A132230 -id:A132230 - cp's OEIS frontend

A030430 Primes of the form 10*n+1.

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

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.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

A039949 Primes of the form 30n - 13.

Original entry on oeis.org

17, 47, 107, 137, 167, 197, 227, 257, 317, 347, 467, 557, 587, 617, 647, 677, 797, 827, 857, 887, 947, 977, 1097, 1187, 1217, 1277, 1307, 1367, 1427, 1487, 1607, 1637, 1667, 1697, 1787, 1847, 1877, 1907, 1997, 2027, 2087, 2207, 2237, 2267, 2297, 2357, 2417

Offset: 1

Felice Russo

This linear form produces the most primes for n between 1 and 1000 (411/1000).
Primes congruent to 17 (mod 30). - Omar E. Pol, Aug 15 2007
Primes ending in 7 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches
Or primes p such that (p mod 3) = (p mod 5) and (p mod 2) <> (p mod 3), (p > 2). - Mikk Heidemaa, Jan 19 2016

C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 173

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A132230-A132236, A141860.

[p: p in PrimesUpTo(3000) | p mod 30 in [17]]; // Vincenzo Librandi, Aug 04 2012

Select[Prime[Range[1000]],MemberQ[{17},Mod[#,30]]&] (* Vincenzo Librandi, Aug 04 2012 *)
Select[Range[17,3000,30],PrimeQ] (* Zak Seidov, Apr 15 2015 *)

select(n->n%30==17, primes(500)) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A158648(n)*30 + 17. - Ray Chandler, Apr 07 2009
Intersection of A030432 and A007528. - Ray Chandler, Apr 07 2009
a(n) = A141860(n+1). - Zak Seidov, Apr 15 2015

Extended by Ray Chandler, Apr 07 2009

A132236 Primes congruent to 29 (mod 30).

Original entry on oeis.org

29, 59, 89, 149, 179, 239, 269, 359, 389, 419, 449, 479, 509, 569, 599, 659, 719, 809, 839, 929, 1019, 1049, 1109, 1229, 1259, 1289, 1319, 1409, 1439, 1499, 1559, 1619, 1709, 1889, 1949, 1979, 2039, 2069, 2099, 2129, 2309, 2339, 2399, 2459, 2549, 2579

Offset: 1

Omar E. Pol, Aug 15 2007

Primes ending in 9 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches

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, A039949, A132230-A132235.

[p: p in PrimesUpTo(3000) | p mod 30 eq 29 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[400]],Mod[#,30]==29&] (* Harvey P. Dale, Dec 25 2011 *)
Select[Prime[Range[1000]],MemberQ[{29},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

is(n)=isprime(n) && n%30==29 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A158850(n)*30 + 29. - Chandler

Intersection of A030433 and A007528. - Chandler

Extended by Ray Chandler, Apr 07 2009

A073102 Primes of the form 210n + 1.

Original entry on oeis.org

211, 421, 631, 1051, 1471, 2311, 2521, 2731, 3361, 3571, 4201, 4621, 4831, 5881, 6091, 6301, 7351, 7561, 8191, 8821, 9241, 9661, 9871, 10501, 10711, 11131, 11551, 11971, 12391, 12601, 13441, 14071, 14281, 15121, 15331, 15541, 16381, 17011

Offset: 1

Zak Seidov Aug 24 2002

			211 = 210*1 + 1 is prime; 1471 = 210*7 + 1 is prime.

Subsequence of A142076, A124826, and A132230.

Cf. A073085.

[ a: n in [0..400] | IsPrime(a) where a is 210*n+1]; // Vincenzo Librandi, Aug 08 2010

210*Flatten[Position[PrimeQ[210Range[100]+1], True]]+1
Select[1+210Range[100],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

select(x->((x % 210)==1), primes(3000)) \\ Michel Marcus, Jan 14 2018

Extended by Ray Chandler, Apr 29 2010

A132231 Primes congruent to 7 (mod 30).

