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.

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A358539 a(n) is the smallest number with exactly n divisors that are n-gonal numbers.

Original entry on oeis.org

6, 36, 210, 1260, 6426, 3360, 351000, 207900, 3749460, 1153152, 15036840, 204204000, 213825150, 11737440, 91797866160, 1006485480, 2310808500, 4966241280, 22651328700, 325269404460, 14266470332400, 11203920000, 256653797400, 45843256859400, 59207908359600, 46822406400
Offset: 3

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Author

Ilya Gutkovskiy, Nov 21 2022

Keywords

Examples

			a(5) = 210 because 210 has 5 pentagonal divisors {1, 5, 35, 70, 210} and this is the smallest such number.

Links

Crossrefs

Cf. A005179, A130279, A130317, A356132, A358540, A358541.

Programs

  • PARI
    a(n) = my(k=1); while (sumdiv(k, d, ispolygonal(d, n)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

Extensions

a(12)-a(13) from Michel Marcus, Nov 21 2022
a(14)-a(16) from Daniel Suteu, Dec 04 2022
a(17)-a(28) from Martin Ehrenstein, Dec 05 2022

A359094 a(n) is the smallest square pyramidal number divisible by exactly n square pyramidal numbers.

Original entry on oeis.org

1, 5, 30, 140, 4900, 155155, 6930, 223300, 3573570, 380380, 340889640, 1801800, 333833500, 711410700, 78963134250, 427826509110, 70836325560, 862289508080, 62366724420, 3975527876320, 2279301054030, 3422848288860, 58264695188700, 4903512426212400
Offset: 1

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Author

Ilya Gutkovskiy, Dec 16 2022

Keywords

Examples

			a(5) = 4900, because 4900 is a square pyramidal number that has 5 square pyramidal divisors {1, 5, 14, 140, 4900} and this is the smallest such number.

Links

Crossrefs

Cf. A000330, A005179, A130279, A358543, A359095.

Extensions

a(11)-a(24) from Jon E. Schoenfield, Dec 17 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

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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.

Links

Crossrefs

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

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

A361418 a(n) is the least number with exactly n noninfinitary divisors.

Original entry on oeis.org

1, 4, 12, 16, 60, 36, 48, 256, 360, 4096, 180, 144, 240, 576, 768, 65536, 2520, 1048576, 12288, 900, 1260, 1296, 720, 2304, 1680, 9216, 2880, 5184, 3840, 147456, 196608, 36864, 27720, 46656, 3145728, 4398046511104, 61440, 3600, 6300, 18014398509481984, 10080, 20736
Offset: 0

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

a(n) is the least number k such that A348341(k) = n.
Since A348341(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

Examples

			a(1) = 4 since 4 is the least number with exactly one noninfinitary divisor, 2.

Links

Crossrefs

Cf. A025487, A348341, A348342.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

Programs

  • Mathematica
    f[1] = 0; f[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]];
seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s];
seq[35, 10^7]
  • PARI
    s(n) = {my(f = factor(n)); numdiv(f) - prod(i = 1, #f~, 2^hammingweight(f[i,2]));}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};
Previous Showing 11-14 of 14 results.

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