A016017 -id:A016017 - cp's OEIS frontend

A016013 Probably an erroneous version of A016017.

Original entry on oeis.org

2, 1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1, 2

Offset: 1

dead

A061284 Duplicate of A016017.

Original entry on oeis.org

1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192

Offset: 1

dead

A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 7, 5, 9, 3, 15, 3, 9, 9, 9, 3, 15, 3, 15, 9, 9, 3, 21, 5, 9, 7, 15, 3, 27, 3, 11, 9, 9, 9, 25, 3, 9, 9, 21, 3, 27, 3, 15, 15, 9, 3, 27, 5, 15, 9, 15, 3, 21, 9, 21, 9, 9, 3, 45, 3, 9, 15, 13, 9, 27, 3, 15, 9, 27, 3, 35, 3, 9, 15, 15, 9, 27, 3, 27

Offset: 1

David Johnson-Davies

Inverse Moebius transform of A034444: Sum_{d|n} 2^omega(d), where omega(n) = A001221(n) is the number of distinct primes dividing n.
Number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1}.
Number of elements in the set {(x,y): lcm(x,y)=n}.
Also gives total number of positive integral solutions (x,y), order being taken into account, to the optical or parallel resistor equation 1/x + 1/y = 1/n. Indeed, writing the latter as X*Y=N, with X=x-n, Y=y-n, N=n^2, the one-to-one correspondence between solutions (X, Y) and (x, y) is obvious, so that clearly, the solution pairs (x, y) are tau(N)=tau(n^2) in number. - Lekraj Beedassy, May 31 2002
Number of ordered pairs of positive integers (a,c) such that n^2 - ac = 0. Therefore number of quadratic equations of the form ax^2 + 2nx + c = 0 where a,n,c are positive integers and each equation has two equal (rational) roots, -n/a. (If a and c are positive integers, but, instead, the coefficient of x is odd, it is impossible for the equation to have equal roots.) - Rick L. Shepherd, Jun 19 2005
Problem A1 on the 21st Putnam competition in 1960 (see John Scholes link) asked for the number of pairs of positive integers (x,y) such that xy/(x+y) = n: the answer is a(n); for n = 4, the a(4) = 5 solutions (x,y) are (5,20), (6,12), (8,8), (12,6), (20,5). - Bernard Schott, Feb 12 2023
Numbers k such that a(k)/d(k) is an integer are in A217584 and the corresponding quotients are in A339055. - Bernard Schott, Feb 15 2023

A. M. Gleason et al., The William Lowell Putnam Mathematical Competitions, Problems & Solutions:1938-1960 Soln. to Prob. 1 1960, p. 516, MAA, 1980.
Ross Honsberger, More Mathematical Morsels, Morsel 43, pp. 232-3, DMA No. 10 MAA, 1991.
Loren C. Larson, Problem-Solving Through Problems, Prob. 3.3.7, p. 102, Springer 1983.
Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Algebra, Prob. 9-9 pp. 143 Dover NY, 1988.
D. O. Shklarsky et al., The USSR Olympiad Problem Book, Soln. to Prob. 123, pp. 28, 217-8, Dover NY.
Wacław Sierpiński, Elementary Theory of Numbers, pp. 71-2, Elsevier, North Holland, 1988.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.
Charles W. Trigg, Mathematical Quickies, Question 194, pp. 53, 168, Dover, 1985.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
Ovidiu Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., Vol. 6 (2003), Article 03.1.6.
John Scholes, Problem A1 of 21st Putnam Competition 1960, 2002. [Wayback Machine link]
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A000005, A034444, Mobius transf. of A035116, A048785, A061391, A018805, A002088, A015614, A018892, A063647, A182139.
Partial sums give A061503.
For similar LCM sequences, see A070919, A070920, A070921.
Cf. A005408, A124010.
For the earliest occurrence of 2n-1 see A016017.
Cf. A001620, A082633, A217584, A339055.

a048691 = product . map (a005408 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Jul 12 2012

A048691[n_]:=DivisorSigma[0,n^2] (* Enrique Pérez Herrero, May 30 2010 *)
DivisorSigma[0,Range[80]^2] (* Harvey P. Dale, Apr 08 2015 *)

numlib::tau (n^2)$ n=1..90 // Zerinvary Lajos, May 13 2008

A048691(n)=prod(i=1,#n=factor(n)[,2],n[i]*2+1) /* or, of course, a(n)=numdiv(n^2) */ \\ M. F. Hasler, Dec 30 2007

