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.

A303123 Numbers whose sum of divisors is the square of one of their divisors.

Original entry on oeis.org

1, 364, 1080, 1782, 8736, 30256, 86800, 90768, 149856, 632400, 828816, 1033560, 2467600, 8182944, 9587160, 10593720, 12239136, 15487600, 16702800, 23194080, 23556960, 25371360, 33330528, 35746920, 35889480, 36036000, 40753440, 44013120, 45890208, 46462800, 49035168
Offset: 1

Views

Author

Paolo P. Lava, May 04 2018

Keywords

Comments

Subset of A090777 and A300906.
From Robert Israel, May 10 2018: (Start)
If m and n are coprime members of the sequence, then m*n is in the sequence.
However, it is not clear whether there are such m and n where neither is 1: in particular, are there odd members other than 1?
Any odd member > 1 is a square greater than 10^14. (End)

Examples

			Divisors of 364 are 1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182, 364 and their sum is 784 = 28^2.

Links

Crossrefs

Cf. A000203, A090777, A300906, A303993, A303994, A303995, A303996.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a) do if sigma(n)=a[k]^2 then print(n); break;
fi; od; od; end: P(10^9);
# Alternative:
filter:= proc(n) local s;
  s:= numtheory:-sigma(n);
  issqr(s) and n^2 mod s = 0
end proc:
select(filter, [$1..10^7]); # Robert Israel, May 10 2018
  • Mathematica
    Reap[For[k = 1, k <= 10^7, k++, If[AnyTrue[Divisors[k], DivisorSigma[1, k] == #^2&], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 05 2020 *)
  • PARI
    isok(n) = (s = sigma(n)) && issquare(s) && !(n % sqrtint(s)); \\ Michel Marcus, May 04 2018

A356410 Numbers k for which k^3 is divisible by sigma(k).

Original entry on oeis.org

1, 6, 28, 30, 84, 102, 120, 364, 420, 496, 672, 840, 1080, 1092, 1320, 1428, 1488, 1782, 2280, 2716, 2760, 3276, 3360, 3444, 3472, 3480, 3720, 4452, 5640, 7080, 7392, 7440, 7560, 8128, 8148, 8736, 8910, 9240, 9480, 10416, 10920, 11880, 12400, 15456, 15960
Offset: 1

Views

Author

Zdenek Cervenka, Aug 05 2022

Keywords

Examples

			30 is a term, because 30^3 = 27000, sigma(30) = 72 and 27000 / 72 = 375.

Links

Crossrefs

Cf. A000578, A000203, A090777.

Programs

  • Maple
    select(t -> t^3 mod numtheory:-sigma(t) = 0, [$1..20000]); # Robert Israel, Sep 16 2022
  • Mathematica
    Select[Range[16000], Divisible[#^3, DivisorSigma[1, #]] &]
Select[Range[16000],PowerMod[#,3,DivisorSigma[1,#]]==0&] (* Harvey P. Dale, Sep 04 2024 *)
  • PARI
    for(k=1,10^6,if(k^3%sigma(k)==0,print1(k,", "))) \\ Alexandru Petrescu, Sep 10 2022

A369093 Numbers k >= 1 such that sigma(k) divides the sum of the triangular numbers T(k) and T(k+1), where sigma(k) = A000203(k) is the sum of the divisors of k.

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, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293
Offset: 1

Views

Author

Claude H. R. Dequatre, Jan 13 2024

Keywords

Comments

k is a term if (k^2+k)/2 + ((k+1)^2+k+1)/2 = k^2+2*k+1 = (k+1)^2 is divisible by sigma(k).
Trivial case: If k is prime, then sigma(k) = k+1 and (k+1)^2 is divisible by k+1, thus all primes are terms of this sequence.
Table with the percentage of primes <= 10^k compared with the number of terms and the number of primes <= 10^k, for k = 2..8:
.
| k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
| 2 | 27 | 25 | 92.59 |
| 3 | 175 | 168 | 96.00 |
| 4 | 1248 | 1229 | 98.48 |
| 5 | 9627 | 9592 | 99.64 |
| 6 | 78565 | 78498 | 99.91 |
| 7 | 664707 | 664579 | 99.98 |
| 8 | 5761724 | 5761455 | 99.99 |
.
The percentage of primes increases asymptotically as 10^k increases.
Conjecture: The asymptotic density of primes in this sequence is 1.
Contains terms like 2, 399, 935, 1539,.. which are not in A210494. Does not contain terms like 775, 819, 3335, 6815,.. which are in A210494. - R. J. Mathar, Jan 18 2024

Examples

			3 is a term since (3+1)^2 = 4^2 = 16 is divisible by sigma(3) = 4.
35 is a term since (35+1)^2 = 36^2 = 1296 is divisible by sigma(35) = 48.
42 is not a term since (42+1)^2 = 43^2 = 1849 is not divisible by sigma(42) = 96.

