A048691 -id:A048691 - cp's OEIS frontend

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.

1, 5, 5, 14, 5, 25, 5, 30, 14, 25, 5, 70, 5, 25, 25, 55, 5, 70, 5, 70, 25, 25, 5, 150, 14, 25, 30, 70, 5, 125, 5, 91, 25, 25, 25, 196, 5, 25, 25, 150, 5, 125, 5, 70, 70, 25, 5, 275, 14, 70, 25, 70, 5, 150, 25, 150, 25, 25, 5, 350, 5, 25, 70, 140, 25, 125, 5, 70, 25, 125, 5

Offset: 1

Vladeta Jovovic, Jul 07 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.

A062367 := proc(n)
    add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
end proc:
seq(A062367(n),n=1..40) ; # R. J. Mathar, May 15 2025

{1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* Michael De Vlieger, Mar 05 2021 *)

a(n) = sumdiv(n, d, numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - R. J. Mathar, Feb 03 2011
G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - R. J. Mathar, Jun 05 2020
Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - Werner Schulte, Feb 07 2021
a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - Peter Bala, Jan 26 2024

A136404 Square numbers with more divisors than any smaller square number.

Original entry on oeis.org

1, 4, 16, 36, 144, 576, 900, 3600, 14400, 32400, 44100, 129600, 176400, 705600, 1587600, 2822400, 6350400, 21344400, 57153600, 85377600, 192099600, 341510400, 768398400, 3073593600, 6915585600, 12294374400, 14428814400, 32464832400, 57715257600, 129859329600

Offset: 1

J. Lowell, Mar 30 2008

nonn

Being the square of a number in A002182 is neither necessary nor sufficient.
Conjecture: square roots of the terms of this sequence are the same terms as A126098.
These are the squares of the indices of the RECORDS of A048691. - R. J. Mathar, Apr 04 2008
The square roots of terms are in A025487. - David A. Corneth, Oct 17 2018
From David A. Corneth, Oct 18 2018: (Start)
Records for largest exponents occur at: 1, 2, 5, 15, 25, 35, 200, 203
Least k such that a(k) divides prime(i)^4: 5, 10, 34, 104, 302
Based on these exponents I made the following dataset:
primorials <= 10^200 (92 such numbers).
Then made products of primorials <= 10^200 where the exponent of prime(11) is at most 2. Then searched records here. The b-file is checking A025487 squared checked. (End)

			900 qualifies because 576 has only 21 divisors and 900 has 27. 1296 does not because 1296 has only 25 divisors as opposed to the 27 of the smaller 900.

Ray Chandler, Table of n, a(n) for n = 1..582 (first 78 terms from Donovan Johnson, terms to 320 from David A. Corneth)
David A. Corneth, Conjectured first 1350 terms

Cf. A002182, A025487.

a := 0 : for n from 1 to 1000000 do ndvs := numtheory[tau](n^2) ; if ndvs > a then printf("%d,",n^2) ; a := ndvs ; fi ; od: # R. J. Mathar, Apr 04 2008
with(numtheory): a:=proc(n) if max(seq(tau(j^2),j=1..n-1))Emeric Deutsch, Apr 04 2008

With[{s = Array[DivisorSigma[0, #^2] &, 10^6]}, Map[FirstPosition[s, #][[1]]^2 &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2018 *)

More terms from R. J. Mathar and Donovan Johnson, Apr 04 2008

A147809 Half the number of proper divisors (> 1) of n^2 + 1, i.e., tau(n^2 + 1)/2 - 1.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Dec 13 2008

For any n > 0, n^2 + 1 cannot be a square and thus has an even number of divisors which always include 1 and n^2 + 1, therefore a(n) = (half that number minus 1) is always a nonnegative integer.

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

Cf. A000005, A002522, A147810, A048691, A063647, A093582.

