A132231 -id:A132231 - cp's OEIS frontend

A007528 Primes of the form 6k-1.

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A030432 Primes of form 10n+7.

Original entry on oeis.org

7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277, 307, 317, 337, 347, 367, 397, 457, 467, 487, 547, 557, 577, 587, 607, 617, 647, 677, 727, 757, 787, 797, 827, 857, 877, 887, 907, 937, 947, 967, 977, 997, 1087, 1097, 1117, 1187, 1217, 1237

Offset: 1

Warut Roonguthai

Union of A132231 and A039949. - Ray Chandler, Apr 07 2009
5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Also primes of the form 5n+2 with positive n. - Danny Rorabaugh, Feb 20 2016
Intersection of A000040 and A017353. - Iain Fox, Dec 30 2017

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Cf. A030430 (10n+1), A030431 (10n+3), A030433 (10n+9).

Cf. A045357, A045380, A049509, A102342.

[n: n in [7..1240 by 10] | IsPrime(n)]; // Bruno Berselli, Apr 06 2011

Select[Prime@Range[210], Mod[ #, 10] == 7 &] (* Ray Chandler, Nov 07 2006 *)

is(n)=n%10==7 && isprime(n) \\ Charles R Greathouse IV, Jul 01 2013

lista(nn) = forprime(p=7, nn, if(p%10==7, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

[10*n+7 for n in range(124) if is_prime(10*n+7)] # Danny Rorabaugh, Feb 20 2016

a(n) = 10*A102342(n) + 7.

a(n) ~ 4n log n. - Charles R Greathouse IV, Jul 01 2013

Extended by Ray Chandler, Nov 07 2006

A132230 Primes congruent to 1 (mod 30).

Original entry on oeis.org

31, 61, 151, 181, 211, 241, 271, 331, 421, 541, 571, 601, 631, 661, 691, 751, 811, 991, 1021, 1051, 1171, 1201, 1231, 1291, 1321, 1381, 1471, 1531, 1621, 1741, 1801, 1831, 1861, 1951, 2011, 2131, 2161, 2221, 2251, 2281, 2311, 2341, 2371, 2521, 2551, 2671

Offset: 1

Omar E. Pol, Aug 15 2007

Also primes congruent to 1 (mod 15). - N. J. A. Sloane, Jul 11 2008

Primes ending in 1 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 04 2009

			From _Muniru A Asiru_, Nov 01 2017: (Start)
31 is a prime and 31 = 30*1 + 1;
61 is a prime and 61 = 30*2 + 1;
151 is a prime and 151 = 30*5 + 1;
211 is a prime and 211 = 30*7 + 1;
241 is a prime and 241 = 30*8 + 1;
271 is a prime and 271 = 30*9 + 1.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A057204, A068228, A129805, A039949, A132231-A132236.

A132230 := Filtered(Filtered([1..10^6], n -> n mod 30 = 1), IsPrime); # Muniru A Asiru, Nov 01 2017

select(isprime, [seq(i,i=1..1000,30)]); # Robert Israel, Jan 19 2016

Select[Range[1, 3000, 30], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)
Select[Prime[Range[400]],Mod[#,30]==1&] (* Harvey P. Dale, May 21 2021 *)

is(n)=isprime(n) && n%30==1 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A111175(n)*30 + 1. - Ray Chandler, Apr 07 2009

Intersection of A030430 and A002476. - Ray Chandler, Apr 07 2009

Edited by Ray Chandler, Apr 07 2009

A132232 Primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 491, 521, 641, 701, 761, 821, 881, 911, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1871, 1901, 1931, 2081, 2111, 2141, 2351, 2381, 2411, 2441, 2531

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A068231, A039949.

Cf. A132230, A132231, A132233, A132234, A132235, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{11},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[11,2600,30],PrimeQ] (* Harvey P. Dale, Mar 07 2020 *)

is(n)=isprime(n) && n%30==11 \\ Charles R Greathouse IV, Jul 01 2016

From Ray Chandler, Apr 07 2009: (Start)
a(n) = A158614(n)*30 + 11.
Intersection of A030430 and A007528. (End)

Extended by Ray Chandler, Apr 07 2009

A132235 Primes congruent to 23 (mod 30).

