A024899 -id:A024899 - cp's OEIS frontend

A061599 Primes p such that the greatest prime divisor of p-1 is 5.

11, 31, 41, 61, 101, 151, 181, 241, 251, 271, 401, 541, 601, 641, 751, 811, 1201, 1601, 1621, 1801, 2161, 2251, 3001, 4001, 4051, 4801, 4861, 6481, 7681, 8101, 8641, 9001, 9601, 9721, 11251, 14401, 15361, 16001, 19441, 21601, 21871, 22501, 23041, 24001

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2Pi/n) is an algebraic number of 5th degree. - Artur Jasinski, Dec 13 2006

The least significant digit of each term is one. - Harvey P. Dale, Jul 07 2024

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

The 3rd in a family of sequences after A019434(=Fermat-primes) and A058383.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A000040.
Cf. A004729, A058383, A125867-A125875, A024899.

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {5} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
Select[Prime[Range[3000]],Max[FactorInteger[#-1][[;;,1]]]==5&] (* Harvey P. Dale, Jul 07 2024 *)

{ default(primelimit, 167772161); n=0; forprime (p=3, 167772161, f=factor(p - 1)~; if (f[1, length(f)]==5, write("b061599.txt", n++, " ", p)) ) } \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), s, t); lim\=1; lim--; for(i=1, logint(lim\2, 5), t=2*5^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, if(isprime(s+1), listput(v, s+1)); s<<=1))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of the form 2^a*3^b*5^c + 1 with a and c > 0.

A125867 Numbers k such that p=6k+1 is prime and cos(2*Pi/p) is an algebraic number of a 3-smooth degree, but not 2-smooth.

Original entry on oeis.org

1, 2, 3, 6, 12, 16, 18, 27, 32, 72, 81, 96, 128, 192, 216, 243, 432, 486, 576, 648, 1728, 2048, 2916, 3072, 6561, 8748, 23328, 24576, 34992, 55296, 78732, 104976, 124416, 131072, 139968, 165888, 196608, 248832, 294912, 331776, 442368, 839808

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

Numbers k such that p=6k+1 is prime and the greatest prime divisor of p-1 is 3.

Cf. A024899, A058383, A125866-A125878.

Do[If[Take[FactorInteger[EulerPhi[6n+1]][[ -1]], 1]=={3} && PrimeQ[6n+1],Print[n]],{n,1,100000}]

Edited by Don Reble, Apr 24 2007

A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 5, 6, 10, 11, 12, 16, 17, 25, 26, 32, 37, 45, 46, 51, 55, 61, 62, 72, 76, 90, 95, 100, 101, 102, 121, 122, 125, 137, 142, 146, 165, 172, 177, 181, 186, 187, 205, 215, 216, 220, 237, 241, 242, 247, 257, 270, 276, 277, 282, 290, 291, 292, 296, 297, 310, 311, 312, 331, 332, 335, 347, 355, 356, 380, 381, 390

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 138 such numbers between 1 and 1000.
Prime pairs that differ by 6 are called "sexy" primes. Other prime pairs that differ by 6 are of the form 6n - 1 and 6n + 5.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd - c - d, 6cd + c + d - 1 or 6cd + c + d, that is, they are not (6c - 1)d - c - 1, (6c - 1)d - c, (6c + 1)d + c - 1 or (6c + 1)d + c.

			a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

For the primes see A023201, A046117.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024899 and A153218.
Cf. also A307561, A307563.

Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* Michael De Vlieger, Apr 15 2019 *)

isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A263309 Numbers k such that p=6k+1 and q=6k+1 are primes.

Original entry on oeis.org

1, 2, 6, 10, 12, 17, 25, 30, 40, 45, 46, 47, 52, 55, 61, 62, 66, 96, 100, 101, 110, 121, 125, 131, 142, 151, 156, 172, 177, 186, 195, 200, 220, 221, 230, 237, 242, 255, 261, 282, 296, 305, 312, 331, 332, 356, 360, 367, 370, 380, 381, 382, 391, 425, 432, 446, 461, 465, 475, 495, 506, 510, 527, 530

Offset: 1

Zak Seidov, Oct 13 2015

nonn

Subsequence of A024899.

The subsequence of primes in this sequence is A023256.

