A051062 -id:A051062 - cp's OEIS frontend

A179000 Array T(n,k) read by antidiagonals: coefficient [x^k] of (1 + n*Sum_{i>=1} x^i)^2, k >= 0.

1, 1, 2, 1, 4, 3, 1, 6, 8, 4, 1, 8, 15, 12, 5, 1, 10, 24, 24, 16, 6, 1, 12, 35, 40, 33, 20, 7, 1, 14, 48, 60, 56, 42, 24, 8, 1, 16, 63, 84, 85, 72, 51, 28, 9, 1, 18, 80, 112, 120, 110, 88, 60, 32, 10

Offset: 1

Gary W. Adamson, Jan 03 2011

Antidiagonal sums are in A136396.

			First few rows of the array:
  1   2   3   4   5   6   7   8   9  10  11  A000027
  1   4   8  12  16  20  24  28  32  36  40  A008574
  1   6  15  24  33  42  51  60  69  78  87  A122709
  1   8  24  40  56  72  88 104 120 136 152  A051062
  1  10  35  60  85 110 135 160 185 210 235
  1  12  48  84 120 156 192 228 264 300 336
  1  14  63 112 161 210 259 308 357 406 455
  1  16  80 144 208 272 336 400 464 528 592
  1  18  99 180 261 342 423 504 585 666 747
Row n=3 is generated by (1 + 3x + 3x^2 + 3x^3 + 3x^4 + ...)^2 = 1 + 6x + 15x^2 + 24x^3 + ..., for example.

Cf. A136396, A179901.

A179000 := proc(n,k) if k = 0 then 1; else 2*n+n^2*(k-1) ; end if; end proc: # R. J. Mathar, Jan 05 2011

T(n,0) = 1; T(n,k) = n*(2+n*(k-1)), k > 0. - R. J. Mathar, Jan 05 2011

A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 4, 0, 10, 8, 4, 8, 4, 8, 16, -5, 4, 20, 4, 8, 16, 8, 4, 0, 10, 8, 20, 8, 4, 32, 4, -14, 16, 8, 16, 20, 4, 8, 16, 0, 4, 32, 4, 8, 40, 8, 4, -20, 10, 20, 16, 8, 4, 40, 16, 0, 16, 8, 4, 32, 4, 8, 40, -28, 16, 32, 4, 8, 16, 32, 4, 0, 4, 8, 40, 8, 16, 32, 4

Offset: 1

Ilya Gutkovskiy, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A007425, A007426, A051062 (positions of 0's), A288571.

Cf. A001620, A082633.

Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 79}]
nmax = 79; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[0, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 79; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[0, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, 1 + (7 - e^2)*e/6, Binomial[e + 3, 3]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1) * sumdiv(d, j, numdiv(j))); \\ Michel Marcus, Sep 04 2018

G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 + x^k), where tau_3() = A007425.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau_3(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = 1 + (7-e^2)*e/6, and a(p^e) = binomial(e+3,3) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s)^4 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ (log(2)/2) * n * (log(n)^2 + (8 * gamma - log(2) - 2) * log(n) + 12 * gamma^2 - 8 * gamma + log(2) + 2 - 4 * gamma * log(2) + log(2)^2/3 - 8 * gamma_1), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). (End)

A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 1, 4, 5, 22, 1, 4, 3, 2, 3, 28, 2, 30, 1, 20, 4, 24, 1, 36, 9, 8, 1, 40, 1, 42, 5, 8, 11, 46, 1, 6, 1, 32, 3, 52, 1, 8, 3, 4, 7, 58, 1, 60, 15, 4, 1, 48, 5, 66, 2, 44, 6, 70, 1, 72, 9, 8, 9, 60, 2, 78

Offset: 1

Amiram Eldar, Jun 01 2024

Numbers that are unitarily divided by n are numbers k such that n is a unitary divisor of k, or equivalently, numbers of the form m*n, with gcd(m, n) = 1.

			Fractions begin with: 1, 1/4, 2/9, 1/8, 4/25, 1/18, 6/49, 1/16, 2/27, 1/25, 10/121, 1/36, ...
For n = 2, the numbers that are unitarily divided by 2 are the numbers of the form 4*k+2 whose asymptotic density is 1/4. Therefore a(2) = numerator(1/4) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.
Cf. A001620, A013661, A306016.
Numbers that are unitarily divided by k: A000027 (k=1), A016825 (k=2), A016051 (k=3), A017113 (k=4), A051062 (k=8), A051063 (k=9).

a[n_] := Numerator[EulerPhi[n]/n^2]; Array[a, 100]

PARI
```
a(n) = numerator(eulerphi(n)/n^2);
```

a(n) = 1 if and only if n is in A090778.
a(n) = A000010(n) if and only if n is a cyclic number (A003277).
Let f(n) = a(n)/A373319(n). Then:
f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).
f(n) = A173557(n)/A064549(n).
f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.
Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).

