A065608 -id:A065608 - cp's OEIS frontend

A363611 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.

0, 0, 0, 1, 4, 10, 20, 36, 56, 88, 120, 176, 220, 306, 368, 491, 560, 746, 816, 1058, 1160, 1450, 1540, 1982, 2028, 2520, 2656, 3232, 3276, 4116, 4060, 4986, 5080, 6016, 6008, 7457, 7140, 8586, 8656, 10232, 9880, 12116, 11480, 13792, 13668, 15730, 15180, 18652, 17316, 20536

Offset: 1

Seiichi Manyama, Jun 11 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001157, A001158, A013662, A065608, A363610.

Cf. A059358, A363604, A363607.

a[n_] := DivisorSum[n, Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - 6*sigma(f, 2) + 11*sigma(f) - 6*numdiv(f)) / 6; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} binomial(k-1,3) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d-1,3).
From Amiram Eldar, Jan 01 2025: (Start)
a(n) = (sigma_3(n) - 6*sigma_2(n) + 11*sigma_1(n) - 6*sigma_0(n)) / 6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) - 6*zeta(s-2) + 11*zeta(s-1) - 6*zeta(s)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A065061 Numbers k such that sigma(k) - tau(k) is a prime.

Original entry on oeis.org

3, 8, 162, 512, 1250, 8192, 31250, 32768, 41472, 663552, 2531250, 3748322, 5120000, 6837602, 7558272, 8000000, 15780962, 33554432, 35701250, 42762752, 45334242, 68024448, 75031250, 78125000, 91125000, 137149922, 243101250, 512000000, 907039232, 959570432

Offset: 1

Jason Earls, Nov 06 2001

nonn

From Kevin P. Thompson, Jun 20 2022: (Start)
Terms greater than 3 must be twice a square (see A064205).
No terms are congruent to 4 or 6 (mod 10) (see A064205).
(End)

			162 is a term since sigma(162) - tau(162) = 363 - 10 = 353, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..265 from Kevin P. Thompson)

I.e., A065608(n) is prime. Cf. A064205.

Cf. A000005, A000203, A023194, A038344, A055813, A062700, A115919, A141242, A229264, A229265, A229266, A229268.

Do[ If[ PrimeQ[ DivisorSigma[1, n] - DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]

{ n=0; for (m=1, 10^9, if (isprime(sigma(m) - numdiv(m)), write("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 2009

from itertools import count, islice
from sympy import isprime, divisor_sigma as s, divisor_count as t
def agen(): # generator of terms
    yield 3
    yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)-t(k)))
print(list(islice(agen(), 30))) # Michael S. Branicky, Jun 20 2022

a(17)-a(28) from Harry J. Smith, Oct 05 2009

a(29)-a(30) from Kevin P. Thompson, Jun 20 2022

A125182 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {p(i)-i, i=1,2,...,n} has exactly k elements (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 12, 7, 1, 4, 38, 54, 23, 1, 8, 77, 248, 303, 83, 1, 6, 160, 824, 2008, 1636, 405, 1, 11, 285, 2320, 9449, 15789, 10352, 2113, 1, 10, 476, 5564, 37237, 102726, 133293, 70916, 12657, 1, 14, 799, 13172, 122708, 536900, 1158368, 1177168, 537373, 82297

Offset: 1

Emeric Deutsch, Nov 24 2006

Row sums are the factorial numbers (A000142). T(n,1)=1 (the identity permutation). T(n,2) = A065608(n) = (sum of divisors of n)-(number of divisors of n). T(n,n) = A099152(n). In the first Maple program define n (<=10) to obtain row n.

T(n,k) is also the number of permutations p of {1,2,...,n} such that the set {p(i) + i, i=1,2,...,n} has exactly k elements (1<=k<=n). Example: T(4,2)=4 because we have 1432, 3412, 2143 and 3214. - Emeric Deutsch, Nov 28 2008

			T(4,2) = 4 because we have 4123, 3412, 2143 and 2341.
Triangle starts:
  1;
  1, 1;
  1, 2,  3;
  1, 4, 12,  7;
  1, 4, 38, 54, 23;
  ...