Cf. A023256, A024899.

isA263309 := proc(n)
    if isprime(6*n+1) then
        if isprime(36*n+7) then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 100 do
    if isA263309(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Oct 17 2015

Select[Range[1000],PrimeQ[p=6*#+1]&& PrimeQ[q=6*p+1]&]

isok(n) = isprime(p=6*n+1) && isprime(6*p+1); \\ Michel Marcus, Oct 17 2015

A061638 Primes p such that the greatest prime divisor of p-1 is 7.

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 281, 337, 379, 421, 449, 491, 631, 673, 701, 757, 883, 1009, 1051, 1373, 1471, 2017, 2269, 2521, 2647, 2689, 2801, 3137, 3361, 3529, 4201, 4481, 5881, 6301, 7001, 7057, 7351, 7561, 7841, 8233, 8821, 10501, 10753, 12097

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2*Pi/n) is an algebraic number of 7th degree. - Artur Jasinski, Dec 13 2006

			For n = {4, 8, 9, 12}, a(n)-1 = {70, 210, 280, 420} = 7*{10, 30, 40, 60}.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 500 terms from Smith)

The 4th in a family of sequences after A019434(=Fermat-primes), A058383, A061599.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A061599.
Cf. A004729, A058383, A125867-A125875, A024899.

Select[Prime[Range[2000]],FactorInteger[#-1][[-1,1]] ==7&]  (* Harvey P. Dale, Mar 12 2011 *)

default(primelimit, 108864001); n=0; forprime (p=3, 108864001, f=factor(p - 1)~; if (f[1, length(f)]==7, write("b061638.txt", n++, " ", p))) \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), t, t5, t7); lim\=1; lim--; for(a=1, logint(lim\2, 7), t7=2*7^a; for(b=0, logint(lim\t7, 5), t5=5^b*t7; for(c=0, logint(lim\t5, 3), t=3^c*t5; while(t<=lim, if(isprime(t+1), listput(v, t+1)); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of form 2^a*3^b*5^c*7^d + 1 with a and d > 1.

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

Original entry on oeis.org

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Eric Desbiaux, Feb 11 2010

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 2 + 3 OR 3* 1 + 4 =  7;
  2* 5 + 3 OR 3* 3 + 4 = 13;
  2* 8 + 3 OR 3* 5 + 4 = 19;
  2*14 + 3 OR 3* 9 + 4 = 31;
  2*17 + 3 OR 3*11 + 4 = 37;
  2*20 + 3 OR 3*13 + 4 = 43;
  2*29 + 3 OR 3*19 + 4 = 61;
  2*32 + 3 OR 3*21 + 4 = 67;
  2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Cf. A067076, A034936, A002476, A024899.

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

More terms from Harvey P. Dale, Aug 25 2016

A263310 Numbers n such that p=6n+1, q=6p+1 and r=6*q+1 are primes.

Original entry on oeis.org

10, 25, 55, 61, 101, 125, 156, 220, 221, 381, 391, 465, 475, 495, 576, 810, 891, 901, 975, 1060, 1145, 1396, 1430, 1630, 1650, 1726, 1795, 1811, 1881, 1885, 1915, 2196, 2265, 2335, 2391, 2405, 2456, 2536, 2575, 2636, 2651, 2820, 2911, 2915, 2951, 2965, 3051, 3211, 3245, 3335

Offset: 1

Zak Seidov, Oct 13 2015

nonn

Subsequence of A263309 (and as such also of A024899).

Cf. A024899, A263309.

isA263310 := proc(n)
    return isprime(6*n+1) and isprime(36*n+7) and isprime(216*n+43) ;
end proc:
for n from 1 to 3000 do
    if isA263310(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Oct 17 2015

Select[Range[10000],PrimeQ[p=6*#+1]&& PrimeQ[q=6*p+1]&& PrimeQ[r=6*q+1]&]

for(n=1, 1e4, if(isprime(p=6*n+1)&&isprime(q=6*p+1)&&isprime(6*q+1), print1(n", "))) \\ Altug Alkan, Oct 17 2015

A263311 Numbers n such that each of p=6n+1, q=6p+1, r=6q+1 and s=6r+1 is prime.

