A062011 -id:A062011 - cp's OEIS frontend

A054584 Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.

2, 4, 6, 6, 4, 12, 4, 8, 10, 8, 4, 18, 4, 8, 12, 10, 4, 20, 4, 12, 12, 8, 4, 24, 6, 8, 14, 12, 4, 24, 4, 12, 12, 8, 8, 30, 4, 8, 12, 16, 4, 24, 4, 12, 20, 8, 4, 30, 6, 12, 12, 12, 4, 28, 8, 16, 12, 8, 4, 36, 4, 8, 20, 14, 8, 24, 4, 12, 12, 16, 4, 40, 4, 8, 18, 12, 8, 24, 4, 20, 18, 8, 4

Offset: 1

John W. Layman, Apr 12 2000

Also the number of subgroups of the group C_n X C_3 (where C_n is the cyclic group of order n). Number of subgroups of the group C_n X C_m is Sum_{i|n,j|m} gcd(i,j).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Hampejs, N. Holighaus, L. Tóth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1211.1797 [math.GR], 2012-2014. - From _N. J. A. Sloane_, Jan 02 2013

Cf. A060710, A060724, A062011, A060648, A035191, A000005.
Cf. A001620, A079978, A051176.
A row of A216624.

a054584 n = a000005 n + 3 * a079978 n * a000005 (a051176 n) + a035191 n
-- Reinhard Zumkeller, Aug 27 2012

for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 2*e+1:else b := e+1:fi:s := s*b:od:printf(`%d,`,2*s); od:

f[d_ /; Mod[d, 3] == 0] = 4; f[] = 2; a[n] := Total[f /@ Divisors[n]]; Table[a[n], {n, 1, 100}](* Jean-François Alcover, Nov 21 2011, after Michael Somos *)
f[p_, e_] := e + 1; f[3, e_] := 2*e + 1; a[1] = 2; a[n_] := 2*Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

a(n)=if(n<1, 0, sumdiv(n,d, (d%3==0)*2+2)) /* Michael Somos, Sep 20 2005 */

a(n) = tau(n)+3*tau(n/3)+A035191(n) if n is congruent to 0 mod 3 else tau(n)+A035191(n), where A035191(n) is the number of divisors of n that are not congruent to 0 mod 3.
a(n)/2 is multiplicative with a(3^e)=2e+1 and a(p^e)=e+1 for p<>3.
Moebius transform is period 3 sequence [2, 2, 4, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k(2+2*x^k+4*x^(2k))/(1-x^(3k)).
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: 2 * zeta(s)^2 * (1 + 1/3^s).
Sum_{k=1..n} a(k) ~ 2*(4*n*log(n) + (8*gamma - 4 - log(3))*n)/3, where gamma is Euler's constant (A001620). (End)

Additional comments from Vladeta Jovovic, Oct 25 2001

A087560 Smallest m > n such that gcd(m, n^2) = n.

Original entry on oeis.org

2, 6, 6, 12, 10, 30, 14, 24, 18, 30, 22, 60, 26, 42, 30, 48, 34, 90, 38, 60, 42, 66, 46, 120, 50, 78, 54, 84, 58, 210, 62, 96, 66, 102, 70, 180, 74, 114, 78, 120, 82, 210, 86, 132, 90, 138, 94, 240, 98, 150, 102, 156, 106, 270, 110, 168, 114, 174, 118, 420, 122

Offset: 1

Reinhard Zumkeller, Oct 24 2003

nonn

Equals n multiplied by the least nontrivial number coprime to n. - Amarnath Murthy, Nov 20 2005

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A053669, A062011, A119416, A249270.

Table[n*Select[Prime[Range[Log2[n] + 1]], ! Divisible[n, #] &][[1]], {n, 61}] (* Ivan Neretin, May 21 2015 *)

a(n) = forprime(p = 2, , if(n%p, return(n*p))); \\ Amiram Eldar, Feb 01 2025

a(n) = n*A053669(n).
A000005(a(n)) = 2*A000005(n) = A062011(n). - Reinhard Zumkeller, May 17 2006
Sum_{k=1..n} ~ c * n^2 / 2, where c = A249270. - Amiram Eldar, Feb 01 2025

A119416 n * (smallest prime greater than largest prime factor of n).