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 21 2021

[sigma(n**2, 0) for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

a(n) = A000005(A000290(n)).
tau(n^2) = Sum_{d|n} mu(n/d)*tau(d)^2, where mu(n) = A008683(n), cf. A061391.
Multiplicative with a(p^e) = 2e+1. - Vladeta Jovovic, Jul 23 2001
Also a(n) = Sum_{d|n} (tau(d)*moebius(n/d)^2), Dirichlet convolution of A000005 and A008966. - Benoit Cloitre, Sep 08 2002
a(n) = A055205(n) + A000005(n). - Reinhard Zumkeller, Dec 08 2009
Dirichlet g.f.: (zeta(s))^3/zeta(2s). - R. J. Mathar, Feb 11 2011
a(n) = Sum_{d|n} 2^omega(d). Inverse Mobius transform of A034444. - Enrique Pérez Herrero, Apr 14 2012
G.f.: Sum_{k>=1} 2^omega(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018
Sum_{k=1..n} a(k) ~ n*(6/Pi^2)*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

Additional comments from Vladeta Jovovic, Apr 29 2001

A061283 Smallest number with exactly 2n-1 divisors.

Original entry on oeis.org

1, 4, 16, 64, 36, 1024, 4096, 144, 65536, 262144, 576, 4194304, 1296, 900, 268435456, 1073741824, 9216, 5184, 68719476736, 36864, 1099511627776, 4398046511104, 3600, 70368744177664, 46656, 589824, 4503599627370496, 82944

Offset: 1

Labos Elemer, May 22 2001

nonn

The terms are always squares (because the divisors of a nonsquare N come in pairs, d and N/d, and so their number is always even - N. J. A. Sloane, Dec 26 2018).

			For n=15, a(15)=144 with 15 divisors: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.

Amiram Eldar, Table of n, a(n) for n = 1..1661 (terms 1..1000 from T. D. Noe)

Cf. A000005, A000290, A005408, A003680, A016017, A037992, A055079, A048691.

Second bisection of A005179.

mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n-1] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)

a(n) = Min{k | A000005(k)=2n-1}.
a((p+1)/2) = 2^(p-1) for odd prime p. [Corrected by Jianing Song, Aug 30 2021]
From Jianing Song, Aug 30 2021: (Start)
a(n) = A016017(n)^2.
a(n) <= 2^(2n-2), where the equality holds if and only if n=1 or 2n-1 is prime. (End)

A015995 a(n) = (tau(n^3)+2)/3.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2, 18, 5, 6, 2, 38, 6, 6

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A004194, A018892, A015996, A015999, A016001, A016002, A016003, A016005-A016009, A016012, A016017, A016018, A016020, A016025, A048785.

Table[(DivisorSigma[0,n^3]+2)/3,{n,90}] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=(numdiv(n^3)+2)/3 \\ Charles R Greathouse IV, May 01 2013

a(n)=my(f=factor(n)[,2]);prod(i=1,#f,3*f[i]+1)\3+1 \\ Charles R Greathouse IV, May 01 2013

a(n) = (2+A048785(n))/3. - R. J. Mathar, May 07 2021

Definition corrected by Vladeta Jovovic, Sep 03 2005

A071571 Smallest number whose square has exactly 2n+1 divisors.

Original entry on oeis.org

1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, 1073741824, 120, 576, 8589934592, 6144, 34359738368, 68719476736, 180, 864

Offset: 0

Lekraj Beedassy, May 31 2002

nonn

Only squares have an odd number of divisors.

			a(4)=6 because it is the smallest number followed by 10,14,15,16,21,22,... whose squares have 2*4 + 1, i.e., 9 divisors.

Amiram Eldar, Table of n, a(n) for n = 0..3325

Cf. A005179.

Identical to A016017 shifted left.

a(n) = {k = 1; while (numdiv(k^2) != (2*n+1), k++); return (k);} \\ Michel Marcus, Jul 27 2013

a(n) <= 2^n, where the equality holds if and only if n=0 or 2n+1 is prime. - Jianing Song, Aug 30 2021

More terms from Vladeta Jovovic, Jun 05 2002

a(0) prepended by Jianing Song, Aug 30 2021

A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 5, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 5, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 2, 3

Offset: 1

Lars Blomberg, Feb 14 2022

nonn

By symmetry, if (i, j) is a solution then so is (j, i). When j=i we get n = 3i, corresponding to the solution 1/2 + 1/2 = 1. Therefore, when 3|n, a(n) > 0 and odd, otherwise a(n) >= 0 and even.
For n < 10^6, the largest term is a(720720) = 285, and 483188 terms are 0.
Other record terms: a(1081080) = 369, a(2162160) = 457, a(3243240) = 481, a(4324320) = 533, a(5405400) = 559, a(6126120) = 597. Record terms with index >= 360360 appear to occur at indices that are multiples of 180180. - Chai Wah Wu, Feb 15 2022

			For n = 3: (i, j) = (1, 1), so a(3) = 1. (1/2 + 1/2 = 1)
For n = 18: (i, j) = (3, 8), (6, 6), (8, 3), so a(18) = 3. (3/15 + 8/10 = 1/5 + 4/5 = 1)
For n = 20: (i, j) = (5, 8), (8, 5), so a(20) = 2.
For n = 36: (i, j) = (6, 16), (8, 15), (12, 12), (15, 8), (16, 6), so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A018892, A016017, A051908, A073546, A227610, A238795, A241883, A297895, A303400.