Original entry on oeis.org

23, 53, 83, 113, 173, 233, 263, 293, 353, 383, 443, 503, 563, 593, 653, 683, 743, 773, 863, 953, 983, 1013, 1103, 1163, 1193, 1223, 1283, 1373, 1433, 1493, 1523, 1553, 1583, 1613, 1733, 1823, 1913, 1973, 2003, 2063, 2153, 2213, 2243, 2273, 2333, 2393

Offset: 1

Omar E. Pol, Aug 15 2007

Primes (excluding 3) ending in 3 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches
The sequence is infinite by Dirichlet's theorem. - Arkadiusz Wesolowski, Apr 02 2014
Terms are non-twin primes A007510. - Omar E. Pol, Jul 25 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Omar E. Pol, Prime numbers in a sieve with 30 columns

Cf. A000040, A039949.

Cf. A132230, A132231, A132232, A132233, A132234, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 23 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[23,2400,30],PrimeQ] (* Harvey P. Dale, Jan 27 2020 *)

is(n)=isprime(n) && n%30==23 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A158791(n)*30 + 23. - Ray Chandler, Apr 07 2009

Intersection of A030431 and A007528. - Ray Chandler, Apr 07 2009

Extended by Ray Chandler, Apr 07 2009

A158573 Numbers k such that 30*k + 7 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 24, 25, 26, 29, 30, 31, 32, 33, 36, 37, 41, 43, 44, 48, 52, 53, 54, 55, 58, 59, 62, 66, 67, 71, 76, 78, 79, 81, 82, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 101, 102, 106, 107, 110, 115, 117, 118, 120, 121, 123, 124, 128

Offset: 1

Ki Punches, Mar 21 2009

nonn

Encoded primes with LSD 7 and (SOD-1)/3 integer, (LSD, least significant digit; SOD, sum of digits). Divide any such number by 30, if the whole number portion of the quotient is in the sequence, the number is prime.

			Example: 3877, with LSD 7 and (SOD-1)/3 = 23 (integer); Then 3877/30 = 129.233, or 129, which is in the sequence, and thus 3877 is prime.

Cf. A111175, A158614, A158648, A158746, A158791, A158806, A158850.

Select[Range[0,200],PrimeQ[30#+7]&] (* Harvey P. Dale, Jul 20 2018 *)

is(n)=isprime(30*n+7) \\ Charles R Greathouse IV, Mar 07 2016

a(n) = (A132231(n) - 7)/30 = floor(A132231(n)/30). - Ray Chandler, Apr 07 2009

a(n) ~ (4/15) n log n. - Charles R Greathouse IV, Mar 07 2016

Edited by Ray Chandler, Apr 07 2009

A211890 Triangle read by rows, where row n starts with n-th prime, followed by n primes in arithmetic progression; T(0,0) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 7, 37, 67, 97, 127, 11, 71, 131, 191, 251, 311, 13, 244243, 488473, 732703, 976933, 1221163, 1465393, 17, 6947, 13877, 20807, 27737, 34667, 41597, 48527, 19, 546859, 1093699, 1640539, 2187379, 2734219, 3281059, 3827899

Offset: 0

Reinhard Zumkeller, Jul 13 2012

T(n,0) = A000040(n) and T(n,k+1) - T(n,k) = A211889(n), 0 <= k < n.

			First 9 rows of triangle:
0:  1
1:  2 3
2:  3 5 7
3:  5 11 17 23
4:  7 37 67 97 127
5:  11 71 131 191 251 311
6:  13 244243 488473 732703 976933 1221163 1465393
7:  17 6947 13877 20807 27737 34667 41597 48527
8:  19 546859 1093699 1640539 2187379 2734219 3281059 3827899 4374739

Chai Wah Wu, Rows n=0..10 of triangle, flattened (rows n = 0..9 from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions

Cf. A007528, A007652, A132231, A214360, A008578, A002262.

a211890 n k = a211890_tabl !! n !! k
a211890_row n = a211890_tabl !! n
a211890_tabl = zipWith3 (\p k row -> map ((+ p) . (* k)) row)
                        a008578_list (0 : a211889_list) a002262_tabl

A117047 Primes of the form 60*n+11.