Crossrefs

Cf. A000203, A000217.
Cf. A090777, A356410, A369096, A369097.
Subsequence: A000040.

Programs

  • Maple
    isA369093 := proc(k)
    if modp((k+1)^2, numtheory[sigma](k)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A369093 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA369093(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
[seq(A369093(n),n=1..100)] ; # R. J. Mathar, Jan 18 2024
  • PARI
    isok(n) = my(x=(n+1)^2,y=sigma(n));!(x%y);

A369096 Numbers k >= 2 such that omega(k) divides the sum of the triangular numbers T(k) and T(k+1), where omega(k) is the number of distinct primes dividing k (A001221).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 110, 111, 113, 115, 117, 119, 121, 123, 125, 127, 128, 129
Offset: 1

Views

Author

Claude H. R. Dequatre, Jan 13 2024

Keywords

Comments

k is a term if (k^2+k)/2 + ((k+1)^2+k+1)/2 = k^2+2*k+1 = (k+1)^2 is divisible by omega(k).
Trivial case: If k is prime, then omega(k) = 1 and (k+1)^2 is always divisible by 1, thus all primes are terms of this sequence.
Table with percentage of primes <= 10^k for k = 2..9:
| k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
| 2 | 55 | 25 | 45.45 |
| 3 | 506 | 168 | 33.20 |
| 4 | 4832 | 1229 | 25.43 |
| 5 | 46675 | 9592 | 20.55 |
| 6 | 456155 | 78498 | 17.21 |
| 7 | 4480617 | 664579 | 14.83 |
| 8 | 44081959 | 5761455 | 13.07 |
| 9 | 433916814 | 50847535 | 11.72 |
The percentage of primes decreases asymptotically as 10^k increases.
Conjecture: the asymptotic density of primes in this sequence is 0.

Examples

			2 is a term since (2+1)^2 = 3^2 = 9 is divisible by omega(2) = 1.
15 is a term since (15+1)^2 = 16^2 = 256 is divisible by omega(15) = 2.
12 is not a term since (12+1)^2 = 13^2 = 169 is not divisible by omega(12) = 2.

Crossrefs

Cf. A000217, A001221.
Cf. A090777, A356410, A369093, A369097.
Subsequence: A000040.

Programs

  • Maple
    isA369096 := proc(k)
    if modp((k+1)^2, A001221(k)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A369096 := proc(n)
    option remember ;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA369096(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
[seq(A369096(n),n=1..100)] ; # R. J. Mathar, Jan 18 2024
  • PARI
    isok(n)=my(x=(n+1)^2,y=omega(n));!(x%y);

A245787 Numbers n such that tau(n)*sigma(n) divides n^2.

Original entry on oeis.org

1, 26208, 56896, 293760, 997920, 9694080, 23569920, 25159680, 29669760, 67858560, 117849600, 132723360, 208565280, 222963840, 276756480, 427714560, 513786240, 578672640, 628992000, 649503360, 688279680, 714954240, 779950080, 830269440, 979102080, 1037266560
Offset: 1

Views

Author

Jaroslav Krizek, Aug 08 2014

Keywords

Comments

a(11) > 10^8. - Derek Orr, Aug 08 2014
Numbers n such that n^2 / (A000005(n) * A000203(n)) is an integer.
Subsequence of A090777 (numbers n such that sigma(n) divides n^2).
Sequence of numbers k(n) = n^2 / (tau(n) * sigma(n)): 1, 104, 889, 612, 945, 7344, …

Links

Crossrefs

Cf. A000005, A000203, A071707, A090777.

Programs

  • Magma
    [n: n in [1..1000000] | Denominator(n^2 / ((#[d: d in Divisors(n)])* SumOfDivisors(n))) eq 1];
  • Mathematica
    a245787[n_Integer] := Select[Range[n], Divisible[#^2, DivisorSigma[0, #]*DivisorSigma[1, #]]&] (* Michael De Vlieger, Aug 09 2014 *)
  • PARI
    for(n=1,10^8,if(n^2%(numdiv(n)*sigma(n))==0,print1(n,", "))) \\ Derek Orr, Aug 08 2014

Extensions

a(7)-a(10) from Derek Orr, Aug 08 2014
a(11)-a(26) from Jens Kruse Andersen, Aug 13 2014
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.