DivisorSigma[0,Range[100]^2+1]/2-1 (* Harvey P. Dale, Feb 11 2015 *)

PARI
```
A147809(n)=numdiv(n^2+1)/2-1
```

a(n) = A000005(A002522(n))/2 - 1 = A147810(n) - 1.

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = 3/(2*Pi) = 0.477464... (A093582). - Amiram Eldar, Dec 01 2023

A217584 Numbers k such that d(k^2)/d(k) is an integer, where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 144, 324, 400, 784, 1936, 2025, 2500, 2704, 3600, 3969, 4624, 5625, 5776, 7056, 8100, 8464, 9604, 9801, 13456, 13689, 15376, 15876, 17424, 19600, 21609, 21904, 22500, 23409, 24336, 26896, 29241, 29584, 30625, 35344, 39204, 41616, 42849, 44944, 48400, 51984

Offset: 1

Michel Marcus, Oct 07 2012

nonn

The ratio d(k^2)/d(k) is: 1 for the number 1, 3 for numbers of the form p^4*q^2, 5 for numbers of the form p^4*q^2*r^2 (p, q, r being different primes).
Primes can't be in the sequence. A prime p has two divisors, while p^2 has three divisors: 1, p, p^2. - Alonso del Arte, Oct 07 2012
All the terms are squares since d(m) is odd if and only if m is a square, so d(k^2) is odd and since d(k)|d(k^2), d(k) is also odd, so k is a square. The ratio d(k^2)/d(k) can take values other than 1, 3, and 5: 1587600 is the least term with a ratio 9, and 192099600 is the least term with a ratio 15. - Amiram Eldar, May 23 2020
From Bernard Schott, May 29 2020 and Nov 22 2020: (Start)
This sequence comes from the 3rd problem, proposed by Belarus, during the 39th International Mathematical Olympiad in 1998 at Taipei (Taiwan) [see the link IMO].
If the prime signature of k is (u_1, u_2, ... , u_q) then d(k^2)/d(k) = Product_{i=1..q} (2*u_i+1)/(u_i+1). Two results:
1) If k is a term such that d(k^2)/d(k) = m, then all numbers that have the same prime signature of k are also terms and give the same ratio (see examples below).
2) The set of the integer values of the ratio d(k^2)/d(k) is exactly the set of all positive odd integers (see Marcin E. Kuczma reference).
Some examples:
For numbers with prime signature = (4, 2) (A189988), the ratio is 3 and the smallest such integer is 144 = 2^4 * 3^2.
For numbers with prime signature = (4, 2, 2) (A179746), the ratio is 5 and the smallest such integer is 3600 = 2^4 * 3^2 * 5^2.
For numbers with prime signature = (4, 4, 2, 2) the ratio is 9 and the smallest such integer is 1587600 = 2^4 * 3^4 * 5^2 * 7^2.
For numbers with prime signature = (8, 4, 4, 2, 2) the ratio is 17 and the smallest such integer is 76839840000 = 2^8 * 3^4 * 5^4 * 7^2 * 11^2 (found by David A. Corneth with other prime signatures). (End)

			d(1^2)/d(1) = d(1)/d(1) = 1 an integer, so 1 belongs to the sequence.
144^2 has 45 divisors: 1, 2, 3, 4, 6, 8, 9, 12, ..., 20736, while 144 has 15 divisors: 1, 2, 3, 4, 6, 8, 9, 12, ..., 144; 45/15 = 3 and so 144 is in the sequence.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 134-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Paolo P. Lava)
The IMO Compendium, Problem 3, 39th IMO 1998.
Kin Y. Li, Problem 3, Mathematical Excalibur, Vol. 4, No. 3, Jan.-Mar. 1999.
Index to sequences related to Olympiads.

Cf. A000005, A003593, A025487, A048691.
Subsequences: A189988 (d(k^2)/d(k) = 3), A179746 (d(k^2)/d(k) = 5).
Cf. A339055 (values taken by d(a(n)^2)/d(a(n))), A339056 (smallest k such that d(k^2)/d(k) = n-th odd).

