A163573 -id:A163573 - cp's OEIS frontend

A163625 Numbers k such that 120*k + 1 is a term in A163573.

106, 141, 162, 204, 414, 513, 868, 1911, 2128, 2164, 2220, 2431, 2485, 2592, 2766, 3495, 3529, 3640, 3724, 3927, 4377, 4440, 4495, 4846, 5005, 5530, 5868, 5916, 6531, 7225, 7540, 7567, 7653, 9283, 9612, 9633, 10017, 10159, 10830, 11152, 11538, 12363

Offset: 1

Zak Seidov, Aug 01 2009

nonn

			a(1)=106 because 120*106 + 1 = 12721 = A163573(1).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A163573, A163623, A163624.

A163573 = {}; Do[p = Prime[n]; If[PrimeQ[(p + 1)/2] && PrimeQ[(p + 2)/3] && PrimeQ[(p + 3)/4], AppendTo[A163573, p]], {n, 2*9!}]; A163573;
Table[(A163573[[n]] - 1)/120, {n, 1, 50}] (* G. C. Greubel, Jul 30 2017 *)

a(n) = (A163573(n) - 1)/120.

A036570 Primes p such that (p+1)/2 and (p+2)/3 are also primes.

Original entry on oeis.org

13, 37, 157, 541, 877, 1201, 1381, 1621, 2017, 2557, 2857, 3061, 4357, 4441, 5077, 5581, 5701, 6337, 6637, 6661, 6997, 7417, 8221, 9181, 9661, 9901, 10837, 11497, 12457, 12601, 12721, 12757, 13681, 14437, 15241, 16921, 17077, 18217

Offset: 1

N. J. A. Sloane

nonn

The prime p is followed by two semiprimes; a third semiprime is not possible. - T. D. Noe, Jul 23 2008
A subsequence of A005383, which has A163573 as a subsequence. - M. F. Hasler, Feb 26 2012
Similarly, the only "prime sandwiched by semiprimes" is 5. - Zak Seidov, Aug 04 2013
For n > 1, a(n) == 1 or (7 mod 10). If a(n) == 3 (mod 10), then (a(n) + 2)/3 == 0 (mod 5) which is a composite number if a(n) > 13. - Chai Wah Wu, Nov 30 2016
All terms are congruent to 1 (mod 12). - Zak Seidov, Feb 16 2017

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005383, A074200, A093553, A147615, A163573.
A278583 is an equivalent sequence.
See also A278585.

lst={};Do[p=Prime[n];If[PrimeQ[(p+1)/2]&&PrimeQ[(p+2)/3],AppendTo[lst,p]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 31 2009 *)

is_A036570(n)={ !(n%3-1) & isprime(n\3+1) & isprime(n\2+1) & isprime(n) }
for(n=1,2e4,is_A036570(n) & print1(n","))  \\ M. F. Hasler, Feb 26 2012

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

Jon E. Schoenfield, Jan 11 2022

nonn

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.

			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

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

Cf. A000005, A163573, A247347.

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).

Position[Plus @@@ Partition[Array[DivisorSigma[0, #] & , 10^5], 4, 1], 16] // Flatten (* Amiram Eldar, Jan 12 2022 *)

isok(k) = numdiv(k) + numdiv(k+1) + numdiv(k+2) + numdiv(k+3) == 16; \\ Michel Marcus, Jan 12 2022

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

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

A050498 Arithmetic progressions of at least 4 terms with common difference 6 having the same value of phi(x) start at these numbers.

Original entry on oeis.org

72, 216, 76326, 101526, 116646, 146886, 298086, 369366, 624966, 1375926, 1532166, 1558086, 1598406, 1750326, 1789206, 1866246, 1991526, 2516406, 2540886, 2620806, 2681286, 2827446, 3151446, 3196806, 3236406, 3489126

Offset: 1

Jud McCranie, Dec 27 1999

nonn

From Wolfdieter Lang, Jan 11 2021: (Start)
Conjecture: a(n) == 0 (mod 6) for n >= 1. After division by 6 the sequence becomes [12, 36, 12721, 16921, 19441, 24481, 49681, 61561, 104161, 229321, 255361, 259681, 266401, 291721, 298201, 311041, 331921, ...].
6*A163573 is a subsequence. See A163573 for the proof. Note that not all a(n), for n >= 3, are obtained by 6*A163573. The first such term is a(115) = 31850496, and a(115)/6 = 5308416 which is not a prime number, hence not a term of A163573. (End)
Wells gives a wrong value of a(3): 76236. - Stefano Spezia, Sep 08 2024

			phi(72) = phi(78) = phi(84) = phi(90) = 24, so 72 is in the sequence.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.

Jud McCranie, Table of n, a(n) for n = 1..10000

Cf. A000010, A039670, A163573.

isok(k) = #Set(vector(4, i, eulerphi(k+(i-1)*6))) == 1; \\ Michel Marcus, Sep 17 2023

A093553 a(n) is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,n.