Cf. A351694, A351695 for records.

a(n)={my(x=n^2, y=2*n); sum(i=1,(n-1)/2, x-=2*n; y-=3; if(x%y==0,1,0))}

from sympy.abc import i, j
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A351532(n):
    return sum(1 for d in diop_quadratic(n**2+3*i*j-2*n*(i+j)) if 0 < d[0] < n and 0 < d[1] < n) # Chai Wah Wu, Feb 15 2022

The original equation can be solved for j giving j = (n(n - 2i)) / (2n - 3i). Varying i from 1 to n-1, a(n) is given by the number of integer values of j, 0 < j < n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A016018 Least k such that (tau(k^3)+2)/3=n.

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 64, 128, 256, 12, 1024, 2048, 4096, 24, 16384, 32768, 36, 48, 262144, 524288, 1048576, 30, 4194304, 72, 16777216, 192, 67108864, 134217728, 268435456, 384, 144, 2147483648, 4294967296, 216, 17179869184, 34359738368, 68719476736

Offset: 0

Robert G. Wilson v

nonn

Cf. A015995, A015996, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016012, A016017, A016020 & A016025

More terms from Robert G. Wilson v, Aug 06 2005
Definition corrected by Vladeta Jovovic, Sep 03 2005
Extended by Ray Chandler, Sep 05 2008

A061708 Smallest number whose square has (2n - 1)^2 divisors.

Original entry on oeis.org

1, 6, 36, 216, 210, 7776, 46656, 1260, 1679616, 10077696, 7560, 362797056, 44100, 18480, 78364164096, 470184984576, 272160, 264600, 101559956668416, 1632960, 3656158440062976, 21936950640377856, 180180, 789730223053602816, 9261000, 58786560, 170581728179578208256

Offset: 1

Labos Elemer, Jun 19 2001

nonn

a(n) <= 6^(n-1); 36^(n-1) has (2n-1)^2 divisors for all n.

			For n = 8, a(8) = 1260 = 2*2*3*3*5*7 and d(1260^2) = d(2*2*2*2*3*3*3*3*5*5*7*7) = 225 = (2*8-1)^2.
For n = 14, a(14) = 18480 and d((2*2*2*2*2*2*2*2*3*5*7*11)^2) = 729 = (2*14-1)^2.

Amiram Eldar, Table of n, a(n) for n = 1..68

Cf. A000005, A000290, A005179, A005408, A016017, A025281, A048691.

a(n) = Min_{x : d(x^2) = (2n-1)^2};

a(n) = Min_{x : A000005(A000290(x)) = A000290(A005408(n))}.

More terms from David Wasserman, Jun 24 2002
Edited by Charlie Neder, Jun 03 2019
a(26)-a(27) from Amiram Eldar, Dec 03 2023

A354530 Numbers k such that k^2 is a minimal number; numbers k whose square is in A007416.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 60, 64, 72, 96, 120, 180, 192, 210, 216, 256, 288, 360, 420, 480, 512, 576, 768, 840, 864, 900, 960, 1080, 1260, 1440, 1536, 1680, 1728, 1800, 2048, 2304, 2520, 2880, 3360, 3840, 4320, 4608, 4620, 5400, 6144, 6300, 6720, 6912, 7200, 7560

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that there is no m < k^2 such that d(m) = d(k^2), d = A000005. Since only squares have an odd number of divisors, also numbers k such that there is no m < k such that d(m^2) = d(k^2).

			8 is a term since 8^2 = 64 has 7 divisors and no smaller number (smaller square) has that many divisors.

David A. Corneth, Table of n, a(n) for n = 1..14008 (first 310 terms from Jianing Song, terms <= 10^20).

Cf. A007416, A000005, A025487.
Square root of A166721. Also A016017 or A071571 sorted.
Cf. also A166722.

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n, ", "); v = Set(concat(v, d))); ); } \\ from Michel Marcus's program for A166721

d(a(n)^2) = A166722(n).

cp's OEIS Frontend

A016013 Probably an erroneous version of A016017.

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A061284 Duplicate of A016017.

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A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

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A061283 Smallest number with exactly 2n-1 divisors.

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A015995 a(n) = (tau(n^3)+2)/3.

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A071571 Smallest number whose square has exactly 2n+1 divisors.

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A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

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A016018 Least k such that (tau(k^3)+2)/3=n.

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