A126246 -id:A126246 - cp's OEIS frontend

A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99, 67, 102

Offset: 1

Amarnath Murthy, Mar 20 2003

First differences of the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011
Last term in n-th row of A080511.
Also A005408 and positive terms of A008585 interleaved. - Omar E. Pol, May 28 2012
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized heptagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A080511, A008619, A126988.

Cf. A005408, A008585, A064455, A085787, A126246

import Data.List (transpose)
a080512 n = if m == 0 then 3 * n' else n  where (n', m) = divMod n 2
a080512_list = concat $ transpose [[1, 3 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Apr 06 2015

[n*(5+(-1)^n)/4: n in [1..60]]; // Vincenzo Librandi, Sep 11 2011

Table[If[EvenQ[n],3n/2,n],{n,68}] (* Jayanta Basu, May 20 2013 *)

a(n) = n if n is odd, a(n) = 3*n/2 if n is even.
a(n)*a(n+3) = -3 + a(n+1)*a(n+2).
From Paul Barry, Sep 04 2003: (Start)
G.f.: (1+3*x+x^2)/((1-x^2)^2);
a(n) = n*(5 + (-1)^n)/4. (End)
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = p^e otherwise. - Christian G. Bower, May 17 2005
Equals A126988 * (1, 1, 0, 0, 0, ...) - Gary W. Adamson, Apr 17 2007
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s). - Amiram Eldar, Oct 25 2023
Sum_{d divides n} mu(n/d)*a(d) = A126246(n), where mu(n) = A008683(n) is the Möbius function. - Peter Bala, Dec 31 2023

A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 6, 7, 4, 10, 6, 12, 6, 8, 12, 16, 7, 18, 12, 12, 10, 22, 12, 21, 12, 21, 18, 28, 8, 30, 24, 20, 16, 24, 21, 36, 18, 24, 24, 40, 12, 42, 30, 28, 22, 46, 24, 43, 21, 32, 36, 52, 21, 40, 36, 36, 28, 58, 24, 60, 30, 42, 48, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the powerfree part (A055231) of gcd(n,k) is 1.

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

Cf. A000010, A001694, A005117, A050873, A055231, A063658, A206369.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), this sequence (powerful), A384040 (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := If[e == 1, p-1, (p^2-p+1)*p^(e-2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, (f[i,1]^2-f[i,1]+1)*f[i,1]^(f[i,2]-2)));}

Multiplicative with a(p^e) = (p^2-p+1)*p^(e-2) if e >= 2, and p-1 otherwise.
a(n) >= A000010(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^4) = 0.66922021803510257394... .

A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 10, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 20, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 20, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 40, 48, 20, 66, 32, 44, 24

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the cubefree part (A360539) of gcd(n,k) is 1.

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

Cf. A005117, A036966, A078429, A360539, A254926.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), this sequence (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := Switch[e, 1, p-1, 2, p^2-p, , (p^3-p^2+1)*p^(e-3)]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, if(f[i,2] == 2, f[i,1]*(f[i,1]-1), (f[i,1]^3-f[i,1]^2+1)*f[i,1]^(f[i,2]-3))));}

Multiplicative with a(p^e) = (p^3-p^2+1)*p^(e-3) if e >= 3, p*(p-1) if e = 2, and p-1 otherwise.
a(n) >= A384039(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^6) = 0.62159731307414305346... .

A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 13, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 25, 21, 29, 30, 31, 27, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 39, 48, 48, 51, 39, 53, 50, 55, 49, 57, 58, 59, 45, 61, 62, 56, 53, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A268335.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), this sequence (exponentially odd), A384042 (5-rough).

f[p_, e_] := ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i,1]^2+f[i,1]-1)*f[i,1]^(f[i,2]-1) - (-1)^f[i,2])/(f[i,1] + 1));}

Multiplicative with a(p^e) = ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1).
a(n) >= A000010(n), with equality if and only if n = 1.
Dirichlet g.f.: (zeta(s-1)*zeta(2*s)/zeta(s)) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/(p^2+1)) = 0.93749428273130025078... .

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 7, 8, 8, 10, 6, 12, 12, 8, 15, 16, 16, 18, 12, 12, 20, 22, 14, 24, 24, 26, 18, 28, 16, 30, 31, 20, 32, 24, 24, 36, 36, 24, 28, 40, 24, 42, 30, 32, 44, 46, 30, 48, 48, 32, 36, 52, 52, 40, 42, 36, 56, 58, 24, 60, 60, 48, 63, 48, 40, 66, 48, 44

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 4 of the values are either 1 or 2, and therefore a(6) = 4.

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

The unitary analog of A126246 (with respect to the definition "the number of integers k from 1 to n such that gcd(n,k) is either 1 or 2").
Cf. A065463, A077610, A138191, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough), this sequence (1 or 2), A385196 (prime), A385197 (noncomposite), A385198 (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; f[2, 1] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,f[i,1]^f[i,2] - if(f[i,1] == 2 && f[i,2] == 1, 0, 1));}

Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and p^e - 1 otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k), the greatest divisor of k that is a unitary divisor of n, is either 1 or a prime power q is a multiplicative function f(n) with f(p^e) = q if p^e = q, and p^e - 1 otherwise.
a(n) = A138191(n) * A047994(n), i.e., a(n) = 2*A047994(n) if n == 2 (mod 4) and A047994(n) otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k) is either 1 or a prime power q is (q/(q-1))*A047994(n) if q is a unitary divisor of n, and A047994(n) otherwise.
Sum_{k=1..n} a(k) ~ (23/40) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.
In general, the average order of the number of integers k from 1 to n such that ugcd(n, k) is either 1 or a prime p is ((p^4+p^3-1)/(p^4+p^3-p^2)) * c * n^2 / 2, where c = A065463.

