A300409 -id:A300409 - cp's OEIS frontend

A300410 Number of centered square numbers dividing n.

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

Offset: 1

Ilya Gutkovskiy, Mar 05 2018

nonn

			a(26) = 2 because 26 has 4 divisors {1, 2, 13, 26} among which 2 divisors {1, 13} are centered square numbers.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Square Number.
Index entries for sequences related to centered polygonal numbers.

Cf. A001844, A046951, A228048, A279496, A300409.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
for k from 1 do
  m:= 2*k*(k+1)+1;
  if m > N then break fi;
  r:= [seq(i,i=m..N,m)];
  V[r]:= map(t->t+1, V[r]);
od:
convert(V,list); # Robert Israel, Mar 05 2018

nmax = 100; Rest[CoefficientList[Series[Sum[x^(2 k (k + 1) + 1)/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=0} x^(2*k*(k+1)+1)/(1 - x^(2*k*(k+1)+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A228048 = 1.440659... . - Amiram Eldar, Jan 02 2024

A358544 a(n) is the smallest number with exactly n divisors that are centered triangular numbers.

Original entry on oeis.org

1, 4, 20, 320, 460, 5440, 14260, 12920, 168640, 103360, 594320, 3878720, 2377280, 9211960, 18423920, 36847840, 125995840, 73695680, 865924240, 976467760, 1952935520, 3463696960, 3905871040, 31246968320, 22946992360

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

Any subsequent terms are > 10^10. - Lucas A. Brown, Dec 24 2022

			a(3) = 20 because 20 has 3 centered triangular divisors {1, 4, 10} and this is the smallest such number.

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to divisors of numbers

Cf. A005179, A005448, A130317, A300409, A358542, A358545.

isct(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ A005448
a(n) = my(k=1); while (sumdiv(k, d, isct(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(14) from Michel Marcus, Nov 21 2022

a(15)-a(25) from Jinyuan Wang, Nov 29 2022

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of divisors of n of the form k*(k^2 + 1)/2 (A006003).

Cf. A006003, A007862, A279495, A300409, A334926.

Cf. A001620, A248177.

nmax = 100; CoefficientList[Series[Sum[x^(k (k^2 + 1)/2)/(1 - x^(k (k^2 + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (A248177 + A001620) = 1.343731... . - Amiram Eldar, Jan 02 2024

A334987 Sum of centered triangular numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 20, 15, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 32, 5, 1, 1, 1, 5, 1, 20, 1, 15, 1, 1, 1, 5, 1, 47, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 20, 1, 1, 15, 1, 32, 1, 69, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 24, 1, 1, 1, 15, 1, 1, 1, 5, 86, 1, 1, 5, 1, 11

Offset: 1

Ilya Gutkovskiy, May 18 2020

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A185027, A280950, A300409, A334988.

nmax = 90; CoefficientList[Series[Sum[(3 k (k - 1)/2 + 1) x^(3 k (k - 1)/2 + 1)/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

isc(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k));
a(n) = sumdiv(n, d, if (isc(d), d)); \\ Michel Marcus, May 19 2020

G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^(3*k*(k - 1)/2 + 1) / (1 - x^(3*k*(k - 1)/2 + 1)).

L.g.f.: log(G(x)), where G(x) is the g.f. for A280950.

A359231 a(n) is the smallest centered triangular number divisible by exactly n centered triangular numbers.

Original entry on oeis.org

1, 4, 64, 5860, 460, 74260, 14260, 1221760, 5567104, 103360, 20120860, 169096960, 1211757760, 31286787760, 31498960, 114183284260, 1553569960, 33186496960, 446613160960, 43581101074960, 274644405760, 64262632960, 121634429663260, 5786547945760

Offset: 1

Ilya Gutkovskiy, Dec 22 2022

nonn

a(25) > 10^15. a(30) = 281149511296960. - Jon E. Schoenfield, Dec 25 2022

			a(5) = 460, because 460 is a centered triangular number that has 5 centered triangular divisors {1, 4, 10, 46, 460} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to divisors of numbers

Cf. A005448, A076983, A300409, A358544, A358861, A359232.

// Note: the program below finds all terms through a(22) except for
//  a(20) = 43581101074960, which would be reached at k = 5390183.
a := [ 0 : n in [ 1 .. 22 ] ];
for k in [ 0 .. 550000 ] do
   c := 3*((k*(k - 1)) div 2) + 1;
   D := Divisors(c);
   n := 0;
   for d in D do
      if d mod 3 eq 1 then
         if IsSquare(((d - 1) div 3)*8 + 1) then
            n +:= 1;
         end if;
      end if;
   end for;
   if a[n] eq 0 then
      a[n] := c;
   end if;
end for;
a; // Jon E. Schoenfield, Dec 25 2022

a(8)-a(24) from Jon E. Schoenfield, Dec 25 2022

cp's OEIS Frontend

A300410 Number of centered square numbers dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358544 a(n) is the smallest number with exactly n divisors that are centered triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A334925 G.f.: Sum_{k>=1} x^(k*(k^2 + 1)/2) / (1 - x^(k*(k^2 + 1)/2)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A334987 Sum of centered triangular numbers dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A359231 a(n) is the smallest centered triangular number divisible by exactly n centered triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Extensions

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).