Original entry on oeis.org

2, 6, 15, 12, 35, 30, 77, 24, 45, 70, 143, 60, 221, 154, 105, 48, 323, 90, 437, 140, 231, 286, 667, 120, 175, 442, 135, 308, 899, 210, 1147, 96, 429, 646, 385, 180, 1517, 874, 663, 280, 1763, 462, 2021, 572, 315, 1334, 2491, 240, 539, 350, 969, 884, 3127, 270

Offset: 1

Reinhard Zumkeller, May 17 2006

nonn

a(n) = n * A117366(n);

A000005(a(n)) = 2*A000005(n) = A062011(n).

			a(10) = (2*5)*7.

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

Cf. A006530, A087560.

Cf. A151800.

a119416 n = n * (a151800 $ a006530 n) -- Reinhard Zumkeller, Apr 06 2015

Insert[Table[n*Prime[PrimePi[FactorInteger[n][[ -1]][[1]]] + 1],{n,2,100}], 2, 1] (* Stefan Steinerberger, May 18 2006 *)

a(n) = n * A151800(A006530(n)). - Reinhard Zumkeller, Apr 06 2015

A152771 a(n) = sigma(n) - 2*d(n) + 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 8, 8, 11, 9, 17, 11, 17, 17, 22, 15, 28, 17, 31, 25, 29, 21, 45, 26, 35, 33, 45, 27, 57, 29, 52, 41, 47, 41, 74, 35, 53, 49, 75, 39, 81, 41, 73, 67, 65, 45, 105, 52, 82, 65, 87, 51, 105, 65, 105, 73, 83, 57, 145

Offset: 1

Omar E. Pol, Dec 14 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A152770, A152772.

Table[DivisorSigma[1, n] - 2 DivisorSigma[0, n] + 1, {n, 60}] (* Ivan Neretin, Sep 30 2017 *)

a(n) = sigma(n) - 2*numdiv(n) + 1; \\ Michel Marcus, Sep 30 2017

a(n) = A000203(n) - 2*A000005(n) + 1 = A000203(n) - A114003(n) = A088580(n) - A062011(n). - Omar E. Pol, Sep 30 2017

G.f.: Sum_{k>=1} x^(3*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Apr 24 2021

A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 3, 0, 2, 1, 2, 0, 2, 2, 2, 4, 0, 3, 2, 4, 2, 2, 0, 2, 3, 4, 4, 4, 4, 0, 4, 2, 6, 4, 8, 4, 3, 0, 2, 4, 4, 6, 8, 8, 7, 4, 0, 4, 2, 8, 4, 12, 8, 14, 8, 2, 0, 3, 4, 4, 8, 8, 12, 14, 16, 12, 6, 0, 4, 3, 8, 4, 16, 8, 21, 16, 24, 14, 2, 0, 2, 4, 6, 8, 8, 16, 14, 24, 24, 28, 21

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
1;
2, 0;
2, 0, 1;
3, 0, 2, 1;
2, 0, 2, 2, 2;
4, 0, 3, 2, 4, 2;
2, 0, 2, 3, 4, 4,  4;
4, 0, 4, 2, 6, 4,  8,  4;
3, 0, 2, 4, 4, 6,  8,  8,  7;
4, 0, 4, 2, 8, 4, 12,  8, 14,  8;
2, 0, 3, 4, 4, 8,  8, 12, 14, 16, 12;
6, 0, 4, 3, 8, 4, 16,  8, 21, 16, 24, 14;
2, 0, 2, 4, 6, 8,  8, 16, 14, 24, 24, 28, 21;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   4   =   4
2      0   *   2   =   0
3      1   *   3   =   3
4      1   *   2   =   2
5      2   *   2   =   4
6      2   *   1   =   2
.           A000005
--------------------------
The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.

Row sums give A138137 (conjectured).
Columns 1, 3 and 4 are A000005.
Column 2 gives A000004.
Columns 5 and 6 give A062011.
Columns 7 and 8 give A145154, n >= 1.
Leading diagonal gives A002865.
Cf. A339304 (irregular or expanded version).
Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.

f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
T(n, k) = numdiv(n-k+1) * f(k-1); \\ Michel Marcus, Jan 13 2021

A349410 Length of cycle reached when iterating the mapping x-> n*A000005(x) on 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 4, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2

Offset: 1

Tejo Vrush, Nov 16 2021

nonn

			For n = 9, 1 --> 9 --> 27 --> 36 --> 81 --> 45 --> 54 --> 72 --> 108 --> 108. The cycle reached has just one term: 108. Therefore, a(9) = 1.

