+z3] is a row.
Generalizing, when summing n rows to another row, the criterion involves the sum of binomial(n,2) versions of id_i/theta_j + id_j/theta_i - 2 as i and j go from 1 to n and i < j. Furthermore, each of these expressions is divided by the product of the y values from rows other than i and j. There are binomial(n,n-2) = binomial(n,2) such products. Formally this is:
If [x1,y1,z1] , ..., [xn,yn,zn] are rows and Sum_{1<=iAll of the above comments are still true when x, y, z, id and theta are complex numbers with y != 1 and x, y, z != 0.
If [x, y, z] is not a row, compute K = y/(y^2 - x*z). Then, K*[x, y, z] is a row if it is an integer triple. Note that if [x, y, z] were a row, K = 1. Furthermore, if [n*x, n*y, n*z] is not a row, compute K' = n*y/((n*y)^2 - (n*x)*(n*z)) = y/(n*(y^2 - x*z)) = K/n.  Then K'*[n*x, n*y, n*z] = K*[x, y, z] as before. When K*[x, y, z] is not a triple for not having integer values, we still have (K*y)^2 - K*y - (K*x)*(K*z) = 0. - David Lovler, Jan 24 2022

			Table begins:
  [ x, y,  z]
-------------
  [ 1, 2,  2];
  [ 2, 2,  1];
  [ 1, 3,  6];
  [ 2, 3,  3];
  [ 3, 3,  2];
  [ 6, 3,  1];
  [ 1, 4, 12];
  [ 2, 4,  6];
  [ 3, 4,  4];
  [ 4, 4,  3];
  [ 6, 4,  2];
  [12, 4,  1];
  [ 1, 5, 20];
  [ 2, 5, 10];
  [ 4, 5,  5];
  [ 5, 5,  4];
  [10, 5,  2];
  [20, 5,  1];
  ...
Example of the distributive rule:
  [x, y, z] = [1, 2, 2]
  f(a,b) = a*b + 2*(a+b) + 2
  theta = -y/x = -2
  f(5,7) = 35 + 2*(5+7) + 2 = 61 which equals
  f(5,3 + 2 -(-2)) = f(5,3) + f(5,2) - (-2) = (15 + 16 + 2) + (10 + 14 + 2) + 2 = 61.
Examples of rows that sum to another row:
  [1, 7, 42] + [2, 8, 28] = [3, 15, 70] because
  id_1/theta_2 + id_2/theta_1 = (-42/7)/(-8/2) + (-28/8)/(-7/1) = 2.
  [42, 7, 1] + [28, 8, 2] = [70, 15, 3] because
  id_1/theta_2 + id_2/theta_1 = (-1/7)/(-8/28) + (-2/8)/(-7/42) = 2.
  [2, 8, 28] + [3, 18, 102] = [5, 26, 130] because
  id_1/theta_2 + id_2/theta_1 = (-28/8)/(-18/3) + (-102/18)/(-8/2) = 2.
  [28, 8, 2] + [102, 18, 3] = [130, 26, 5] because
  id_1/theta_2 + id_2/theta_1 = (-2/8)/(-18/102) + (-3/18)/(-8/28) = 2.
Examples of three rows that sum to a row:
  [1, 2, 2] + [1, 2, 2] + [1, 5, 20] = [3, 9, 24] because
  (id_1/theta_2 + id_2/theta_1 - 2)/y3 + (id_1/theta_3 + id_3/theta_1 - 2)/y2 + (id_2/theta_3 + id_3/theta_2 - 2)/y1 = ((-2/2)/(-2/1) + (-2/2)/(-2/1) - 2)/5 + ((-2/2)/(-5/1) + (-20/5)/(-2/1) - 2)/2 + ((-2/2)/(-5/1) + (-20/5)/(-2/1) - 2)/2 = 0. In this example no two of the rows sum to another row.
  [1, 7, 42] + [2, 8, 28] + [3, 10, 30] = [6, 25, 100] because
  (id_1/theta_2 + id_2/theta_1 - 2)/y3 + (id_1/theta_3 + id_3/theta_1 - 2)/y2 + (id_2/theta_3 + id_3/theta_2 - 2)/y1 = ((-42/7)/(-8/2) + (-28/8)/(-7/1) - 2)/10 + ((-42/7)/(-10/3) + (-30/10)/(-7/1) - 2)/8 + ((-28/8)/(-10/3) + (-30/10)/(-8/2) - 2)/7 = 0. In this example [1, 7, 42] and [2, 8, 28] sum to [3, 15, 70], another row.
For n=4,
  if (id_1/theta_2 + id_2/theta_1 - 2)/(y3*y4) + (id_1/theta_3 + id_3/theta_1 - 2)/(y2*y4) + (id_1/theta_4 + id_4/theta_1 - 2)/(y2*y3) + (id_2/theta_3 + id_3/theta_2 - 2)/(y1*y4) + (id_2/theta_4 + id_4/theta_2 - 2)/(y1*y3) + (id_3/theta_4 + id_4/theta_3 - 2)/(y1*y2) = 0,
  then [x1+x2+x3+x4 , y1+y2+y3+y4 , z1+z2+z3+z4] is a row.
[x, y, z] = [3, 2, 1] is not a row, but (y/(y^2 - x*z))*[x, y, z] = (2/(2^2 - 3*1))*[3, 2, 1] = [6, 4, 2] which is a row. - _David Lovler_, Jan 22 2022

David Lovler, Table of n, a(n) for n = 1..2898
David Lovler, The first 8492 triples for y up to 300
David Lovler, Comments and examples

The number of rows for each y beginning with y=2 is A092517.

Cf. A332083.

for (y = 2, 8, fordiv (y^2-y, x, print([x, y, (y^2-y)/x]) ) ) \\ David Lovler, Mar 12 2021

x = (y^2 - y)/z.
y = (1 + sqrt(1 + 4*x*z))/2.
z = (y^2 - y)/x.

Table format in example edited by David Lovler, Feb 14 2022

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cp's OEIS Frontend

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

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A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

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A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

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A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

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A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

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A063123 Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).

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A133947 a(n) = the number of "non-isolated divisors" of n(n+1). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

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