A132345 -id:A132345 - cp's OEIS frontend

A078147 First differences of sequence of nonsquarefree numbers, A013929.

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

Offset: 1

Labos Elemer, Nov 26 2002

Run lengths in A132345, apart from initial run of zeros. - Reinhard Zumkeller, Apr 22 2012

The asymptotic density of the occurrences of 1 in this sequence is density(A068781)/density(A013929) = (1 - 2 * A059956 + A065474)/A229099 = 0.272347... - Amiram Eldar, Mar 09 2021

			a(1) = 4 = 8 - 4.

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

Cf. A068781, A013929, A132345.

Cf. A059956, A065474, A229099.

a078147 n = a078147_list !! (n-1)
a078147_list = zipWith (-) (tail a013929_list) a013929_list
-- Reinhard Zumkeller, Apr 22 2012

t=Flatten[Position[Table[MoebiusMu[w], {w, 1, 1000}], 0]]; t1=Delete[RotateLeft[t]-t, -1]
Differences[Select[Range[300],!SquareFreeQ[#]&]] (* Harvey P. Dale, May 07 2012 *)

lista(nn) = {my(prec=0); for (n=1, nn, if (!issquarefree(n), if (prec, print1(n-prec, ", ")); prec = n;););} \\ Michel Marcus, Mar 26 2020

from math import isqrt
from sympy import mobius, factorint
def A078147(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values())) # Chai Wah Wu, Sep 10 2024

a(n) = A013929(n+1) - A013929(n).
a(n) = 1, 2, 3 or 4 since n = 4*k is always nonsquarefree.
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/(Pi^2-6) = 2.550546... - Amiram Eldar, Oct 21 2020

Offset fixed by Reinhard Zumkeller, Apr 22 2012

A132188 Number of 3-term geometric progressions with no term exceeding n.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 12, 17, 18, 19, 22, 23, 24, 25, 32, 33, 38, 39, 42, 43, 44, 45, 48, 57, 58, 63, 66, 67, 68, 69, 76, 77, 78, 79, 90, 91, 92, 93, 96, 97, 98, 99, 102, 107, 108, 109, 116, 129, 138, 139, 142, 143, 148, 149, 152, 153, 154, 155, 158

Offset: 1

Gerry Myerson, Nov 21 2007

nonn

a(n) = number of pairs (i,j) in [1..n] X [1..n] with integral geometric mean sqrt(i*j). Cf. A000982, A362931. - N. J. A. Sloane, Aug 28 2023
Also the number of 2 X 2 symmetric singular matrices with entries from {1, ..., n} - cf. A064368.
Rephrased: Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=x*y. See A211422. - Clark Kimberling, Apr 14 2012

			a(4) counts these six (w,x,y) - triples: (1,1,1), (2,1,4), (2,4,1), (2,2,2), (3,3,3), (4,4,4). - _Clark Kimberling_, Apr 14 2012

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette 35 (3) (2008) pp. 189--194 (pages 47--52 in PDF).

Cf. A057918, A064368, A120486, A132189, A132345, A211422, A338894.

Cf. also A000982, A362931.

a132188 0 = 0
a132188 n = a132345 n + (a120486 $ fromInteger n)
-- Reinhard Zumkeller, Apr 21 2012

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
      1+2*add(`if`(issqr(i*n), 1, 0), i=1..n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 28 2023

t[n_] := t[n] = Flatten[Table[w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* Clark Kimberling, Apr 14 2012 *)

from sympy.ntheory.primetest import is_square
def A132188(n): return n+(sum(1 for x in range(1,n+1) for y in range(1,x) if is_square(x*y))<<1) # Chai Wah Wu, Aug 28 2023

a(n) = Sum [sqrt(n/k)]^2, where the sum is over all squarefree k not exceeding n.
If we call A120486, this sequence and A132189 F(n), P(n) and S(n), respectively, then P(n) = 2 F(n) - n = S(n) + n. The Finch-Sebah paper cited at A000188 proves that F(n) is asymptotic to (3 / pi^2) n log n. In the reference, we prove that F(n) = (3 / pi^2) n log n + O(n), from which it follows that P(n) = (6 / pi^2) n log n + O(n) and similarly for S(n).
a(n) = Sum_{1 <=x,y <=n} A010052(x*y). - Clark Kimberling, Apr 14 2012
a(n) = n+2*Sum_{1<=xA010052(x*y). - Chai Wah Wu, Aug 28 2023