Select[Range[1000], IntegerQ[DivisorSigma[0, #^2]/DivisorSigma[0, #]] &] (* Alonso del Arte, Oct 07 2012 *)
Select[Range[228]^2, Divisible[DivisorSigma[0, #^2], DivisorSigma[0, #]] &] (* Amiram Eldar, May 23 2020 *)

dn2dn(n)= {for (i=1, n, if (denominator(numdiv(i^2)/numdiv(i))==1, print1(i,", ");););}

A276678 Number of divisors of the n-th pentagonal number.

Original entry on oeis.org

1, 2, 6, 4, 4, 4, 8, 6, 6, 4, 10, 16, 4, 4, 16, 8, 6, 6, 12, 8, 8, 8, 8, 12, 6, 8, 32, 8, 4, 8, 8, 20, 12, 4, 24, 12, 8, 4, 16, 24, 4, 16, 14, 8, 12, 4, 16, 32, 6, 6, 24, 16, 4, 16, 16, 12, 16, 4, 16, 16, 8, 8, 24, 12, 8, 8, 18, 16, 8, 16, 8, 36, 4, 8, 60, 8

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(7) = 8 because the 7th pentagonal number is 70, which has 8 divisors: 1,2,5,7,10,14,35,70.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A000326 (pentagonal numbers).

Cf. A063440 (m=3), A048691 (m=4), A276679 (m=6), A276680 (m=7), A276681 (m=8), A276682 (m=9), A276683 (m=10).

DivisorSigma[0,PolygonalNumber[5,Range[80]]] (* Harvey P. Dale, Jul 19 2025 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(100, n, numdiv(pg(5,n)))

a(n) = A000005(A000326(n)). - Omar E. Pol, Sep 13 2016

A276679 Number of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 4, 4, 6, 6, 8, 4, 16, 6, 8, 8, 12, 6, 16, 8, 10, 8, 24, 4, 24, 8, 8, 12, 16, 9, 16, 8, 24, 8, 16, 4, 36, 16, 8, 16, 18, 4, 24, 16, 16, 10, 16, 8, 24, 12, 16, 8, 40, 6, 36, 8, 12, 16, 16, 8, 32, 8, 16, 12, 48, 6, 16, 24, 14, 16, 16, 8, 48, 8, 16, 8, 48, 8

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(6) = 8 because the 6th hexagonal number is 66, which has 8 divisors: 1,2,3,6,11,22,33,66.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A000384.

Number of divisors of m-gonal numbers (m = 3..10): A063440, A048691, A276678, this sequence, A276680, A276681, A276682, A276683.

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(100, n, numdiv(pg(6,n)))

a(n) = A000005(A000384(n)). - Omar E. Pol, Sep 13 2016

A276680 Number of divisors of the n-th heptagonal number.

Original entry on oeis.org

1, 2, 6, 4, 4, 5, 10, 6, 8, 4, 8, 12, 4, 4, 24, 16, 4, 8, 8, 8, 12, 4, 16, 24, 6, 4, 20, 8, 4, 18, 12, 10, 12, 4, 16, 16, 8, 8, 36, 12, 4, 16, 8, 16, 16, 4, 12, 24, 9, 12, 32, 8, 4, 10, 32, 12, 12, 8, 8, 40, 4, 4, 48, 12, 16, 12, 8, 8, 16, 8, 20, 48, 4, 4

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(3) = 6 because the 3rd heptagonal number is 18, which has 6 divisors: 1,2,3,6,9,18.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A000566.

Cf. A063440 (m=3), A048691 (m=4), A276678 (m=5), A276679 (m=6), A276681 (m=8), A276682 (m=9), A276683 (m=10).

DivisorSigma[0,PolygonalNumber[7,Range[80]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 09 2017 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(100, n, numdiv(pg(7,n)))

a(n) = A000005(A000566(n)).

A276681 Number of divisors of the n-th octagonal number.

