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 1-5 of 5 results.

A122815 Terms in A122375 that differ from corresponding term in A122810.

Original entry on oeis.org

2, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 4, 5, 5, 4, 6, 4, 6, 4, 6, 5, 5, 4, 6, 5, 6, 6, 4, 6, 5, 4, 6, 5, 6, 4, 5, 7, 6, 4, 6, 4, 6, 5, 5, 4, 7, 6, 4, 7, 7, 5, 4, 7, 6, 4, 5, 7, 6, 4, 6, 4, 7, 4, 5, 5, 6, 6, 4, 7, 5, 4, 7, 5, 7, 4, 7, 6, 6, 5, 8, 4, 4, 6, 6, 4, 7, 5, 4, 4, 4, 6, 6, 5, 8, 7, 5, 6, 4, 8
Offset: 1

Views

Author

Ray Chandler, Sep 22 2006

Keywords

Crossrefs

Cf. A122375, A122810, A122812, A122816.

Formula

a(n) = A122375(A122812(n)).

A122816 Terms in A122810 that differ from corresponding term in A122375.

Original entry on oeis.org

3, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 5, 5, 6, 5, 6, 6, 5, 7, 5, 7, 5, 7, 6, 6, 5, 7, 6, 7, 7, 5, 7, 6, 5, 8, 6, 7, 5, 6, 8, 7, 5, 7, 5, 7, 6, 6, 5, 8, 7, 5, 8, 8, 6, 5, 8, 7, 5, 6, 8, 7, 5, 7, 5, 8, 5, 6, 6, 7, 7, 5, 8, 6, 5, 8, 6, 8, 5, 8, 7, 7, 6, 9, 5, 5, 7, 7, 5, 8, 6, 5, 5, 5, 7, 7, 6, 9, 8, 6, 7, 5, 9
Offset: 1

Views

Author

Ray Chandler, Sep 22 2006

Keywords

Crossrefs

Cf. A122375, A122810, A122812, A122815.

Formula

a(n) = A122810(A122812(n)).

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

Views

Author

Reinhard Zumkeller, May 20 2007

Keywords

Comments

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

Links

Crossrefs

Cf. A001221, A005179, A046951, A046952, A122375, A122376, A353282.
Cf. A357450 (similar, but with odd squares divisors).

Programs

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

Formula

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)

A122810 Number of distinct prime factors of the smallest odd number with exactly n divisors.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 5, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 5, 1, 3, 3, 4, 1, 3, 1, 4, 3
Offset: 1

Views

Author

Ray Chandler, Sep 22 2006

Keywords

Comments

a(n) = 1 iff n is prime.

Links

Crossrefs

Cf. A001221, A038547, A122375.

Programs

  • Mathematica
    PrimeNu[Table[Select[Range[1, 532000, 2], DivisorSigma[0, #] == k + 1 &, 1], {k, 0, 15}] // Flatten] (* G. C. Greubel, May 07 2017 *)

Formula

a(n) = omega(A038547(n)), where omega(n) = A001221(n).

Extensions

Typo in A_number fixed by Matthew Vandermast, May 03 2014

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

Original entry on oeis.org

8, 24, 48, 64, 72, 80, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848, 864, 896
Offset: 1

Views

Author

Ray Chandler, Sep 22 2006

Keywords

Comments

Where the prime signature of the least number with exactly k divisors differs from the prime signature of the least odd number with exactly k divisors.
Also where A122375(k) differs from A122810(k).
Also where A122376(k) differs from A122811(k).

Links

Crossrefs

Cf. A005179, A038547, A046523, A122375, A122376, A122810-A122819.
Showing 1-5 of 5 results.

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