A046952 -id:A046952 - cp's OEIS frontend

A046951 a(n) is the number of squares dividing n.

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

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A056624 Number of unitary square divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 08 2000

Unitary analog of A046951.

The number of exponential divisors (A322791) of n that are cubefree (A004709). - Amiram Eldar, Jun 03 2025

			n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
n=108 has 12 divisors, 4 square divisors: {1,4,9,36} of which 1 and 4 are unitary, 9 and 36 are not. So a(108)=2. The largest unitary square divisor of 108 is 4 with 1 prime divisor so a(108) = 2^1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A004709, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A322791, A350388.

isA056624 := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d and igcd(n/d^2, d) = 1:
a := n -> nops(select(k -> isA056624(n, k), [seq(1..n)])):  # Peter Luschny, Jun 13 2025

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2022 *)

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ Michel Marcus, Jul 28 2017

from sympy import factorint
def A056624(n): return 1<Chai Wah Wu, Aug 03 2024

def is_A056624(n, d): return gcd(n, d) == d and gcd(n//d, d) == d and gcd(n//(d*d), d) == 1
def a(n): return len([k for k in range(1, n+1) if is_A056624(n, k)])
print([a(n) for n in range(1, 106)])  # Peter Luschny, Jun 13 2025

(define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

a(n) = 2^r, where r is the number of prime factors of the largest unitary square divisor of n.
Multiplicative with a(p^e) = 2^(1-(e mod 2)). - Vladeta Jovovic, Dec 13 2002
Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - Werner Schulte, Apr 03 2018
Sum_{k=1..n} a(k) ~ n*Pi^2/(6*zeta(3)) + sqrt(n)*zeta(1/2)/zeta(3/2). - Vaclav Kotesovec, Feb 07 2019
a(n) = 2^A162641(n). - Amiram Eldar, Sep 26 2022
a(n) = A034444(A350388(n)). - Amiram Eldar, Sep 09 2023

More terms from Vladeta Jovovic, Dec 13 2002

A130279 Smallest number having exactly n square divisors.

Original entry on oeis.org

1, 4, 16, 36, 256, 144, 4096, 576, 1296, 2304, 1048576, 3600, 16777216, 36864, 20736, 14400, 4294967296, 32400, 68719476736, 57600, 331776, 9437184, 17592186044416, 129600, 1679616, 150994944, 810000, 921600, 72057594037927936

Offset: 1

Reinhard Zumkeller, May 20 2007

nonn

A046951(a(n)) = n and A046951(m) <> n for m < a(n);

all terms are smooth squares: if prime(k) is a factor of a(n) then also prime(i) are factors, i

a(p) = 2^(2*(p-1)) for primes p;

if prime(j) is the greatest prime factor of a(n) then a(2*n) = a(n)*prime(j+1)^2;

A001221(a(n)) = A122375(n); A001222(a(n)) = 2*A122376(n).

a(n+1) is the smallest nonsquarefree number m such that Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m has exactly n solutions, for n >= 0 (A353282); example: a(4) = 36 and 36 is the smallest number m such that equation S(x,y) = m has exactly 3 solutions: (9,1), (8,2), (5,5). - Bernard Schott, Apr 13 2022

a(n) is the square of the smallest integer having exactly n divisors (see formula with proof). - Bernard Schott, Oct 01 2022

Amiram Eldar, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A001221, A005179, A046951, A046952, A122375, A122376, A353282.

Cf. A357450 (similar, but with odd squares divisors).

a(n) = my(k=1); while(sumdiv(k, d, issquare(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2019

From Bernard Schott, Oct 01 2022: (Start)
a(n) = A005179(n)^2.
Proof: Suppose a(n) = Product p_i^(2*e_i), where the p_i are primes. Then the n square divisors are all of the form d = Product p_i^(2*k_i) with 0 <= k_i <= e_i. As a(n) = Product (p_i^e_i)^2 = (Product (p_i^e_i))^2, we get that sqrt(a(n)) = Product (p_i^e_i). This is the prime decomposition of sqrt(a(n)). As there is a bijection between prime factors p_i^(2*k_i) and (p_i^k_i), there is also bijection between square divisors of a(n) and divisors of sqrt(a(n)). We conclude that sqrt(a(n)) is the smallest integer that has exactly n divisors. (End)

A350756 Integers whose number of divisors that are triangular numbers sets a new record.

Original entry on oeis.org

1, 3, 6, 30, 90, 180, 210, 420, 630, 1260, 2520, 6930, 13860, 27720, 41580, 83160, 138600, 180180, 360360, 540540, 1081080, 1413720, 2162160, 3063060, 6126120, 12252240, 18378360, 36756720, 73513440, 91891800, 116396280, 183783600, 232792560, 349188840

Offset: 1

Bernard Schott, Jan 13 2022

nonn

Terms that are triangular: 1, 3, 6, 210, 630, 2162160, ...