Original entry on oeis.org

10, 1060, 1795, 1885, 2965, 3245, 3335, 4065, 4325, 5015, 5875, 6985, 7605, 7905, 9785, 11315, 12045, 12360, 14390, 14970, 15285, 15500, 15885, 17195, 18220, 20670, 20695, 22160, 24915, 25645, 25955, 26025, 29410, 29910, 32925, 35530, 36280

Offset: 1

Zak Seidov, Oct 13 2015

nonn

Each p is a starting prime of a complete generalized Cunningham chain p(k)=6*p(k-1)+1.
All terms are multiples of 5. Hence t = 6s+1 = 1555+7776n are always composite, and the chains are indeed "complete."
Subsequence of A263310 (and as such of A263309 and of A024899).

Zak Seidov, Table of n, a(n) for n = 1..20000
Wikipedia, Cunningham chain

Cf. A024899, A263309, A263310.

isA263311 := proc(n)
    return isprime(6*n+1) and isprime(36*n+7) and isprime(216*n+43) and isprime(1296*n+259) ;
end proc:
for n from 1 to 30000 do
    if isA263311(n) then
        printf("%d,",n);
    end if;
end do; # R. J. Mathar, Oct 17 2015

Select[Range[10,100000,5],PrimeQ[p=6*#+1]&&PrimeQ[q=6*p+1]&&PrimeQ[r=6*q+1]&&PrimeQ[s=6*r+1]&]

for(n=1, 1e5, if(isprime(p=6*n+1) && isprime(q=6*p+1) && isprime(r=6*q+1) && isprime(6*r+1), print1(n", "))) \\ Altug Alkan, Oct 17 2015

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

Original entry on oeis.org

1530, 4260, 25410, 26040, 78540, 111720, 174990, 211050, 214830, 395430, 403260, 409290, 459690, 487830, 512820, 711120, 779790, 910560, 1023750, 1135950, 1280370, 1312350, 1451520, 1464810, 1487070, 1563510, 1623360, 1698060, 1824330, 1933680, 2006340, 2097480

Offset: 1

Pedro Caceres, May 04 2020

nonn

All terms are multiples of 6.

All terms are multiples of 30. - Robert Israel, Jun 17 2020

			25410 is in the sequence because 25411, 50821, 76231, 101641, 152461 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Pedro Caceres, 30 Ideas about Prime Numbers

Cf. A006093, A005097, A024892, A087370, A005098, A005099, A024898, A024899.

select(t -> andmap(isprime, [t+1,2*t+1,3*t+1,4*t+1,6*t+1]), [seq(i,i=30..3*10^6,30)]); # Robert Israel, Jun 17 2020

isok(m)={for(i=1, 6, if(i<>5&&!isprime(i*m+1), return(0))); 1}
{ forstep(n=0, 3*10^6, 6, if(isok(n), print1(n, ", "))) } \\ Andrew Howroyd, May 04 2020

A368202 Least k such that 6nk+1 is a prime.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 3, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 4, 1, 3, 3, 4, 5, 1, 1, 1, 2, 3, 2, 1, 1, 10, 4, 1, 1, 6, 1, 2, 5, 1, 1, 1, 2, 3, 1, 4, 1, 2, 1, 2, 1, 1, 4, 4, 1, 1, 2, 3, 7, 1, 6, 1, 2, 2, 2

Offset: 1

Robert Price, Dec 16 2023

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A016014, A024899.

A070850 lists the corresponding primes.

A368202 = {};
Do[k=1; While[!PrimeQ[9 n k+1], k++]; AppendTo[A368202, k], {n, 86}];
A368202

a(n) = my(k=1); while (!isprime(6*n*k+1), k++); k; \\ Michel Marcus, Dec 16 2023

cp's OEIS Frontend

A061599 Primes p such that the greatest prime divisor of p-1 is 5.

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A125867 Numbers k such that p=6k+1 is prime and cos(2*Pi/p) is an algebraic number of a 3-smooth degree, but not 2-smooth.

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A307562 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.

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A263309 Numbers k such that p=6k+1 and q=6k+1 are primes.

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A061638 Primes p such that the greatest prime divisor of p-1 is 7.

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A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

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A263310 Numbers n such that p=6*n+1, q=6*p+1 and r=6*q+1 are primes.

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A263311 Numbers n such that each of p=6*n+1, q=6*p+1, r=6*q+1 and s=6*r+1 is prime.

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A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.

A263310 Numbers n such that p=6n+1, q=6p+1 and r=6*q+1 are primes.

A263311 Numbers n such that each of p=6n+1, q=6p+1, r=6q+1 and s=6r+1 is prime.

A368202 Least k such that 6nk+1 is a prime.