A189835 -id:A189835 - cp's OEIS frontend

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

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).

A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Bruno Berselli, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

Equivalently, numbers m such that ( m*sigma_0(m)+sigma_2(m) ) / (2*sigma_1(m)) = (A038040(m) + A001157(m))/A074400(m) is an integer.

All odd primes belong to the sequence. In fact, if p is an odd prime, (p*sigma_0(p)+sigma_2(p))/(2*sigma_1(p)) = (p+1)/2, therefore p is a biharmonic number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)
Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016, pages 6-14.
Umberto Cerruti, Numeri Armonici e Numeri Perfetti (in Italian), 2013. The sequence is on page 13.

Cf. A001599 (harmonic numbers), A020487 (antiharmonic numbers), A038040 (n*sigma_0(n)), A001157 (sigma_2(n)), A074400 (2*sigma_1(n)), A230214 (nonprime terms of A210494).

Cf. A189835.

a210494 n = a210494_list !! (n-1)
a210494_list = filter
   (\x -> (a001157 x + a038040 x) `mod` a074400 x == 0) [1..]
-- Reinhard Zumkeller, Jan 21 2014

IsInteger := func; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2,n))/(2*SumOfDivisors(n)))];

Maple

with(numtheory); P:=proc(q) local a,k,n; for n from 1 to q do a:=divisors(n); if type((n*tau(n)+add(a[k]^2,k=1..nops(a)))/(2*sigma(n)),integer) then print(n); fi; od; end; P(1000); # Paolo P. Lava, Oct 11 2013

Mathematica

B[n_] := (n DivisorSigma[0, n] + DivisorSigma[2, n])/(2 DivisorSigma[1, n]); Select[Range[300], IntegerQ[B[#]] &]

PARI

isok(n) = denominator((n*sigma(n,0) + sigma(n,2))/(2*sigma(n)))==1; \\ Michel Marcus, Jan 14 2016

A344485 a(n) = Sum_{d|n} (n-d) * phi(n/d).

Original entry on oeis.org

0, 1, 4, 8, 16, 21, 36, 44, 60, 73, 100, 104, 144, 157, 180, 208, 256, 261, 324, 328, 376, 421, 484, 476, 560, 601, 648, 680, 784, 765, 900, 912, 984, 1057, 1108, 1128, 1296, 1333, 1396, 1420, 1600, 1569, 1764, 1768, 1836, 1981, 2116, 2064, 2268, 2305, 2436, 2504, 2704, 2673

Offset: 1

Wesley Ivan Hurt, May 20 2021

nonn

a(n) is the sum of the (n - 1)-th antidiagonal in A074712. - Ctibor O. Zizka, Mar 14 2025

Möbius transform of A189835(n). - Wesley Ivan Hurt, Jul 16 2025

			a(6) = Sum_{d|6} (6-d) * phi(6/d) = 5*phi(6) + 4*phi(3) + 3*phi(2) + 0*phi(1) = 5*2 + 4*2 + 3*1 + 0*1 = 21.

Cf. A000010, A000290, A018804, A074712, A189835.

with(numtheory): seq(add((n-d)*phi(n/d), d in divisors(n)), n=1..80); # Ridouane Oudra, Jan 21 2024

Table[Sum[(n - k)*EulerPhi[n/k^(1 - Ceiling[n/k] + Floor[n/k])] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

a(n) = sumdiv(n, d, (n-d) * eulerphi(n/d)); \\ Michel Marcus, May 21 2021

a(n) = A000290(n) - A018804(n). - Ridouane Oudra, Jan 21 2024
From Wesley Ivan Hurt, Jul 16 2025: (Start)
a(n) = Sum_{d|n} A189835(d) * mu(n/d).
a(p^k) = p^(2*k)-p^k-k*p^k+k*p^(k-1) for p prime and k>=1. (End)

A375035 Number of representations of n as 2wx + xy + yz + zu + ux where w, x, y, z>0, u>=0 are integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 3, 11, 4, 21, 10, 30, 15, 51, 14, 71, 28, 86, 36, 119, 36, 150, 55, 178, 59, 219, 68, 269, 91, 278, 105, 367, 110, 396, 116, 469, 153, 504, 162, 587, 190, 608, 210, 752, 199, 759, 253, 906, 261, 879, 296, 1085, 325, 1068, 322, 1249, 378, 1239, 406, 1466, 435, 1425, 431, 1695, 464

Offset: 1

Seiichi Manyama, Jul 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 275.

Cf. A000005, A000203, A001157, A189835.

a(n) = sum(w=1, n\2, sum(x=1, n\(2*w), sum(y=1, n\x, sum(z=1, n\y, sum(u=0, n\z, 2*w*x+x*y+y*z+z*u+u*x==n)))));

If n is an odd, a(n) = (sigma_2(n) + 4*sigma(n) - (4*n+1)*sigma_0(n))/8.

cp's OEIS Frontend

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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A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.

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A344485 a(n) = Sum_{d|n} (n-d) * phi(n/d).

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A375035 Number of representations of n as 2*w*x + x*y + y*z + z*u + u*x where w, x, y, z>0, u>=0 are integers.

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

A375035 Number of representations of n as 2wx + xy + yz + zu + ux where w, x, y, z>0, u>=0 are integers.