A300410 -id:A300410 - cp's OEIS frontend

A300409 Number of centered triangular numbers dividing n.

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

Offset: 1

Ilya Gutkovskiy, Mar 05 2018

nonn

			a(20) = 3 because 20 has 6 divisors {1, 2, 4, 5, 10, 20} among which 3 divisors {1, 4, 10} are centered triangular numbers.

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

Cf. A005448, A007862, A279495, A300410, A306324.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
for k from 1 do
  m:= 3*k*(k+1)/2+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^(3 k (k + 1)/2 + 1)/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=0} x^(3*k*(k+1)/2+1)/(1 - x^(3*k*(k+1)/2+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A306324 = 1.5670651... . - Amiram Eldar, Jan 02 2024

A358545 a(n) is the smallest number with exactly n divisors that are centered square numbers.

Original entry on oeis.org

1, 5, 25, 325, 1625, 1105, 5525, 27625, 160225, 1022125, 801125, 5928325, 8491925, 29641625, 42459625, 444215525, 314201225, 2003613625, 1571006125, 14826740825, 12882250225, 127081657625, 64411251125, 88717383625

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

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

			a(3) = 25 because 25 has 3 centered square divisors {1, 5, 25} and this is the smallest such number.

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

Cf. A001844, A005179, A130279, A300410, A358543, A358544.

iscsq(n) = issquare(2*n-1); \\ A001844
a(n) = my(k=1); while (sumdiv(k, d, iscsq(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(12)-a(15) from Michel Marcus, Nov 21 2022
a(16) from Martin Ehrenstein, Dec 02 2022
a(17)-a(24) from Jinyuan Wang, Dec 02 2022

A334926 G.f.: Sum_{k>=1} x^(k(2k^2 + 1)/3) / (1 - x^(k(2k^2 + 1)/3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of octahedral numbers (A005900) dividing n.

Eric Weisstein's World of Mathematics, Octahedral Number.

Cf. A005900, A061704, A175577, A279495, A279496, A300410, A334925.

nmax = 100; CoefficientList[Series[Sum[x^(k (2 k^2 + 1)/3)/(1 - x^(k (2 k^2 + 1)/3)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A175577 = 1.278185... . - Amiram Eldar, Jan 02 2024

A359232 a(n) is the smallest centered square number divisible by exactly n centered square numbers.

Original entry on oeis.org

1, 5, 25, 925, 1625, 1105, 47125, 350285, 493025, 3572465, 47074105, 13818025, 4109345825, 171921425, 294346585, 130334225125, 190608050165, 2687125303525, 2406144489125, 5821530534625, 49723952067725, 1500939251825, 665571884367325, 8362509238504525, 1344402738869125

Offset: 1

Ilya Gutkovskiy, Dec 22 2022

nonn

From Jon E. Schoenfield, Dec 24 2022: (Start)
For all n > 22, a(n) > 5*10^14.
For all n in 10..22, the prime factors of a(n) include 5, 13, and 17. Every index k such that 5*13*17=1105 divides the k-th centered square number satisfies k == { 23, 231, 418, 431, 673, 686, 873, 1081 } (mod 1105), so a search for upper bounds for larger terms can be facilitated by testing only such indices k.
Some known upper bounds: a(23) <= 665571884367325, a(24) <= 8362509238504525, a(25) <= 1344402738869125, a(26) <= 49165090920807485, a(27) <= 4384711086003625, a(30) <= 13148945184367525, a(33) <= 179899779754020625. (End)

			a(5) = 1625, because 1625 is a centered square number that has 5 centered square divisors {1, 5, 13, 25, 1625} and this is the smallest such number.

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

Cf. A001844, A130279, A300410, A358545, A358861, A359231.

a := [ 0 : n in [ 1 .. 17 ] ];
for k in [ 0 .. 310000 ] do
   c := 2*k*(k+1)+1;
   D := Divisors(c);
   n := 0;
   for d in D do
      if IsSquare(2*d - 1) then
         n +:= 1;
      end if;
   end for;
   if a[n] eq 0 then
      a[n] := c;
   end if;
end for;
a; // Jon E. Schoenfield, Dec 24 2022

a(n) = for(k=0, oo, my(t=2*k*(k+1)+1); if(sumdiv(t, d, issquare(2*d-1)) == n, return(t))); \\ Daniel Suteu, Dec 31 2022

a(10)-a(22) from Jon E. Schoenfield, Dec 24 2022

a(23)-a(25) confirmed by Daniel Suteu, Dec 31 2022

cp's OEIS Frontend

A300409 Number of centered triangular numbers dividing n.

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A358545 a(n) is the smallest number with exactly n divisors that are centered square numbers.

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A334926 G.f.: Sum_{k>=1} x^(k*(2*k^2 + 1)/3) / (1 - x^(k*(2*k^2 + 1)/3)).

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Mathematica

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A359232 a(n) is the smallest centered square number divisible by exactly n centered square numbers.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

Extensions

A334926 G.f.: Sum_{k>=1} x^(k(2k^2 + 1)/3) / (1 - x^(k(2k^2 + 1)/3)).