The number of triangular divisors of a(n) is A007862(a(n)): 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, ...

			1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.

Cf. A000217, A007862, A130317.

Similar for A046952 (squares), A053624 (odd), A093036 (palindromes), A181808 (even), A340548 (repdigits), A340549 (repunits) divisors.

max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)

lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 2022

A355304 Integers whose number of normal undulating divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1080, 1260, 1440, 1680, 2160, 2520, 5040, 7560, 10080, 15120, 21840, 28080, 32760, 56160, 65520, 98280, 131040, 196560, 393120, 589680, 786240, 1113840, 1670760, 2227680, 3341520, 6683040, 13366080, 20049120

Offset: 1

Bernard Schott, Jun 30 2022

Normal undulating integers are in A355301.
The first 14 terms are also the first 14 highly composite numbers in A002182, then A002182(15) = 840 while a(15) = 1080. Indeed, 840 is the smallest integer that has 32 divisors of which only 28 are normal undulating integers, while 1080 has also 32 divisors of which 30 are normal undulating integers.
Corresponding records of number of normal undulating divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, ...

			a(6) = 24 is in the sequence because A355302(24) is larger than any earlier value in A355302.

Cf. A002182, A355301, A355302, A355303.

Similar, but with divisors that are: A046952 (squares), A053624 (odd), A181808 (even), A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A350756 (triangular).

nuQ[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; dm = -1; seq = {}; Do[If[(d = DivisorSum[n, 1 &, nuQ[#] &]) > dm, dm = d; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jun 30 2022 *)

More terms from Amiram Eldar, Jun 30 2022

A353282 a(n) is the number of solutions (x,y) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) when x >= y > 1 and y | x.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Apr 09 2022

nonn

This is the generalization of a problem proposed by Yakov Perelman for A013929(93) = 243 (references, links and example).
a(n) is the number of squares > 1 dividing A013929(n), so, there is no solution (x,y) for S(x,y) = m when m is a squarefree number (A005117).
Also, number of times where A013929(n) appears in table A351381.
The smallest nonsquare number m such that equation S(x,y) = m has exactly n solutions, for n >= 0, is A130279(n+1).
Integers k for which number of solutions to the equation S(x,y) = k sets a new record are in A046952.

			For S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(2) = 8, the unique solution is (2,1) because (2+1) + (2-1) + (2*1) + (2/1) = 8, hence a(2) = 1.
For S(x,y) = A013929(93) = 243, the two solutions are (24,8) and (54,2) because S(24,8) = S(54,2) = 243, hence a(93) = 2 (problem from Perelman's book).

I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
Wikipedia, Yakov Perelman.

Cf. A046951, A005117, A013929, A046952, A130279, A351381.

f[p_, e_] := 1 + Floor[e/2]; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - 1; s /@ Select[Range[250], ! SquareFreeQ[#] &] (* Amiram Eldar, Apr 09 2022 *)

a(n) = A046951(A013929(n)) - 1.

More terms from Amiram Eldar, Apr 09 2022

A358253 Numbers with a record number of non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 10368, 20736, 28800, 41472, 64800, 115200, 259200, 518400, 1036800, 2073600, 4147200, 8294400, 9331200, 12700800, 25401600, 50803200, 101606400, 203212800, 406425600, 457228800, 635040000, 812851200, 914457600, 1270080000

Offset: 1

Amiram Eldar, Nov 05 2022

nonn

Numbers m such that A056626(m) > A056626(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 20, 22, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..283
Amiram Eldar, Table of n, a(n), A056626(a(n)) for n = 1..283

Cf. A056626, A358252.
Subsequence of A025487.
Similar sequences: A002182 (all divisors), A002110 (unitary), A037992 (infinitary), A046952 (square divisors), A053624 (odd divisors), A293185 (bi-unitary), A309141 (non-unitary), A318278 (exponential).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; s = {}; fmax = -1; Do[If[(fn = f[n]) > fmax, fmax = fn; AppendTo[s, n]], {n, 1, 10^5}]; s

s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(nmax) = {my(smax = -1, sn); for(n = 1, nmax, sn = s(n); if(sn > smax, smax = sn; print1(n, ", "))); }

A325839 Exponentially-odd coreful highly composite numbers: numbers with record values of the number of exponentially odd coreful divisors (A325837).

