cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A358543 a(n) is the smallest number with exactly n divisors that are square pyramidal numbers.

Original entry on oeis.org

1, 5, 30, 140, 420, 1540, 4620, 13860, 78540, 157080, 471240, 1141140, 3603600, 3423420, 13693680, 30630600, 58198140, 116396280, 214414200, 428828400, 581981400, 1163962800, 5354228880, 4073869800, 8147739600, 26771144400, 36082846800, 80313433200, 93699005400, 187398010800
Offset: 1

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Author

Ilya Gutkovskiy, Nov 21 2022

Keywords

Comments

Any terms for n > 25 exceed 10^10. - Lucas A. Brown, Dec 24 2022
a(25) <= 8147739600, a(26) <= 26771144400, a(27) <= 36082846800, a(28) <= 80313433200. - Jon E. Schoenfield, Dec 16 2022

Examples

			a(3) = 30 because 30 has 3 square pyramidal divisors {1, 5, 30} and this is the smallest such number.
		

Crossrefs

Programs

  • PARI
    issqpyr(n) = my(m = sqrtnint(3*n, 3)); n==m*(m+1)*(2*m+1)/6; \\ A253903
    a(n) = my(k=1); while (sumdiv(k, d, issqpyr(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

Extensions

a(15) from Michel Marcus, Nov 21 2022
a(16)-a(20) from Jinyuan Wang, Nov 28 2022
a(21)-a(22) from Lucas A. Brown, Dec 14 2022
a(23)-a(24) from Lucas A. Brown, Dec 18 2022
a(25) from Lucas A. Brown, Dec 22 2022
a(26)-a(30) from Bert Dobbelaere, May 18 2025

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

Views

Author

Ilya Gutkovskiy, Nov 21 2022

Keywords

Comments

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

Examples

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

Crossrefs

Programs

  • PARI
    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

Extensions

a(14) from Michel Marcus, Nov 21 2022
a(15)-a(25) from Jinyuan Wang, Nov 29 2022

A358541 a(n) is the smallest number with exactly n divisors that are centered n-gonal numbers.

Original entry on oeis.org

20, 325, 912, 43771, 234784, 11025, 680680, 9143308361, 2470852896
Offset: 3

Views

Author

Ilya Gutkovskiy, Nov 21 2022

Keywords

Comments

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

Examples

			a(5) = 912 because 912 has 5 centered pentagonal divisors {1, 6, 16, 76, 456} and this is the smallest such number.
		

Crossrefs

Extensions

a(10)-a(11) from Martin Ehrenstein, Dec 04 2022

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

Views

Author

Ilya Gutkovskiy, Dec 22 2022

Keywords

Comments

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)

Examples

			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.
		

Crossrefs

Programs

  • Magma
    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
    
  • PARI
    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

Extensions

a(10)-a(22) from Jon E. Schoenfield, Dec 24 2022
a(23)-a(25) confirmed by Daniel Suteu, Dec 31 2022
Showing 1-4 of 4 results.