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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A350686 Numbers k such that tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16, where tau is the number of divisors function A000005.

Original entry on oeis.org

12, 17, 19, 20, 26, 31, 211, 716, 1226, 1436, 2306, 2731, 2971, 5636, 8011, 12146, 12721, 16921, 18266, 19441, 24481, 24691, 25796, 28316, 30026, 34651, 35876, 37171, 45986, 49681, 51691, 56036, 58676, 61561, 67531, 77276, 98731, 98996, 104161, 104756, 108571
Offset: 1

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Author

Jon E. Schoenfield, Jan 11 2022

Keywords

Comments

It can be shown that if tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16, the quadruple (tau(k), tau(k+1), tau(k+2), tau(k+3)) must be one of the following, each of which might plausibly occur infinitely often:
(2, 4, 4, 6), which first occurs at k = 12721, 16921, 19441, 24481, ... (A163573);
(2, 6, 4, 4), which first occurs at k = 19, 31, 211, 2731, ...;
(4, 4, 6, 2), which first occurs at k = 26, 1226, 2306, 12146, ...;
(6, 4, 4, 2), which first occurs at k = 20, 716, 1436, 5636, ...; ({A247347(n)-3}, other than its first term)
or one of the following, each of which occurs only once:
(2, 6, 2, 6), which occurs only at k = 17; and
(6, 2, 4, 4), which occurs only at k = 12.
Tau(k) + tau(k+1) + tau(k+2) + tau(k+3) >= 16 for all sufficiently large k; the only numbers k for which tau(k) + tau(k+1) + tau(k+2) + tau(k+3) < 16 are 1..11, 13, 14, and 16.

Examples

			The table below includes all terms k such that at least one of the four numbers k, k+1, k+2, k+3 has no prime factor > 5; each such number appears in parentheses in the columns under "factorization".
The table also includes, for each of the patterns (tau(k), tau(k+1), tau(k+2), tau(k+3)) that continues to appear for large k, the smallest such k for which each of the four numbers k, k+1, k+2, k+3 has a prime factor > 5. For each such quadruple, each of the four numbers is the product of a distinct multiplier m from 1..4 and a prime > 5, and each pattern corresponds to a distinct value of k mod 120: the tau patterns (2, 4, 4, 6), (2, 6, 4, 4), (4, 4, 6, 2), and (6, 4, 4, 2) correspond to k mod 120 = 1, 91, 26, and 116, respectively.
.
                                factorization as
                # divisors of     m*(prime > 5)
   n  a(n)=k    k  k+1 k+2 k+3    k  k+1 k+2 k+3   k mod 120
   -  ------   --- --- --- ---   --- --- --- ---   ---------
   1      12    6   2   4   4    (12)  q  2r  3s       12
   2      17    2   6   2   6      p (18)  r  4s       17
   3      19    2   6   4   4      p (20) 3r  2s       19
   4      20    6   4   4   2    (20) 3q  2r   s       20
   5      26    4   4   6   2     2p (27) 4r   s       26
   6      31    2   6   4   4      p (32) 3r  2s       31
   7     211    2   6   4   4      p  4q  3r  2s       91
   8     716    6   4   2   2     4p  3q  2r   s      116
   9    1226    4   4   6   2     2p  3q  4r   s       26
  17   12721    2   4   4   6      p  2q  3r  4s        1
		

Crossrefs

Numbers k such that Sum_{j=0..N-1} tau(k+j) = 2*Sum_{k=1..N} tau(k): A000040 (N=1), A350593 (N=2), A350675 (N=3), (this sequence) (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).

Programs

  • Mathematica
    Position[Plus @@@ Partition[Array[DivisorSigma[0, #] & , 10^5], 4, 1], 16] // Flatten (* Amiram Eldar, Jan 12 2022 *)
  • PARI
    isok(k) = numdiv(k) + numdiv(k+1) + numdiv(k+2) + numdiv(k+3) == 16; \\ Michel Marcus, Jan 12 2022
    
  • Python
    from sympy import divisor_count as tau
    print([k for k in range( 1, 108572) if tau(k) + tau(k+1) + tau(k+2) + tau(k+3) == 16]) # Karl-Heinz Hofmann, Jan 12 2022

Formula

{ k : tau(k) + tau(k+1) + tau(k+2) + tau(k+3) = 16 }.

A247348 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3, 4.

Original entry on oeis.org

174599, 334319, 535919, 671039, 907199, 2129399, 2298119, 3103799, 3369959, 4351199, 4598159, 5697599, 6184799, 6446159, 7224839, 7943759, 7957319, 8148839, 8346959, 8656919, 9096359, 9339119, 9463319, 9511199, 10514159, 10780559, 11816999, 12424319, 13781039
Offset: 1

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Author

Keywords

Comments

Could be called 4-safe primes, or safe primes of order 4, as the safe primes are the primes such that (p-1)/2 is prime.
Obviously a subsequence of the k-safe primes for k < 4 : A005385 (safe primes, k=1), A181841 (supersafe primes, k=2), A247347 (k=3).
a(n) = 119 (mod 120) for all n.
These numbers generate sequences 5-4-3-2-1 in A052126.

Crossrefs

Cf. A005385 (safe primes), A181841 (supersafe primes), A247347 (3-safe primes), A163573 (similar definition with (p+k)/(k+1) as primes).

Programs

  • Mathematica
    lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[(p-2)/3]&&PrimeQ[(p-3)/4]&&PrimeQ[(p-4)/5], AppendTo[lst, p]], {n, 2*9!}]; lst
    Select[Prime[Range[900000]],AllTrue[Table[(#-k)/(k+1),{k,4}],PrimeQ]&] (* Harvey P. Dale, Jul 07 2025 *)
  • PARI
    isokp(v) = (type(v) == "t_INT") && isprime(v);
    lista(nn) = {forprime(p=2, nn, if (isokp((p-1)/2) && isokp((p-2)/3) && isokp((p-3)/4) && isokp((p-4)/5), print1(p, ", ")););} \\ Michel Marcus, Sep 15 2014
    
  • Python
    from _future_ import division
    from sympy import prime, isprime
    A247348_list = [p for p in (5*prime(n)+4 for n in range(1,10**6)) if not ((p-1) % 2 or (p-2) % 3 or (p-3) % 4) and isprime(p) and isprime((p-1)//2) and isprime((p-2)//3) and isprime((p-3)//4)] # Chai Wah Wu, Sep 18 2014

Extensions

More terms from Michel Marcus, Sep 15 2014
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