A057918 Number of pairs of numbers (r,s) each less than n such that (r,s,n) is in geometric progression.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 3, 6, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 7, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 4, 1, 0, 0, 0, 3, 8, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 6, 2, 9, 0, 0, 0, 1, 0

Offset: 1

Henry Bottomley, Nov 22 2000

Also, the number of integers k in {1,2,...,n-1} such that k*n is square. - John W. Layman, Sep 08 2011

			a(72)=5 since (2,12,72), (8,24,72), (18,36,72), (32,48,72), (50,60,72) are the possible three term geometric progressions.

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

Cf. A000188, A005117, A133466.

Cf. A132345 (partial sums).

a057918 n = sum $ map ((0 ^) . (`mod` n) . (^ 2)) [1..n-1]
-- Reinhard Zumkeller, Mar 27 2012

a(n) = A000188(n) - 1.

a(A005117(n)) = 0; a(A013929(n)) > 0; A008966(n) = A000007(a(n)); a(A133466(n)) = 1; a(A195085(n)) = 2. - Reinhard Zumkeller, Mar 27 2012

A120486 Partial sums of A000188.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, 28, 29, 31, 32, 33, 34, 36, 41, 42, 45, 47, 48, 49, 50, 54, 55, 56, 57, 63, 64, 65, 66, 68, 69, 70, 71, 73, 76, 77, 78, 82, 89, 94, 95, 97, 98, 101, 102, 104, 105, 106, 107, 109, 110, 111, 114, 122, 123, 124, 125, 127, 128

Offset: 1

Gerry Myerson, Nov 21 2007

nonn

This sequence can also be described as the number of 3-term nondecreasing geometric progressions with no term exceeding n.

a(n) = A132188(n) - A132345(n). - Reinhard Zumkeller, Apr 21 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 35 (3) 2008, p 189-194.

Cf. A000188, A132188, A132189, A132345, A010052.

a120486 n = a120486_list !! (n - 1)
a120486_list = scanl1 (+) a000188_list
-- Reinhard Zumkeller, Apr 22 2012

with(numtheory): seq(add(phi(k)*floor(n/k^2), k=1..floor(sqrt(n))), n=1..100); # Ridouane Oudra, Aug 18 2019

a(n) = 3n log(n) / Pi^2 + O(n). - Griffin N. Macris, Jan 28 2017
a(n) ~ 3*n*((log(n) + (3*gamma - 1))/ Pi^2 - 12*(Zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{k=1..floor(sqrt(n))} phi(k)*floor(n/k^2), where phi is the Euler totient function A000010. - Ridouane Oudra, Aug 18 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 26 2021
From Ridouane Oudra, Oct 05 2024: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..i} A010052(i*j).
a(n) = A132345(n) + n.
a(n) = (1/2)*A132189(n) + n.
a(n) = (1/2)*(A132188(n) + n). (End)

A132189 Number of non-constant 3-term geometric progressions with no term exceeding n.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 2, 4, 8, 8, 8, 10, 10, 10, 10, 16, 16, 20, 20, 22, 22, 22, 22, 24, 32, 32, 36, 38, 38, 38, 38, 44, 44, 44, 44, 54, 54, 54, 54, 56, 56, 56, 56, 58, 62, 62, 62, 68, 80, 88, 88, 90, 90, 94, 94, 96, 96, 96, 96, 98, 98, 98, 102, 116, 116, 116, 116, 118, 118, 118

Offset: 1

Gerry Myerson, Nov 21 2007

nonn

In racetrack language, this is the number of trifectas in geometric progression in an n-horse race.