Original entry on oeis.org

1, 8, 32, 128, 512, 864, 3456, 7776, 13824, 31104, 124416, 279936, 497664, 1119744, 1990656, 3888000, 10077696, 15552000, 34992000, 62208000, 139968000, 248832000, 388800000, 559872000, 1259712000, 1555200000, 2239488000, 3499200000, 6220800000, 8957952000

Offset: 1

Amiram Eldar, Sep 07 2019

nonn

The corresponding record numbers of exponentially-odd divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, ... (see the link for more terms).
The even version of this sequence is A046952 which is the sequence of numbers with record number of square divisors (only even exponents, A046951).
Numbers with record values of the number of exponentially odd divisors are the same as the numbers with record values of the number of semi-unitary divisors (A322484). - Amiram Eldar, Sep 08 2023

Amiram Eldar, Table of n, a(n) for n = 1..290
Amiram Eldar, Table of n, a(n), A325837(a(n)) for n = 1..290

Cf. A033634, A046951, A046952, A268335, A322484, A325837.

Subsequence of A025487.

fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); am = 0; s={}; Do[a1=a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 1, 300000}]; s

Name corrected by Amiram Eldar, Sep 08 2023

A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 144, 400, 900, 3600, 11025, 28224, 78400, 254016, 705600, 2286144, 6350400, 25401600, 85377600, 250905600, 768398400, 3073593600, 10600761600, 32464832400, 129859329600, 456536705625

Offset: 0

Andrei Cretu, Nov 11 2014

nonn

The number of binary strings, counted up to isomorphism, that can be constructed by taking an equal number (n) of 0's and 1's and partitioning both the 0's and the 1's into m runs using the same partition, can be written as:
dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
where m_j is the multiplicity of runs of length j.
The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
The strings obtained in this manner are a subset of those in A247651.
Both the finest and coarsest partitions of n minimize dualseq[partition[n]]. In this sense, dualseq[partition[n]] is another relative measure of the complexity of the partition and the associated binary strings.
a[n] is the number of strings, counted up to isomorphism, that can be generated based on the partition that maximizes dualseq[partition[n]].

			n=0 gives the empty string.
n=1 and the only possible partition generate 01 (and the isomorphic 10).
For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
For n=3, the optimal partition is {1,2}, generating, up to isomorphism, 4 strings: 001011 (110100), 001101 (110010), 010011 (101100) and 011001 (100110).
For n=4, the optimal partition is {1,1,2}, generating, up to isomorphism, 9 strings: 00101011 (11010100), 00101101 (11010010), 00110101 (11001010), 01001011 (10110100), 01001101 (10110010), 01010011 (10101100), 01011001 (10100110), 01100101 (10011010) and 01101001 (10010110).

Cf. A247651, A046952, A092266, A136404, A176471, A065886.

dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
Table[a[n],{n,0,25}] (* after A130670 *)

a[n]=Max[m!^2/(Prod_j(m_j!))^2] where Sum_j(m_j)=m, Sum_j(j*m_j)=n, over all partitions of n.

A356179 Positions of records in A279497, i.e., integers whose number of pentagonal divisors sets a new record.

Original entry on oeis.org

1, 5, 35, 70, 210, 420, 2310, 4620, 18480, 32340, 60060, 120120, 240240, 720720, 1141140, 2042040, 4084080, 4564560, 13693680, 19399380, 38798760, 77597520, 232792560, 387987600

Offset: 1

Bernard Schott, Jul 28 2022

The first fourteen terms are the same as A356132; then a(15) = 1141140 while A356132(15) = 1261260.

Corresponding records of number of pentagonal divisors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...

			210 is in the sequence because A279497(210) = 5 is larger than any earlier value in A279497.

Cf. A000326, A279497, A356132.

Similar sequences: A046952, A093036, A350756, A355595.

f[n_] := DivisorSum[n, 1 &, IntegerQ[(1 + Sqrt[1 + 24*#])/6] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 28 2022 *)

lista(nn) = my(m=0); for (n=1, nn, my(new = sumdiv(n, d, ispolygonal(d, 5))); if (new > m, m = new; print1(n, ", "));); \\ Michel Marcus, Jul 28 2022

a(23)-a(24) from David A. Corneth, Jul 28 2022

Showing 1-10 of 10 results.

cp's OEIS Frontend

A046951 a(n) is the number of squares dividing n.

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A056624 Number of unitary square divisors of n.

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A130279 Smallest number having exactly n square divisors.

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A350756 Integers whose number of divisors that are triangular numbers sets a new record.

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A355304 Integers whose number of normal undulating divisors sets a new record.

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A353282 a(n) is the number of solutions (x,y) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) when x >= y > 1 and y | x.

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