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 1-3 of 3 results.

A280076 Numbers n such that Sum_{d|n} tau(d) = Product_{d|n} tau(d).

Original entry on oeis.org

1, 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481
Offset: 1

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Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

Union of 1 and A001248 (squares of primes).
Numbers n such that A007425(n) = A211776(n).
Numbers n such that A007425(n) = Sum_{d|n} tau(d) = A211776(n) = Product_{d|n} tau(d) = 6.
Also squares of noncomposite numbers (A008578).
Subsequence of A350343. - Lorenzo Sauras Altuzarra, Sep 18 2022

Examples

			9 is a term because Sum_{d|9} tau(d) = 1+2+3 = Product_{d|9} tau(d) = 1*2*3 = 6.

Links

Crossrefs

Cf. A001248, A007425, A008578, A211776, A331294, A350343.

Programs

  • Magma
    [n: n in [1..1000000] | &*[NumberOfDivisors(d): d in Divisors(n)]  eq &+[NumberOfDivisors(d): d in Divisors(n)]]
  • Mathematica
    Select[Range@ 37500, Total@ # == Times @@ # &@ Map[DivisorSigma[0, #] &, Divisors@ #] &] (* Michael De Vlieger, Dec 25 2016 *)
  • PARI
    isok(n) = my(d = divisors(n), nd = apply(numdiv, d)); vecsum(nd) == prod(k=1, #nd, nd[k]); \\ Michel Marcus, Jun 26 2017

Formula

A007425(a(n)) = A211776(a(n)) = 6.
Apparently, a(n) = A331294(n + 3) if n > 5. - Lorenzo Sauras Altuzarra, Sep 18 2022

A350342 Numbers k such that k^2 is an abelian order.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 209
Offset: 1

Views

Author

Jianing Song, Dec 25 2021

Keywords

Comments

Such k must be squarefree. Actually, such k must be a cyclic number (A003277).
Number of the form p_1*p_2*...*p_r where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest term with exactly n distinct prime factors is given by A350340.
From the term 5 on, no term can be divisible by 2 or 3.

Examples

			For primes p, p is a term since every group of order p^2 is abelian. Such group is isomorphic to either C_{p^2} or C_p X C_p.
For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p*q is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or C_{p*q} X C_{p*q}.

Links

Crossrefs

Cf. A051532 (abelian orders), A003277 (cyclic numbers), A350343, A350340.
A350344 is the subsequence of composite numbers.

Programs

  • PARI
    isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350342(n) = isA051532(n^2)

Formula

A350343(n) = a(n)^2.

A350345 Squares of composite numbers k that are abelian orders.

Original entry on oeis.org

1225, 4225, 5929, 7225, 13225, 14161, 17689, 20449, 25921, 34225, 34969, 43681, 46225, 47089, 48841, 55225, 61009, 67081, 70225, 89401, 101761, 104329, 108241, 112225, 116281, 133225, 137641, 142129, 152881, 162409, 165649, 170569, 172225, 182329, 190969
Offset: 1

Views

Author

Jianing Song, Dec 25 2021

Keywords

Comments

Square numbers k that are abelian orders with at least 4 groups.
Number of the form (p_1*p_2*...*p_r)^2 where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest square number k that is an abelian order with at least 8 groups is A350341(3) = 354025.
No term can be divisible by 2 or 3.

Examples

			For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p^2*q^2 is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or C_{p*q} X C_{p*q}.

Links

Crossrefs

Cf. A051532 (abelian orders), A050384, A350341.
Equals A350343 \ ({1} U A001248).
A350323 is a subsequence.

Programs

  • PARI
    isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350345(n) = issquare(n) && (n>1) && !isprime(sqrtint(n)) && isA051532(n^2)

Formula

a(n) = A350344(n)^2.
Showing 1-3 of 3 results.

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