It appears that geometric progression like (k,0,0) are excluded. - Stefan Steinerberger, Nov 24 2007

Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, Volume 35 Number 3 July 2008 pp. 189-194.

Cf. A132345.

a = {}; For[n = 1, n < 80, n++, c = 0; For[j = 1, j < n + 1, j++, For[h = 1, h < n + 1, h++, If[Not[h == j], If[IntegerQ[j*(h/j)^2], If[j*(h/j)^2 < n + 1, c++ ]]]]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Nov 24 2007 *)

a(n) = 2 * A132345(n). - Alois P. Heinz, Jun 08 2025

More terms from Stefan Steinerberger, Nov 24 2007

A339026 Number of pairs (x,y): 1 <= x < y <= nn, such that xy is a square.

Original entry on oeis.org

0, 1, 4, 8, 16, 27, 40, 58, 80, 105, 128, 158, 198, 237, 278, 336, 382, 435, 502, 574, 632, 699, 798, 868, 968, 1063, 1176, 1262, 1362, 1505, 1590, 1732, 1880, 2007, 2150, 2272, 2434, 2567, 2746, 2930, 3062, 3265, 3444, 3654, 3832, 4035, 4284, 4454, 4668, 4875, 5142, 5364, 5590, 5889, 6078, 6336, 6614, 6881, 7166

Offset: 1

Edward Krogius, Nov 19 2020

nonn

			For n = 3, we have the following solutions: (1,4), (1,9), (2,8), (4,9), therefore a(3) = 4.
For n = 4, we have the following solutions: (1,4), (1,9), (1,16), (2,8), (3,12), (4,9), (4,16), (9,16), therefore a(4) = 8.

Edward Krogius, Table of n, a(n) for n = 1..1000
Edward Krogius, Illustration of 105 solutions in 100x100 grid

Cf. A000010, A015614, A018805, A132188, A132189, A132345, A338894.

Array[Sum[EulerPhi[j] Floor[(#^2)/(j^2)], {j, 2, #}] &, 59] (* Michael De Vlieger, Dec 11 2020 *)

A339026(n) = sum(i=2,n,floor(n^2/i^2)*eulerphi(i)); \\ Antti Karttunen, Nov 23 2020

a(n) = Sum_{j=2..n} phi(j) * floor(n^2/j^2).
a(n) = (A338894(n) - n^2)/2.
a(n) = A132189(n^2)/2. - Antti Karttunen, Nov 23 2020

A365047 a(n) is the number of three-term geometric progressions, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k) and a(n-1-2k), where k >= 1 and n - 1 - 2k >= 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 4, 2, 0, 0, 4, 1, 0, 1, 0, 2, 1, 0, 3, 0, 5, 0, 4, 1, 0, 2, 0, 2, 0, 5, 0, 4, 1, 3, 0, 4, 1, 1, 1, 2, 1, 4, 2, 0, 4, 1, 0, 3, 0, 3, 0, 2, 2, 1, 4, 0, 5, 0, 3, 0, 6, 0, 3, 1, 3, 0, 5, 0, 6, 0, 5, 0, 6, 0, 6, 0, 8, 0, 8, 0, 9, 1, 2, 1, 1, 2

Offset: 0

Scott R. Shannon, Oct 21 2023

nonn

The sequence is dominated by the count of three-term progressions consisting of three 0's. The 0 terms alternate between long runs on the even and odd n values, so the larger nonzero terms alternate between counting the progressions on these two subsequences, leading to two interrupted lines on the graph of the terms, along with the much lower counts of other three-term subsequences. See the attached image.

			a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
a(8) = 1 as a(7) = a(5) = a(3) = 1 for a progression with ratio 1 separated by two terms.
a(12) = 2 as a(11) = a(8) = a(5) = 1 form a progression with ratio 1 separated by three terms, while a(11) = a(7) = a(3) = 1 form a progression with ratio 1 separated by four terms.
a(20) = 2 as a(19) = 4, a(15) = 2, a(11) = 1 form a progression with ratio 1/2 separated by four terms, while a(19) = 4, a(12) = 2, a(5) = 1  form a progression with ratio 1/2 separated by seven terms.
a(170) = 1 as a(169) = 16, a(131) = 12, a(93) = 9 form a progression with ratio 3/4 separated by thirty-eight terms. This is the first series with a ratio that is not an integer or an integer reciprocal.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 50000 terms.