Original entry on oeis.org

2, 3, 13, 12721, 19441, 5516281, 5516281, 7321991041, 363500177041, 2394196081201, 3163427380990801, 22755817971366481, 3788978012188649281, 2918756139031688155201

Offset: 1

Farideh Firoozbakht, Apr 14 2004

This sequence is A074200(n) + 1. See that entry for more information. - N. J. A. Sloane, May 04 2009
It is obvious that this sequence is increasing and each term is prime. If n > 3 then a(n) == 1 (mod 10).
From Jean-Christophe Hervé, Sep 14 2014: (Start)
a(n) == 1 (mod 120) for all n > 3 (see A163573).
a(4) = 12721 is a quite remarkable number: it is a palindromic prime, its 5 (prime) digits sum to 13, still a prime number (and the preceding element in this sequence, among other things), and as the fourth element of this sequence, it is the smallest prime such that (p-1)/2, (p-2)/3 and (p-3)/4 are also prime, and many other properties. (End)

			a(9)=363500177041 because all the nine numbers 363500177041,
(363500177041+1)/2, (363500177041+2)/3, (363500177041+3)/4,
(363500177041+4)/5, (363500177041+5)/6, (363500177041+6)/7,
(363500177041+7)/8 and (363500177041+8)/9 are primes and
363500177041 is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,9.

Walter Nissen, Calculation without Words : Doric Columns of Primes, Up for the Count !

Cf. A072875.

A163623 Primes of the form 120*k + 1.

Original entry on oeis.org

241, 601, 1201, 1321, 1801, 2161, 2281, 2521, 3001, 3121, 3361, 4201, 4441, 4561, 4801, 5281, 5521, 5641, 5881, 6121, 6361, 6481, 6841, 6961, 7321, 7561, 7681, 8161, 8521, 8641, 8761, 9001, 9241, 9601, 9721, 10321, 11161, 12241, 12601, 12721, 12841

Offset: 1

Zak Seidov, Aug 01 2009

nonn

a(n) = 120*A163624(n) + 1.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A163573, A163624.

[a: n in [0..250]|IsPrime(a) where a is 120*n+1] // Vincenzo Librandi, Dec 13 2010

Select[120*Range[120]+1,PrimeQ] (* Harvey P. Dale, Dec 05 2013 *)

lista(nn) = forprime(p=2, nn, if (!((p-1) % 120), print1(p, ", "))); \\ Michel Marcus, Jul 31 2017

A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

Original entry on oeis.org

12720, 16920, 19440, 24480, 49680, 61560, 104160, 229320, 255360, 259680, 266400, 291720, 298200, 311040, 331920, 419400, 423480, 436800, 446880, 471240, 525240, 532800, 539400, 581520, 600600, 663600, 704160, 709920, 783720, 867000, 904800, 908040, 918360

Offset: 1

N. J. A. Sloane, Nov 30 2016

nonn

a(n) == 0 mod 120 (see comment in A163573). - Chai Wah Wu, Nov 30 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Equals A163573(n) - 1.
Cf. A036570, A074200.
Positions of terms >= 4 in A278500, thus a subsequence of A278583, A089965 and A006093.

Select[Range[920000],AllTrue[{#+1,(#+2)/2,(#+3)/3,(#+4)/4},PrimeQ]&] (* Harvey P. Dale, Aug 08 2021 *)

is(k)=k%120==0 && isprime(k+1) && isprime(k/2+1) && isprime(k/3+1) && isprime(k/4+1) \\ Charles R Greathouse IV, Dec 03 2016

from sympy import prime, isprime
A278585_list = [4*q-4 for q in (prime(i) for i in range(1,10000)) if isprime(4*q-3) and isprime(2*q-1) and (not (4*q-1) % 3) and isprime((4*q-1)//3)] # Chai Wah Wu, Nov 30 2016

A204592 Primes p such that (p+1)/2, (p+2)/3, (p+3)/4 and (p+4)/5 are also prime.

Original entry on oeis.org

19441, 266401, 423481, 539401, 600601, 663601, 908041, 1113961, 1338241, 1483561, 1657441, 1673401, 2578801, 3109681, 3150841, 3336601, 3613681, 4112761, 4160641, 4798081, 5114881, 5412961, 5516281, 5590201, 5839681, 6078361, 7660801, 8628481, 9362641, 9388801, 9584401, 9733081

Offset: 1

M. F. Hasler, Feb 26 2012

nonn

Equivalently, primes p in A163573 such that p+4 is a semiprime. (Since all p in A163573 are of the form p=120k+1, p+4 is necessarily a multiple of 5. The other prime factor is then (p+4)/5 = 24k+1.)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[Prime[Range[700000]],AllTrue[{(#+1)/2,(#+2)/3,(#+3)/4,(#+4)/5},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 05 2017 *)

{my(p=1); until(, isprime(p+=120) || next; for( j=2,5, isprime(p\j+1) || next(2)); print1(p","))}

forprime(p=2,1e7,if(p%120==1&&isprime((p+1)/2)&&isprime((p+2)/3)&& isprime((p+3)/4)&&isprime((p+4)/5),print1(p", "))) \\ Charles R Greathouse IV, Feb 26 2012

A204592 = A163573 intersect A136061.

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

cp's OEIS Frontend

A163625 Numbers k such that 120*k + 1 is a term in A163573.

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A036570 Primes p such that (p+1)/2 and (p+2)/3 are also primes.

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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.

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A050498 Arithmetic progressions of at least 4 terms with common difference 6 having the same value of phi(x) start at these numbers.

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A093553 a(n) is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,n.

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A163623 Primes of the form 120*k + 1.

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A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

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A204592 Primes p such that (p+1)/2, (p+2)/3, (p+3)/4 and (p+4)/5 are also prime.

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