A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 1, 1, 1;
  4, 0, 0, 1, 1;
  4, 3, 1, 0, 1, 1;
  6, 0, 0, 0, 0, 1, 1;
  6, 2, 1, 1, 0, 0, 1, 1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)

A128317 Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.

Original entry on oeis.org

1, 0, 1, 3, -2, 1, 0, 4, -3, 1, 5, -4, 6, -4, 1, 0, 5, -9, 10, -5, 1, 7, -6, 15, -20, 15, -6, 1, 0, 12, -24, 36, -35, 21, -7, 1, 9, -12, 30, -56, 70, -56, 28, -8, 1, 0, 9, -30, 80, -125, 126, -84, 36, -9, 1, 11, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

			First few rows of the triangle:
  1;
  0,  1;
  3, -2,  1;
  0,  4, -3,   1;
  5, -4,  6,  -4,  1;
  0,  5, -9,  10, -5,  1;
  7, -6, 15, -20, 15, -6, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000984, A001700, A007318, A054523, A130595.

Sums include: A000010 (row sums), A126246.

A128317:= func< n,k | (&+[(-1)^(d+k)*EulerPhi(Floor(n/d))*Binomial(d-1, k-1) : d in Divisors(n)]) >;
[A128317(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

A128317[n_, k_]:= DivisorSum[n, (-1)^(#+k)*EulerPhi[n/#]*Binomial[#-1, k-1]  &];
Table[A128317[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 24 2024 *)

def A128317(n,k): return sum((-1)^(k+j)*euler_phi(n/j)*binomial(j-1, k-1) for j in (1..n) if (j).divides(n))
flatten([[A128317(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 24 2024

Equals A054523 * signed A007318 as infinite lower triangular matrices. A007318 is signed by columns: (+, -, +, ...).
Sum_{k=1..n} T(n, k) = A000010(n) (row sums).
From G. C. Greubel, Jun 24 2024: (Start)
T(n, k) = A054523 * A130595, as a lower triangular matrix.
T(n, k) = Sum_{j=k..n} (-1)^(k+j)*A054523(n,j)*binomial(j-1, k-1).
T(n, k) = Sum_{d|n} (-1)^(d+k)*EulerPhi(n/d)*binomial(d-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
T(2*n-2, n-1) = (-1)^n*A001700(n-2), n >= 2.
Sum_{k=1..n} k*T(n, k) = A126246(n). (End)

A209974 a(n) = A209973(n)/4.

Original entry on oeis.org

0, 0, 3, 5, 9, 13, 19, 25, 33, 39, 51, 61, 69, 81, 99, 107, 123, 139, 157, 175, 191, 203, 233, 255, 271, 291, 327, 345, 369, 397, 421, 451, 483, 503, 551, 575, 599, 635, 689, 713, 745, 785, 821, 863, 903, 927, 993, 1039, 1071, 1113, 1173

Offset: 0

Clark Kimberling, Mar 16 2012

nonn

For a guide to related sequences, see A210000.

Cf. A210000.

Cf. A000010, A126246.

(See the Mathematica section at A210000.)

Apparently, a(n) = a(n-1) + 2*A126246(n) - A000010(n) for n >= 2. - Pontus von Brömssen, Jun 28 2021

A346018 Triangle read by rows: T(n,k) is the number of integers m such that 1 <= m <= n and gcd(m,n) <= k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 4, 5, 2, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 4, 6, 6, 7, 7, 7, 7, 8, 6, 6, 8, 8, 8, 8, 8, 8, 9, 4, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 4, 6, 8, 10, 10, 11, 11, 11, 11, 11, 11, 12

Offset: 1

Pontus von Brömssen, Jul 01 2021

			Triangle begins:
   n\k  1  2  3  4  5  6  7  8  9 10
  ----------------------------------
   1:   1
   2:   1  2
   3:   2  2  3
   4:   2  3  3  4
   5:   4  4  4  4  5
   6:   2  4  5  5  5  6
   7:   6  6  6  6  6  6  7
   8:   4  6  6  7  7  7  7  8
   9:   6  6  8  8  8  8  8  8  9
  10:   4  8  8  8  9  9  9  9  9 10

Cf. A000010 (column k=1), A126246 (column k=2), A032742.

T(n,k) = sum(m=1, n, gcd(m, n) <= k); \\ Michel Marcus, Jul 01 2021

T(n,1) = A000010(n).
T(n,2) = A126246(n).
T(n,k) = n-1 for A032742(n) <= k <= n-1.
T(n,n) = n.

cp's OEIS Frontend

A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

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A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

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A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

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A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

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A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

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A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

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A128317 Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.

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