Cf. A366907 (length>=3), A132345, A365677, A308638, A078651, A051336.

A366472 Irregular triangle read by rows: T(n,k) (n >= 1, k >= 1) = number of increasing geometric progressions in {1,2,3,...,n} of length k with rational ratio.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 6, 1, 5, 10, 1, 6, 15, 1, 7, 21, 1, 8, 28, 2, 1, 9, 36, 4, 1, 10, 45, 4, 1, 11, 55, 4, 1, 12, 66, 5, 1, 13, 78, 5, 1, 14, 91, 5, 1, 15, 105, 5, 1, 16, 120, 8, 2, 1, 17, 136, 8, 2, 1, 18, 153, 10, 2, 1, 19, 171, 10, 2, 1, 20, 190, 11, 2, 1, 21, 210, 11, 2, 1, 22, 231, 11, 2, 1, 23, 253, 11, 2, 1, 24, 276, 12, 3, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 23 2023

			Triangle begins:
  [1],
  [2, 1],
  [3, 3],
  [4, 6, 1],
  [5, 10, 1],
  [6, 15, 1],
  [7, 21, 1],
  [8, 28, 2, 1],
  [9, 36, 4, 1],
  [10, 45, 4, 1],
  [11, 55, 4, 1],
  [12, 66, 5, 1],
  [13, 78, 5, 1],
  [14, 91, 5, 1],
  [15, 105, 5, 1],
  [16, 120, 8, 2, 1],
...

Row sums give A366471.
First three columns are A000027, A000217, A132345.
Cf. A000010.

with(numtheory);
A366472 := proc(n) local v,u2,u1,k,i,p;
v := Array(1..100, 0);
v[1] := n;
u1 := 1+floor(log(n)/log(2));
for k from 2 to u1 do
   u2 := floor(n^(1/(k-1)));
   v[k] := add(phi(p)*floor(n/p^(k-1)),p=2..u2);
  od;
[seq(v[i],i=1..u1)];
end;
for n from 1 to 36 do lprint(A366472(n)); od:

T(n,k) = Sum_{p=2..floor(n^(1/(k-1)))} phi(p)*floor(n/p^(k-1)) where phi is the Euler phi-function A000010 and k runs from 1 to 1+floor(log_2(n)).

A366907 a(n) is the number of geometric progressions with three or more terms, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k), a(n-1-2k),...,a(n-1-tk) where k>=1, t>=2, and n-1-t*k>=0.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 1, 0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 6, 0, 7, 0, 9, 0, 13, 0, 12, 0, 15, 0, 21, 0, 20, 0, 22, 0, 30, 0, 30, 0, 31, 0, 38, 0, 39, 0, 43, 0, 47, 0, 46, 0, 53, 0, 61, 0, 57, 0, 59, 0, 69, 0, 72, 0, 72, 0, 78, 0, 79, 0, 84, 0, 91, 0, 90, 0, 96, 0, 103, 0, 98, 0, 105, 0, 116

Offset: 0

Scott R. Shannon, Oct 27 2023

nonn

The sequence is dominated by the count of progressions consisting of three or more 0's. Very rarely the count of these zero-progressions forms a new progression of its own, which forms a short series of small terms and resets the subsequent count of the zero-progressions to a lower value. In the first 10^5 terms this only happens three times - at a(10) (which is not readily noticeable on the graph of the terms), a(644), and a(61434). See the attached images.

			a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
a(7) = 2 as a(6) = a(4) = a(2) = 0 form a three-term progression with ratio 1 separated by two terms, while a(6) = a(4) = a(2) = a(0) = 0 form a four-term progression with ratio 1 separated by two terms.
a(10) = 1 as a(9) = 4, a(7) = 2, a(5) = 1 form a three-term progression with ratio 1/2 separated by two terms.