Original entry on oeis.org

1, 4, 4, 8, 4, 12, 4, 10, 9, 16, 4, 16, 4, 20, 8, 12, 6, 24, 8, 16, 8, 16, 4, 40, 6, 16, 8, 16, 8, 40, 8, 14, 8, 24, 8, 24, 4, 24, 16, 20, 6, 32, 4, 32, 24, 20, 4, 24, 12, 24, 8, 32, 4, 56, 8, 20, 12, 16, 12, 32, 4, 20, 24, 32, 8, 48, 4, 16, 16, 48, 4, 30, 8

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(4) = 8 because the 4th octagonal number is 40, which has 8 divisors: 1,2,4,5,8,10,20,40.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A000567.

Cf. A063440 (m=3), A048691 (m=4), A276678 (m=5), A276679 (m=6), A276680 (m=7), A276682 (m=9), A276683 (m=10).

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(150, n, numdiv(pg(8,n)))

a(n) = A000005(A000567(n)).

A276682 Number of divisors of the n-th 9-gonal number.

Original entry on oeis.org

1, 3, 8, 4, 6, 4, 8, 12, 6, 6, 18, 8, 4, 8, 16, 8, 8, 9, 14, 24, 8, 4, 16, 12, 8, 8, 24, 8, 12, 12, 8, 20, 8, 4, 48, 24, 4, 12, 16, 24, 8, 12, 12, 16, 18, 4, 20, 16, 9, 16, 40, 8, 8, 8, 24, 36, 8, 4, 24, 24, 4, 16, 24, 12, 24, 8, 16, 16, 8, 12, 16, 18, 8, 16

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(2) = 3 because the 2nd 9-gonal number is 9, which has 3 divisors: 1,3,9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A001106.

Cf. A063440 (m=3), A048691 (m=4), A276678 (m=5), A276679 (m=6), A276680 (m=7), A276681 (m=8), A276683 (m=10).

DivisorSigma[0,PolygonalNumber[9,Range[80]]] (* Harvey P. Dale, Dec 02 2024 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(150, n, numdiv(pg(9,n)))

a(n) = A000005(A001106(n)).

A276683 Number of divisors of the n-th 10-gonal number.

Original entry on oeis.org

1, 4, 4, 6, 4, 12, 6, 8, 8, 8, 4, 24, 6, 8, 12, 10, 8, 16, 4, 24, 12, 16, 4, 24, 6, 8, 20, 12, 4, 32, 6, 24, 12, 16, 8, 24, 8, 8, 16, 16, 8, 48, 6, 12, 16, 8, 8, 50, 6, 12, 12, 24, 8, 20, 16, 32, 24, 8, 4, 36, 4, 24, 16, 28, 8, 32, 8, 12, 24, 16, 4, 64, 6, 8

Offset: 1

Colin Barker, Sep 13 2016

nonn

			a(4) = 6 because the 4th 10-gonal number is 52, which has 6 divisors: 1,2,4,13,26,52.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A001107.

Cf. A063440 (m=3), A048691 (m=4), A276678 (m=5), A276679 (m=6), A276680 (m=7), A276681 (m=8), A276682 (m=9).

DivisorSigma[0,PolygonalNumber[10,Range[80]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 05 2021 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
vector(150, n, numdiv(pg(10,n)))

a(n) = A000005(A001107(n)).

cp's OEIS Frontend

A062367 Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.

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A136404 Square numbers with more divisors than any smaller square number.

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A147809 Half the number of proper divisors (> 1) of n^2 + 1, i.e., tau(n^2 + 1)/2 - 1.

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A217584 Numbers k such that d(k^2)/d(k) is an integer, where d(k) is the number of divisors of k.

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Mathematica

PARI

A276678 Number of divisors of the n-th pentagonal number.

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Mathematica

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A276679 Number of divisors of the n-th hexagonal number.

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A276680 Number of divisors of the n-th heptagonal number.

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A276681 Number of divisors of the n-th octagonal number.

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.