Cf. A000005, A000040, A062011.

a[n_] := Module[{s = NestWhileList[n * DivisorSigma[0, #] &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

f(n, x) = n*numdiv(x);
find(nm, v) = {forstep (n=#v-1, 1, -1, if (v[#v] == v[n], return(#v-n);););}
a(n) = {my(list = List(), found=0, m=n); listput(list, m); while (! found, my(nm = f(n, m)); listput(list, nm); found = find(nm, list); m = nm;); found;} \\ Michel Marcus, Nov 17 2021

from sympy import divisor_count
terms = []
for n in range(1, 101):
    s, t = [1], True
    while t:
        for i in range(2, len(s)):
            if s[-i] == s[-1]:
                t = False
                terms.append(i - 1)
                break
        s.append(n*divisor_count(s[-1]))
print(terms) # Gleb Ivanov, Nov 17 2021

a(A000040(n)) = 1.

A146208 a(n) is the number of arithmetic progressions of 2 or more integers with product = n.

Original entry on oeis.org

4, 6, 6, 4, 10, 4, 11, 8, 10, 4, 12, 4, 8, 12, 12, 4, 12, 4, 12, 10, 8, 4, 26, 6, 8, 9, 14, 4, 16, 4, 13, 8, 8, 10, 20, 4, 8, 8, 20, 4, 18, 4, 12, 16, 8, 4, 26, 6, 12, 8, 12, 4, 16, 10, 16, 8, 8, 4, 26, 4, 8, 14, 19, 8, 18, 4, 12, 8, 16, 4, 24, 4, 8, 12

Offset: 2

Naohiro Nomoto, Oct 28 2008

a(n)=number of all integer triples (x,y,z) such that Product_{k=0..z} (x + (y*k)) = n, where n>1, z>0.

			a(8) = 11 as we can have
 (x=-8,y=7,z=1; -8 * -1),
 (x=-4,y=2,z=1; -4 * -2),
 (x=-4,y=3,z=2; -4 * -1 * 2),
 (x=-2,y=-2,z=1; -2 * -4),
 (x=-1,y=-7,z=1; -1 * -8),
 (x=1,y=7,z=1; 1 * 8),
 (x=2,y=-3,z=2; 2 * -1 * -4),
 (x=2,y=0,z=2; 2 * 2 * 2),
 (x=2,y=2,z=1; 2 * 4),
 (x=4,y=-2,z=1; 4 * 2),
 (x=8,y=-7,z=1; 8 * 1). - Example added by _Antti Karttunen_, Feb 28 2023
a(9) = 8 as we can have
 (x=-3,y=0,z=1; -3 * -3),
 (x=3,y=0,z=1; 3 * 3),
 (x=-9,y=8,z=1; -9 * -1),
 (x=1,y=8,z=1; 1 * 9),
 (x=-1,y=-8,z=1; -1 * -9),
 (x=9,y=-8,z=1; 9 * 1),
 (x=3,y=-2,z=3; 3 * 1 * -1 * -3),
 (x=-3,y=2,z=3; -3 * -1 * 1 * 3).

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A062011, A361015.

A146208(n) = sum(x=-n,n,sum(y=-n,n,sum(z=1,n,n==prod(k=0,z,x+(y*k))))); \\ (Slow!) - Antti Karttunen, Feb 28 2023

from sympy import divisors
def A146208(n):
    ds = divisors(n)
    c, s = 0, [-d for d in ds[::-1]]+ds
    for x in s:
        d2 = [d//x for d in ds if d%x==0]
        for y in (f-x for f in [-d for d in d2[::-1]]+d2):
            m, k = x*(z:=x+y), 1
            while n >= abs(m) and k<=n:
                if n == m:
                    c += 1
                z += y
                m *= z
                k += 1
    return c # Chai Wah Wu, May 11 2023

a(n) = A062011(n) + A361015(n). - Antti Karttunen, Feb 28 2023

A361015 Number of arithmetic progressions of 3 or more integers whose product is equal to n.

Original entry on oeis.org

0, 2, 0, 0, 2, 0, 3, 2, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 5, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20

Offset: 2

Antti Karttunen, Feb 28 2023

nonn

Number of all integer triples (x,y,z) such that Product_{k=0..z} (x + (y*k)) = n, where n > 1, z > 1.