Original entry on oeis.org

11, 71, 131, 191, 251, 311, 431, 491, 911, 971, 1031, 1091, 1151, 1451, 1511, 1571, 1811, 1871, 1931, 2111, 2351, 2411, 2531, 2591, 2711, 3011, 3191, 3251, 3371, 3491, 3671, 3851, 3911, 4091, 4211, 4271, 4391, 4451, 4691, 4751, 4871, 4931, 5051, 5171

Offset: 1

Roger L. Bagula, Apr 17 2006

nonn

a(n) = A211890(5,n-1) for n <= 6. - Reinhard Zumkeller, Jul 13 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007528, A132231, A214360, A010051.

a117047 n = a117047_list !! (n-1)
a117047_list = [x | k <- [0..], let x = 60 * k + 11, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

Flatten[Table[If[PrimeQ[60*n + 11], 60*n + 11, {}], {n, 0, 100}]]
Select[60Range[0,100]+11,PrimeQ] (* Harvey P. Dale, Feb 16 2024 *)

Wrong formula removed by Reinhard Zumkeller, Jul 13 2012

A132237 Primes congruent to {7, 23} mod 30.

Original entry on oeis.org

7, 23, 37, 53, 67, 83, 97, 113, 127, 157, 173, 233, 263, 277, 293, 307, 337, 353, 367, 383, 397, 443, 457, 487, 503, 547, 563, 577, 593, 607, 653, 683, 727, 743, 757, 773, 787, 863, 877, 907, 937, 953, 967, 983, 997, 1013, 1087, 1103

Offset: 1

Omar E. Pol, Aug 15 2007

Up to 4913, there are more primes of this form than composites. See also A132231 and A227869 (congruent to 7 only). - M. F. Hasler, Nov 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A007510, A039949, A068229, A129807.

[ p: p in PrimesUpTo(1300) | p mod 30 in [7, 23] ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{7,23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

is_A132237(n)=setsearch([7,23],n%30)&&isprime(n) \\ - M. F. Hasler, Nov 02 2013

A271114 Expansion of (1+x)*(2+x)/(1-x)^2.

Original entry on oeis.org

2, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325

Offset: 0

Colin Barker, Mar 31 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016921, A270700.

Cf. A016969, A082369, A100764, A128470, A132231, A156376.

Join[{2}, LinearRecurrence[{2, -1}, {7, 13}, 100]] (* G. C. Greubel, Mar 31 2016 *)

PARI
```
Vec((1+x)*(2+x)/(1-x)^2 + O(x^70))
```

G.f.: (1+x)*(2+x)/(1-x)^2.
a(n) = A270700(n)/6.
a(n) = 6*n+1 = A016921(n) for n>0.
a(n) = 2*a(n-1)-a(n-2) for n>2.
E.g.f.: 1 + (1+6*x)*exp(x). - G. C. Greubel, Mar 31 2016
From Bruno Berselli and G. C. Greubel, Mar 31 2016: (Start)
a(5*m+1) = 30*m + 7 = A132231(m+1).
a(5*m+2) = 30*m + 13 = A082369(m+1).
a(5*m+3) = 30*m + 19 = A156376(m).
a(5*m+4) = 30*m + 25 = 5*A016969(m).
a(5*m+5) = 30*m + 31 = A128470(m+1). (End)
a(n) = A100764(n+3) for n >= 1. - Georg Fischer, Oct 30 2018

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A030432 Primes of form 10n+7.

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A132230 Primes congruent to 1 (mod 30).

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A132232 Primes congruent to 11 (mod 30).

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A132235 Primes congruent to 23 (mod 30).

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A158573 Numbers k such that 30*k + 7 is prime.

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