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.

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);

A373618 Least prime starting a run of n consecutive primes p_i, i=1..n, such that p_i + 1 is squarefree and p_(n+1) + 1 is not squarefree.

Original entry on oeis.org

2, 37, 397, 389, 11617, 11597, 11593, 2048509, 2772409, 5193997, 33933701, 125624813, 125624809, 432787781, 432787777, 4762221193, 4762221181, 182839149373, 547414016069, 551900822513
Offset: 1

Views

Author

Jean-Marc Rebert, Jun 11 2024

Keywords

Examples

			a(1) = 2, because 2 is the least prime starting a run of 1 prime such that 2+1 is squarefree and 3+1 = 4 = 2^2 is not squarefree.
For n=4 the first run of 4 squarefree p+1 starts at a(4) = 389, and no run of n=3 so a(3) = 397 is the ending 3 of this run.
  p              = 389, 397, 401, 409, 419
  p+1 squarefree = yes  yes  yes  yes  no
  n=4 run          \----------------/
  n=3 run               \-----------/

Crossrefs

Cf. A005117, A072875, A086560, A369097.

Programs

  • Mathematica
    a[n_]:=Module[{k=1}, While[pr=Product[Boole[SquareFreeQ[Prime[k+i-1]+1]], {i, n}]==0||pr&& Boole[SquareFreeQ[Prime[k+n]+1]]==1, k++]; Prime[k]]; Array[a, 8] (* Stefano Spezia, Jun 11 2024 *)

A373533 Least starting prime of exactly n consecutive primes p_i (i = 1..n) such that omega(p_i + 1) = 1 + i.

Original entry on oeis.org

5, 23, 499, 13093, 501343, 162598021, 25296334003
Offset: 1

Views

Author

Jean-Marc Rebert, Jun 08 2024

Keywords

Examples

			a(1) = 5, because omega(5+1) = 2, and no lesser number has this property.
For n=3, the primes starting at a(3) = 499 are as follows and are a run of exactly 3 omega = i+1,
  i          =  1    2    3
  p          = 499, 503, 509, 521
  omega(p+1) =  2,   3,   4,   3
               \-----------/

Crossrefs

Cf. A001221, A086560, A369097.

Programs

  • Mathematica
    a[n_]:=Module[{k=1},While[Product[Boole[PrimeNu[Prime[k+i-1]+1]==1+i],{i,n}]==0, k++]; Prime[k]]; Array[a,5] (* Stefano Spezia, Jun 10 2024 *)

A373626 Least prime of a run of n consecutive primes p_i, i = 1..n, such that bigomega(p_i + 1) = omega(p_i + 1) + i and bigomega(p_(n+1) + 1) <> omega(p_(n+1) + 1) + n + 1, or -1 if no such prime exists.

Original entry on oeis.org

3, 19, 739, 76913, 4510333, 746264059, 290623032907
Offset: 1

Views

Author

Jean-Marc Rebert, Jun 11 2024

Keywords

Examples

			19 starts a run of 2 consecutive primes 19 and 23, bigomega(19+1) = 2 = omega(19+1) + 1, bigomega(23+1) = 4 = omega(23+1) + 2 and bigomega(29+1) = 3 <> omega(29+1) + 3. So a(2) = 19.
Let i, p, b and w be the indices, the primes p_i, bigomega(p_i + 1) and omega(p_i + 1).
i: [ 1  2  3]
p: [19 23 29]
b: [ 3  4  3]
w: [ 2  2  3]
a(2) = 19
i: [  1   2   3   4]
p: [739 743 751 757]
b: [  4   5   5   2]
w: [  3   3   2   2]
a(3) = 739

Crossrefs

Cf. A001221, A001222, A072875, A086560, A369097, A373618.
Showing 1-5 of 5 results.

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