			a(3) = 2 as we have solutions (x=-3,y=2,z=2; -3 * -1 * 1) and (x=1,y=-2,z=2; 1 * -1 * -3).
a(8) = 3 as we have solutions  (x=-4,y=3,z=2; -4 * -1 * 2), (x=2,y=-3,z=2; 2 * -1 * -4), and (x=2,y=0,z=2; 2*2*2).
a(27) = 1 as we have a unique solution (x=3,y=0,z=2; 3*3*3).
a(32) = 1 as we have a unique solution (x=2,y=0,z=4; 2*2*2*2*2).
a(64) = 5 as we have solutions (x=-8,y=6,z=2; -8 * -2 * 4), (x=-2,y=0,z=5; (-2)^6), (x=2,y=0,z=5; 2^6), (x=4,y=-6,z=2; 4 * -2 * -8), and (x=4,y=0,z=2; 4*4*4).
a(81) = 4 as we have solutions (x=-9,y=6,z=2; -9 * -3 * 3), (x=-3,y=0,z=3; -3 * -3 * -3 * -3), (x=3,y=-6,z=2; 3 * -3 * -9), and (x=3,y=0,z=3; 3*3*3*3).
a(300) = 2 as we have solutions (x=-25,y=13,z=2; -25 * -12 * 1) and (x=1,y=-13,z=2; 1 * -12 * -25).

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A062011, A146208.

A361015(n) = sum(x=-n,n,sum(y=-n,n,sum(z=2,n,n==prod(k=0,z,x+(y*k))))); \\ (Slow!)

from sympy import divisors
def A361015(n):
    ds = divisors(n)
    c, s = -len(ds)<<1, [-d for d in ds[::-1]]+ds
    for x in s:
        d2 = [d//x for d in ds if d%x==0]
        for y in (f-x for f in [-d for d in d2[::-1]]+d2):
            m, k = x*(z:=x+y), 1
            while n >= abs(m) and k<=n:
                if n == m:
                    c += 1
                z += y
                m *= z
                k += 1
    return c # Chai Wah Wu, May 11 2023

a(n) = A146208(n) - A062011(n).

A131089 a(n) = Sum_{d|n} (2 - mu(d)).

Original entry on oeis.org

1, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16

Offset: 1

Gary W. Adamson, Jun 14 2007

nonn

Apart from the first term the same as A062011. - Andrew Howroyd, Aug 09 2018

			a(4) = 6 = (2 + 3 + 0 + 1), row 4 of A131088.

Row sums of triangle A131088.

Cf. A062011, A051731, A131090, A228483.

[1] cat [2*NumberOfDivisors(n) : n in [2..100] ]; // Vincenzo Librandi, Aug 10 2018

Join[{1}, Table[2 DivisorSigma[0, n], {n, 100}]] (* Vincenzo Librandi, Aug 10 2018 *)

a(n)={2*numdiv(n) - (n==1)} \\ Andrew Howroyd, Aug 09 2018

a(n) = 2*tau(n) = A062011(n) for n > 1. - Andrew Howroyd, Aug 09 2018

Inverse Moebius transform of A228483.

Name changed and terms a(10) and beyond from Andrew Howroyd, Aug 09 2018

A307179 Numbers k such that k = ij = 6i + j, where i and j are integers.

Original entry on oeis.org

-25, -8, -3, 0, 24, 27, 32, 49

Offset: 1

Scott R. Shannon, Mar 27 2019

The sequence can be found by solving the equality i*j = 6*i + j. Re-arranging for j gives j = 6 + 6/(i-1). As both i and j must be integers this implies i - 1 must divide 6, thus the only values for i are -5,-2,-1,0,2,3,4,7. Finding the corresponding j and multiplying gives the 8 sequences values.
In general if we replace 6 by n, then the number of solutions will be 2*A000005(n), the lowest value will be -(n - 1)^2, and the highest value will be (n + 1)^2.
For values k>=0 this sequence gives the possible point scores in Australian Rules Football which equal the corresponding number of goals (worth six points) times the number of behinds (worth one point).
The number of solutions, in this case 8, is given by A062011(6). Robert G. Wilson v, Apr 10 2019

			The 8 solutions are:
--------------
i   j    k
--------------
-5   5   -25
-2   4   -8
-1   3   -3
0   0    0
2   12   24
3   9    27
4   8    32
7   7    49

Cf. A000005

cp's OEIS Frontend

A054584 Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.

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A087560 Smallest m > n such that gcd(m, n^2) = n.

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A119416 n * (smallest prime greater than largest prime factor of n).

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A152771 a(n) = sigma(n) - 2*d(n) + 1.

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A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

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A349410 Length of cycle reached when iterating the mapping x-> n*A000005(x) on 1.

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A146208 a(n) is the number of arithmetic progressions of 2 or more integers with product = n.

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A307179 Numbers k such that k = ij = 6i + j, where i and j are integers.