A181841 -id:A181841 - cp's OEIS frontend

A350675 Numbers k such that tau(k) + tau(k+1) + tau(k+2) = 10, where tau is the number of divisors function A000005.

6, 11, 13, 17, 21, 37, 57, 157, 177, 381, 501, 541, 717, 877, 1201, 1317, 1381, 1437, 1621, 1821, 2017, 2557, 2577, 2857, 2901, 3061, 3117, 3777, 4281, 4357, 4441, 4677, 4701, 5077, 5097, 5581, 5637, 5701, 5937, 6337, 6637, 6661, 6717, 6997, 7417, 8221, 8781

Offset: 1

Jon E. Schoenfield, Jan 10 2022

nonn

Since tau(k) + tau(k+1) + tau(k+2) = 10 and no three consecutive integers include more than one square, the triple (tau(k), tau(k+1), tau(k+2)) must consist of three even numbers, so it must be one of (2, 2, 6), (2, 4, 4), (2, 6, 2), (4, 2, 4), (4, 4, 2), and (6, 2, 2). Of these, (2, 2, 6) and (6, 2, 2) are impossible. Of the remaining patterns:
(2, 4, 4) requires that k be an odd prime other than 3, followed by two semiprimes, so k is a prime p such that (p+1)/2 and (p+2)/3 are also prime, and such primes are 13, 37, 157, 541, ... (A036570);
(2, 6, 2) requires that (k, k+2) be a twin prime pair whose average has exactly 6 divisors, and is thus either 12 or 18, so k is 11 or 17;
(4, 2, 4) requires that k+1 be an odd prime, with both k and k+2 having exactly 4 divisors, even though one of them is a multiple of 4, so that one is k+2 = 2^3 = 8, so k = 6;
(4, 4, 2) requires that k+2 be an odd prime > 3, preceded by two semiprimes, so k+2 is a prime p such that (p-1)/2 and (p-2)/3 are also prime, so k+2 is in {23, 59, 179, 383, ...} (which is A181841, after its first two terms, 7 and 11), so k is in {A181841(n) - 2} \ {5, 9}, i.e., k is in {21, 57, 177, 381, ...}.
Tau(k) + tau(k+1) + tau(k+2) >= 10 for all sufficiently large k; the only numbers k for which tau(k) + tau(k+1) + tau(k+2) < 10 are 1..5, 7, and 9.

			Each of the patterns (tau(k), ..., tau(k+2)) that appears repeatedly for large k corresponds to one of the two possible orders in which the multipliers m=1..3 can appear among 3 consecutive integers of the form m*prime. E.g., k=37 begins a run of 3 consecutive integers having the form (p, 2*q, 3*r), where p, q, and r are distinct primes > 3; k=57 begins a similar run, but there the 3 consecutive integers have the form (3*p, 2*q, r).
For each of the patterns of tau values that does not occur repeatedly for large k, one or more of the three consecutive integers in k..k+2 has no prime factor > 3; in the table below, each such integer appears in parentheses in the columns on the right.
.
                              factorization as
              # divisors of    m*(prime > 3)
  n  a(n)=k     k  k+1 k+2      k   k+1  k+2
  -  ------    --- --- ---    ---- ---- ----
  1      6      4   2   4      (6)   q   (8)
  2     11      2   6   2       p  (12)   r
  3     13      2   4   4       p   2q   3r
  4     17      2   6   2       p  (18)   r
  5     21      4   4   2      3p   2q    r
  6     37      2   4   4       p   2q   3r
  7     57      4   4   2      3p   2q    r

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000005, A036570, A181841, A307120, A317670.

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), (this sequence) (N=3), A350686 (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).

Position[Plus @@@ Partition[Array[DivisorSigma[0, #] & , 10^4], 3, 1], 10] // Flatten (* Amiram Eldar, Jan 11 2022 *)

isok(k) = numdiv(k) + numdiv(k+1) + numdiv(k+2) == 10; \\ Michel Marcus, Jan 16 2022

{ k : tau(k) + tau(k+1) + tau(k+2) = 10 }.
UNION({6}, {11, 17}, A036570, {A181841(n) - 2} \ {5, 9}).
a(n) = A317670(n) - 1.

A247347 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3.

Original entry on oeis.org

11, 23, 719, 1439, 5639, 25799, 28319, 35879, 56039, 58679, 77279, 98999, 104759, 121559, 166919, 174599, 206639, 253679, 334319, 350159, 424079, 433439, 451679, 452759, 535919, 539159, 582719, 595319, 645839, 671039, 743279, 818999, 824039

Offset: 1

Jean-Christophe Hervé, Sep 14 2014

nonn

Could be called 3-safe primes, or safe primes of order 3, as the safe primes are the primes such that (p-1)/2 is prime.
Obviously a subsequence of the safe primes A005385 and of the supersafe primes A181841; thus (a(n)-1)/2 is a Sophie Germain prime (cf. A005384).
These numbers generate sequences 4-3-2-1 in A052126.
a(n) == -1 (mod 120) for n > 2: because (a(n)-1)/2, (a(n)-2)/3 and (a(n)-3)/4 must be integer, a(n) = -1 (mod 12), thus a(n) = -1 (mod 24) or a(n) = 11 mod(24) for all n; if a(n) = 11 (mod 24), (a(n)-3)/4 = 2 (mod 24) and would be even and not prime unless n=1; thus a(n) = -1 (mod 24) for n > 1. Now, if a(n) = 23 or 47 or 71 or 95 (mod 120), one of the (a(n)-k)/k is a multiple of 5 and thus not prime unless n = 2 and a(2) = 23 (in which case (23-3)/4 is exactly 5); thus a(n) == -1 (mod 120) for n > 2.

			a(1) = 11 because 11, (11-1)/2 = 5, (11-2)/3 = 3 and (11-3)/4 = 2 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..45 from Jean-Christophe Hervé)

Cf. A005384 (Sophie Germain primes), A005385 (safe primes), A181841 (supersafe primes), A247348 (4-safe primes), A163573 (similar definition with (p+k)/(k+1) as primes).

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[(p-2)/3]&&PrimeQ[(p-3)/ 4], AppendTo[lst, p]], {n, 2*9!}]; lst
Select[Prime[Range[70000]],AllTrue[Table[(#-k)/(k+1),{k,3}],PrimeQ]&] (* Harvey P. Dale, Mar 09 2024 *)

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), print1(p, ", ")););} \\ Michel Marcus, Sep 15 2014

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

Jean-Christophe Hervé, Sep 14 2014

nonn

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.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

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

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 *)

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

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

More terms from Michel Marcus, Sep 15 2014

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A350675 Numbers k such that tau(k) + tau(k+1) + tau(k+2) = 10, where tau is the number of divisors function A000005.

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A247347 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3.

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

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