Alois P. Heinz, Rows n = 1..12, flattened
M. Alekseyev, E. Deutsch, and J. H. Steelman, Problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465. - _Emeric Deutsch_, Apr 23 2009

Cf. A000142, A065608, A099152, A125183, A306455.

n:=7: with(combinat): P:=permute(n): for j from 1 to n! do c[j]:=0 od: for j from 1 to n! do if nops({seq(P[j][i]-i,i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i],i=1..n);
# second Maple program:
b:= proc(p, s) option remember; `if`(p={}, x^nops(s),
      add(b(p minus {t}, s union {t+nops(p)}), t=p))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b({$1..n}, {})):
seq(T(n), n=1..9);  # Alois P. Heinz, May 04 2014; revised, Sep 08 2018

b[i_, p_List, s_List] := b[i, p, s] = If[p == {}, x^Length[s], Sum[b[i+1, p ~Complement~ {t}, s ~Union~ {t+i}], {t, p}]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][b[1, Range[n], {}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A306455(n). - Alois P. Heinz, Feb 16 2019

A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

2, 11, 353, 1013, 2333, 16369, 58579, 65519, 123733, 1982273, 7089683, 5778653, 12795053, 10500593, 22586027, 19980143, 24126653, 67108837, 72494713, 90781993, 106199593, 203275951, 164118923, 183105421, 320210549, 259997173, 794091653, 1279963973

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Second term of A065061 is 8 and sigma(8) - tau(8) = 15 - 4 = 11 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065608, A141242, A229264, A229266

with(numtheory); P:=proc(q) local a,n; a:= sigma(n)-tau(n); for n from 1 to q do
if isprime(a) then print(a); fi; od; end: P(10^6);

Join[{2}, Select[(DivisorSigma[1, #] - DivisorSigma[0, #]) & /@ (2*Range[20000]^2), PrimeQ]] (* Amiram Eldar, Dec 06 2022 *)

a(n) = A000203(A065061(n)) - A000005(A065061(n)). - Michel Marcus, Sep 21 2013

a(n) = A065608(A065061(n)). - Amiram Eldar, Dec 06 2022

More terms from Michel Marcus, Sep 21 2013

A263730 Irregular triangle read by rows in which row n > 1 lists k such that (k^2 + k*n)/(k + 1) is an integer.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 3, 0, 4, 0, 1, 2, 5, 0, 6, 0, 1, 3, 7, 0, 2, 8, 0, 1, 4, 9, 0, 10, 0, 1, 2, 3, 5, 11, 0, 12, 0, 1, 6, 13, 0, 2, 4, 14, 0, 1, 3, 7, 15, 0, 16, 0, 1, 2, 5, 8, 17, 0, 18, 0, 1, 3, 4, 9, 19, 0, 2, 6, 20, 0, 1, 10, 21, 0, 22, 0, 1, 2, 3, 5, 7, 11, 23, 0, 4, 24, 0, 1, 12, 25

Offset: 2

Juri-Stepan Gerasimov, Oct 25 2015

Minimal value of k is 0 and maximal value of k is n - 2 for n-th row.
Length of rows gives A000005.
Sum of rows gives A065608.

			  n\k| 0   1   2   3   4   5   6   7   8   9  10
  ---+------------------------------------------
   0 | 0
   1 | 0   1   2   3   4   5   6   7   8   9  10
   2 | 0
   3 | 0   1
   4 | 0   2
   5 | 0   1   3
   6 | 0   4
   7 | 0   1   2   5
   8 | 0   6
   9 | 0   1   3   7
  10 | 0   2   8
  11 | 0   1   4   9
  12 | 0  10
  13 | 0   1   2   3   5  11

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A000005, A065608, A027750, A263729.

Table[Select[Range[0,n-2],Divisible[#^2+n #,#+1]&],{n,30}]//Flatten (* Harvey P. Dale, Dec 27 2016 *)

tabf(nn) = {for (n=2, nn, for (k=0, n, if (!((k^2 + k*n) % (k+1)), print1(k, ", "));); print(););} \\ Michel Marcus, Oct 25 2015

a(n) = A027750(n) - 1.