Scott R. Shannon, Table of n, a(n) for n = 0..10000.
Scott R. Shannon, Image of the first 1000 terms.
Scott R. Shannon, Image of the first 100000 terms.

Cf. A365047 (length=3), A132345, A365677, A308638, A078651, A051336.

A280844 Number of 2 X 2 matrices with entries in {-n,..,0,..,n} with no entries repeated having permanent = trace^n.

Original entry on oeis.org

0, 0, 0, 4, 16, 20, 16, 36, 56, 60, 72, 76, 80, 100, 112, 100, 136, 124, 152, 172, 192, 196, 224, 196, 232, 236, 264, 252, 288, 276, 288, 308, 344, 332, 344, 332, 384, 388, 416, 404, 456, 428, 456, 444, 496, 468, 512, 468, 536, 556, 648

Offset: 0

Indranil Ghosh, Jan 09 2017

nonn

a(n) is also equal to the number of 2 X 2 matrices with entries in {-n,..,0,..n} with no elements repeated having determinant = trace^n except a(4). For permanent = trace^n, a(4) = 16 but for determinant = trace^n, a(4) = 24.
a(n) mod 4 = 0.
All solutions have trace -1, 0, or 1. The number of solutions with trace 0 is 8 * A132345(n), and the number of solutions with trace -1 is equal to the number of solutions with trace 1 when n is even. - David Radcliffe, Jun 08 2025

			For n = 5, the possible matrices are [-3,-5,-1,2], [-3,-1,-5,2],
[-3,1,5,2], [-3,5,1,2], [-2,-4,-1,2], [-2,-1,-4,2], [-2,1,4,2], [-2,4,1,2], [-1,1,3,2], [-1,3,1,2], [2,-5,-1,-3], [2,-4,-1,-2],
[2,-1,-5,-3], [2,-1,-4,-2], [2,1,3,-1], [2,1,4,-2], [2,1,5,-3],
[2,3,1,-1], [2,4,1,-2] and [2,5,1,-3].
Here each of the matrices is defined as M = [a,b,c,d] where a = M[1][1], b = M[1][2], c = M[2][1], d = M[2][2]. There are 20 possibilities. So, for n = 5, a(n) = 20.

David Radcliffe, Table of n, a(n) for n = 0..10000 (terms 0..103 from Indranil Ghosh).

Cf. A278933, A278902, A279018, A132345

def t(n):
    s=0
    for a in range(-n,n+1):
        for b in range(-n,n+1):
            if a!=b:
                for c in range(-n,n+1):
                    if a!=c and b!=c:
                        for d in range(-n,n+1):
                            if d!=a and d!=b and d!=c:
                                if (a*d+b*c)==(a+d)**n:
                                    s+=1
    return s
for i in range(0,24):
    print(t(i), end=', ')

cp's OEIS Frontend

A078147 First differences of sequence of nonsquarefree numbers, A013929.

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A132188 Number of 3-term geometric progressions with no term exceeding n.

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A057918 Number of pairs of numbers (r,s) each less than n such that (r,s,n) is in geometric progression.

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A120486 Partial sums of A000188.

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A132189 Number of non-constant 3-term geometric progressions with no term exceeding n.

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A339026 Number of pairs (x,y): 1 <= x < y <= n*n, such that x*y is a square.

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A365047 a(n) is the number of three-term geometric progressions, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k) and a(n-1-2*k), where k >= 1 and n - 1 - 2*k >= 0.

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A339026 Number of pairs (x,y): 1 <= x < y <= nn, such that xy is a square.

A365047 a(n) is the number of three-term geometric progressions, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k) and a(n-1-2k), where k >= 1 and n - 1 - 2k >= 0.

A366907 a(n) is the number of geometric progressions with three or more terms, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k), a(n-1-2k),...,a(n-1-tk) where k>=1, t>=2, and n-1-t*k>=0.