A334566 Number of solutions of the Diophantine equation z^2 - y^2 - x^2 = n > 0 when the positive integers, x, y and z, are consecutive terms of an arithmetic progression.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 1, 3, 0, 0, 4, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 1, 0, 0, 1, 3, 0, 0, 2, 0

Offset: 1

Bernard Schott, May 06 2020

nonn

Inspired by the 135th and 136th problems of Project Euler (see links).
If d is the common difference of the arithmetic progression (x, y, z), then the Diophantine equation becomes (y+d)^2 - y^2 - (y-d)^2 = n <==> y^2 - 4dy + n = 0 <==> n = y * (4d-y).
If y is the average term, then y divides n.
Offset is 1 because for n = 0, every (x, y, z)= (3d, 4d, 5d) with d>0 would be solution.

			a(3) = 1 because 4^2 - 3^2 - 2^2 = 3.
a(15) = 3 because 5^2 - 3^2 - 1^2 = 7^2 - 5^2 - 3^2 = 19^2 - 15^2 - 11^2 = 15.
If n = 4q+3, q >= 0 then (3q+2, 4q+3, 5q+4) is a solution.
If n = 16q, q >= 1 then (3q-1, 4q, 5q+1) is a solution.
If n = 16q+4, q >= 0 then (6q+1, 8q+2, 10q+3) is a solution.
If n = 16q+12, q >= 0 then (6q+4, 8q+6, 10q+8) is a solution.

Robert Israel, Table of n, a(n) for n = 1..10000
Project Euler, Problem 135: Same differences
Project Euler, Problem 136: Singleton difference

Cf. A001248, A002145, A004767, A008598, A016813, A016825, A051062.

Cf. A334567 (least value of n such that a(n) = k>0).

f:= proc(n) local r; r:= floor(sqrt(n/3));
nops(select(t -> n/t + t mod 4 = 0 and t > r, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jul 31 2020

a[n_] := Length@ Solve[(4 d - x) x == n  && x>0 && x-d>0 && x+d>0, {d, x}, Integers]; Array[a, 90] (* Giovanni Resta, May 06 2020 *)

a(n) = 0 iff n = 4q+1 (A016813), n = 4q+2 (A016825), n = 16q+8 (A051062), q>= 0.
a(n) >= 1 iff n = 4q+3, q >=0 (A004767), n = 16q, q>=1 (A008598), n = 16q+4, q>=0 (A119413), n = 16q+12, q>=0 (A098502).
a(4*q^2) >= 1, for q >= 1, since (q, 2q, 3q) is a solution.
a(p) = 1 for p = 4q+3 prime (A002145).
a(p^2) = 0 for p an odd prime (A065091).

More terms from Giovanni Resta, May 06 2020

A370596 Numbers k such that A007814(k) is a prime number.

Original entry on oeis.org

4, 8, 12, 20, 24, 28, 32, 36, 40, 44, 52, 56, 60, 68, 72, 76, 84, 88, 92, 96, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 160, 164, 168, 172, 180, 184, 188, 196, 200, 204, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276

Offset: 1

Amiram Eldar, Feb 23 2024

Numbers whose binary representation has a prime number of trailing 0's.
a(n)-1 is the sequence of numbers whose binary representation has a prime number of trailing 1's.
Numbers of the form (2^(p+1))*k + 2^p = 2^p * (2*k + 1), where p is prime and k >= 0.
All the terms are divisible by 4.
The asymptotic density of this sequence is Sum_{p prime} 1/2^(p+1) = 0.20734125492555583012... = A051006 / 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A051006, A359794.

Subsequences: A017113, A051062.

Select[Range[300], PrimeQ[IntegerExponent[#, 2]] &]

PARI
```
is(n) = isprime(valuation(n, 2));
```

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 0, 10, 12, 6, 0, 14, 20, 18, 8, 0, 18, 28, 30, 24, 10, 0, 22, 36, 42, 40, 30, 12, 0, 26, 44, 54, 56, 50, 36, 14, 0, 30, 52, 66, 72, 70, 60, 42, 16, 0, 34, 60, 78, 88, 90, 84, 70, 48, 18, 0, 38, 68, 90, 104, 110, 108, 98, 80, 54, 20, 0, 42, 76, 102

Offset: 0

Philippe Deléham, Dec 05 2013

Row sums are A006331(n).
Diagonal sums are A212964(n+1).
T(2n,n)=A002943(n).

			Triangle begins:
  0
  0, 2
  0, 6, 4
  0, 10, 12, 6
  0, 14, 20, 18, 8
  0, 18, 28, 30, 24, 10

Cf. columns: A000004, A016825, A017113, A017593, A051062, 10*A005408, A073762.

Cf. diagonals: A005843, A008588, A008592, A008596, A008600, A008604.

T(n+k,k) = A005843(k)*A005408(n).

Sum_{k=0..n} T(n,k) = n*(n+1)*(2*n+1)/3 = A006331(n).

cp's OEIS Frontend

A179000 Array T(n,k) read by antidiagonals: coefficient [x^k] of (1 + n*Sum_{i>=1} x^i)^2, k >= 0.

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A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

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A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

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A334566 Number of solutions of the Diophantine equation z^2 - y^2 - x^2 = n > 0 when the positive integers, x, y and z, are consecutive terms of an arithmetic progression.

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A370596 Numbers k such that A007814(k) is a prime number.

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A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2*k*(2*n+1).

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A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).