A077478 Rectangular array R read by antidiagonals: R(i,j) is the number of integers k that divide both i and j (i >= 1, j >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1

Offset: 1

Clark Kimberling, Nov 08 2002

Antidiagonal sums of R, alias row sums of T, are essentially A065608. Diagonal elements of R comprise A000203 (sums of divisors of n).
Antidiagonals of an array formed by A051731 * A051731 (transposed). - Gary W. Adamson, Nov 12 2007
If R(n) is the n X n Redheffer matrix (A143104) and Rt(n) is its transposed matrix, then this sequence seems to be formed by R(n)*Rt(n). - Enrique Pérez Herrero, Feb 21 2012

			First few rows of the array R are:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, ...
  1, 1, 2, 1, 1, 2, 1, ...
  1, 2, 1, 3, 1, 2, 1, ...
  1, 1, 1, 1, 2, 1, 1, ...
  1, 2, 2, 2, 1, 4, 1, ...
  ...
First few rows of the triangle T are:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 1, 1;
  1, 2, 2, 2, 1;
  1, 1, 1, 1, 1, 1;
  1, 2, 1, 3, 1, 3, 1;
  1, 1, 2, 1, 1, 2, 1, 1;
  1, 2, 1, 2, 2, 2, 1, 2, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1;
  ...
R(4,2)=2 since 1|2, 1|4 and 2|2, 2|4.

Stefano Spezia, First 150 antidiagonals of the array, flattened

Cf. A051194, A077049, A077051.

Cf. A051731, A065608.

T[n_,k_]:=DivisorSigma[0,GCD[n,k]]; Flatten[Table[T[n-k+1,k],{n,14},{k,n}]] (* Stefano Spezia, May 23 2021 *)

R=U*V, where U and V are the summatory matrices (A077049, A077051). The triangle T(n, k) formed by antidiagonals: T(n, k)=tau(gcd(k, n+1-k)) for 1<=k<=n, where tau(m)=A000005(m). [Corrected by Leroy Quet, Apr 08 2009]

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} tau(gcd(n,k))/n^s/k^c = zeta(s)*zeta(c)* zeta(s + c). - Mats Granvik, May 19 2021

Edited by N. J. A. Sloane, Jan 11 2009

A191822 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 = n, with all xi >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 8, 16, 20, 32, 36, 58, 58, 86, 92, 125, 122, 178, 164, 228, 224, 286, 268, 382, 330, 436, 424, 534, 474, 660, 556, 740, 692, 840, 752, 1043, 846, 1094, 1032, 1276, 1078, 1476, 1204, 1582, 1458, 1710, 1480, 2070, 1628, 2096, 1924, 2332, 1946, 2652, 2148, 2770, 2480, 2908, 2480, 3512

Offset: 1

N. J. A. Sloane, Jun 17 2011

nonn

Related to "Liouville's Last Theorem".

			G.f.: x^4 + 2 x^5 + 6 x^6 + 8 x^7 + 16 x^8 + 20 x^9 + 32 x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_4(n).
E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

Cf. A000005, A002133, A065608, A191832.
Cf. A001157, A038040, A055507.
Cf. A189835.

with(numtheory);
D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
L4:=n->sigma[2](n)-n*sigma[0](n)-D00(n);
[seq(L4(n),n=1..60)];

a[ n_] := Length @ FindInstance[{x1 > 0, x2 > 0, x3 > 0, x4 > 0, x5 > 0, n == x1 x2 + x2 x3 + x3 x4 + x4 x5}, {x1, x2, x3, x4, x5}, Integers, 10^9]; (* Michael Somos, Nov 12 2016 *)

a(n) = sigma_2(n) - n*sigma_0(n) - A055507(n-1).

A342343 Number of strict compositions of n with alternating parts strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 18, 27, 32, 44, 55, 73, 97, 121, 151, 194, 240, 299, 384, 465, 576, 706, 869, 1051, 1293, 1572, 1896, 2290, 2761, 3302, 3973, 4732, 5645, 6759, 7995, 9477, 11218, 13258, 15597, 18393, 21565, 25319, 29703, 34701, 40478, 47278, 54985

Offset: 0

Gus Wiseman, Apr 01 2021

nonn

These are finite odd-length sequences q of distinct positive integers summing to n such that q(i) > q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)
                          (3,2)  (4,2)    (3,4)    (3,5)
                          (4,1)  (5,1)    (4,3)    (5,3)
                                 (2,3,1)  (5,2)    (6,2)
                                 (3,1,2)  (6,1)    (7,1)
                                 (3,2,1)  (2,4,1)  (2,5,1)
                                          (4,1,2)  (3,4,1)
                                          (4,2,1)  (4,1,3)
                                                   (4,3,1)
                                                   (5,1,2)
                                                   (5,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The non-strict case is A000041 (see A342528 for a bijective proof).
The non-strict odd-length case is A001522.
Strict compositions in general are counted by A032020
The non-strict even-length case is A064428.
The case of reversed partitions is A065033.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A027193 counts odd-length compositions.
A034008 counts even-length compositions.
A064391 counts partitions by crank.
A064410 counts partitions of crank 0.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
A325548 counts compositions with strictly decreasing differences.
A342194 counts strict compositions with equal differences.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A000670, A008965, A062968, A065608, A109613, A114921, A325546, A332304, A332305, A342532.

ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],ici]],{n,0,15}]

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, binomial(k, k\2) * polcoef(p,k,y)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} binomial(k,floor(k/2)) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

A348915 a(n) = Sum_{d|n} d^(d mod 2).

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 8, 4, 13, 8, 12, 8, 14, 10, 24, 5, 18, 16, 20, 10, 32, 14, 24, 10, 31, 16, 40, 12, 30, 28, 32, 6, 48, 20, 48, 19, 38, 22, 56, 12, 42, 36, 44, 16, 78, 26, 48, 12, 57, 34, 72, 18, 54, 44, 72, 14, 80, 32, 60, 32, 62, 34, 104, 7, 84, 52, 68, 22, 96, 52, 72, 22, 74

Offset: 1

Wesley Ivan Hurt, Nov 03 2021

nonn

For each divisor d of n, add d if d is odd, otherwise add 1.

Inverse Möbius transform of n^(n mod 2). - Wesley Ivan Hurt, Mar 31 2025

			For n = 12, the divisors of 12 are 1, 2, 3, 4, 6, 12 with corresponding summands 1, 1, 3, 1, 1, 1, respectively. The sum is then a(12) = 1 + 1 + 3 + 1 + 1 + 1 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000005 (tau), A000203 (sigma), A000593, A065608, A183063.

f:= proc(n) local d;
  add(d^(d mod 2), d = numtheory:-divisors(n))
end proc;
map(f, [$1..100]); # Robert Israel, Jun 01 2025

a[n_] := DivisorSum[n, #^Mod[#, 2] &]; Array[a, 100] (* Amiram Eldar, Nov 04 2021 *)

a(n) = sumdiv(n, d, if (d%2, d, 1)); \\ Michel Marcus, Nov 04 2021

from math import prod
from sympy import factorint
def A348915(n):
    f = factorint(n>>(m:=(~n&n-1).bit_length())).items()
    d = prod(e+1 for p,e in f)
    s = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return s+d*m # Chai Wah Wu, Jul 16 2022

a(n) = A000593(n) + A183063(n).
a(n) = A065608(2n) - 2*A065608(n).
a(p) = p+1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021
a(n) = A000203(A000265(n))+A000005(A000265(n))*A007814(n). - Chai Wah Wu, Jul 16 2022
a(n) = A000203(n) if n is odd. - Robert Israel, Jun 01 2025

A184396 a(n) = number of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n, where sigma = A000203.

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 137, 66, 120, 92, 136, 70, 183, 72, 110, 118, 134, 92, 160, 78, 176, 116

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Sequence is not the same as A065608(n): a(66) = 137, A065608(66) = 136.

			For n = 4, sigma(4) = 7, from numbers 1 - 7 there are four numbers k such that k is not equal to sigma(d) for any divisor d of n: 2, 4, 5, 6; a(4) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A065608, A184395.

f[n_] := Block[{c = 0, k = 1, lmt = DivisorSigma[1, n] + 1, sd = DivisorSigma[1, #] & /@ Divisors@ n}, While[k < lmt, If[! MemberQ[sd, k], c++]; k++]; c]; Array[f, 67]

A184395(n) = length(vecsort(apply(d->sigma(d),divisors(n)), , 8));
A184396(n) = (sigma(n) - A184395(n)); \\ Antti Karttunen, Aug 24 2017

a(n) = A000203(n) - A184395(n).

More terms from Antti Karttunen, Aug 24 2017

cp's OEIS Frontend

A363611 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.

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A065061 Numbers k such that sigma(k) - tau(k) is a prime.

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A125182 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {p(i)-i, i=1,2,...,n} has exactly k elements (1<=k<=n).

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A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

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A263730 Irregular triangle read by rows in which row n > 1 lists k such that (k^2 + k*n)/(k + 1) is an integer.

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A077478 Rectangular array R read by antidiagonals: R(i,j) is the number of integers k that divide both i and j (i >= 1, j >= 1).

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A191822 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 = n, with all xi >= 1.

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A191822 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 = n, with all xi >= 1.