Original entry on oeis.org

7, 37, 67, 97, 127, 157, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 937, 967, 997, 1087, 1117, 1237, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1987, 2017, 2137, 2287, 2347, 2377, 2437, 2467, 2557, 2617, 2647

Offset: 1

Omar E. Pol, Aug 15 2007

Primes ending in 7 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 07 2009
Intersection of A030432 and A002476. - Ray Chandler, Apr 07 2009
Only from 4927 on, there are more composite numbers than primes in {7+30k}, see A227869. - M. F. Hasler, Nov 02 2013
Terms are non-twin primes A007510, except for 7. - Jonathan Sondow, Oct 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages
Omar E. Pol, Prime numbers in a sieve with 30 columns
Omar E. Pol, Sobre el patrón de los números primos

Cf. A007510, A007528, A010051, A068229, A117047, A129807, A039949, A132230-A132236, A214360.

a132231 n = a132231_list !! (n-1)
a132231_list = [x | k <- [0..], let x = 30 * k + 7, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

[p: p in PrimesUpTo(3000) | p mod 30 eq 7 ]; // Vincenzo Librandi, Aug 14 2012

Select[30*Range[0,100]+7,PrimeQ] (* Harvey P. Dale, Feb 01 2012 *)
Select[Prime[Range[1000]],MemberQ[{7},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

forstep(p=7,1999,30,isprime(p)&&print1(p",")) \\ M. F. Hasler, Nov 02 2013

a(n) = A158573(n)*30 + 7. - Ray Chandler, Apr 07 2009

a(n) = A211890(4,n-1) for n <= 5. - Reinhard Zumkeller, Jul 13 2012

Extended 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

A111175 Numbers n such that 30*n + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 9, 11, 14, 18, 19, 20, 21, 22, 23, 25, 27, 33, 34, 35, 39, 40, 41, 43, 44, 46, 49, 51, 54, 58, 60, 61, 62, 65, 67, 71, 72, 74, 75, 76, 77, 78, 79, 84, 85, 89, 91, 93, 95, 99, 100, 102, 104, 106, 109, 110, 111, 112, 113, 117, 118, 119, 121, 123, 131, 134, 135

Offset: 1

Parthasarathy Nambi, Oct 21 2005

nonn

Encoded primes with LSD 1 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: 2671, with LSD 1 and (SOD-1)/3 = 2 (integer); Then 2671/30 = 89.033, or 89, which is in the sequence, and thus 2671 is prime. - Ki Punches, Mar 18 2009

			If n=99 then 30*n + 1 = 2971 (prime).

Cf. A158573, A158614, A158648, A158746, A158791, A158806, A158850.

is(n)=isprime(30*n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132230(n) - 1)/30 = Floor[A132230(n)/30]. - Chandler

Extended by Ray Chandler, Apr 07 2009

A132233 Primes congruent to 13 (mod 30).

Original entry on oeis.org

13, 43, 73, 103, 163, 193, 223, 283, 313, 373, 433, 463, 523, 613, 643, 673, 733, 823, 853, 883, 1033, 1063, 1093, 1123, 1153, 1213, 1303, 1423, 1453, 1483, 1543, 1663, 1693, 1723, 1753, 1783, 1873, 1933, 1993, 2053, 2083, 2113, 2143, 2203, 2293, 2383

Offset: 1

Omar E. Pol, Aug 15 2007

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

Subsequence of primes of A082369. - Michel Marcus, Jan 23 2016

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, A039949, A082369, A132230-A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 13 ]; // Vincenzo Librandi, Aug 14 2012

select(isprime, [seq(30*i+13,i=0..1000)]); # Robert Israel, Jan 24 2016

Select[Prime[Range[1000]],MemberQ[{13},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

lista(nn) = forprime(p=2, nn, if(p % 30 == 13, print1(p, ", "))); \\ Altug Alkan, Jan 23 2016

a(n) = A158746(n)*30 + 13. - Ray Chandler, Apr 07 2009

Intersection of A030431 and A002476. - Ray Chandler, Apr 07 2009

Extended by Ray Chandler, Apr 07 2009

A132234 Primes congruent to 19 (mod 30).

Original entry on oeis.org

19, 79, 109, 139, 199, 229, 349, 379, 409, 439, 499, 619, 709, 739, 769, 829, 859, 919, 1009, 1039, 1069, 1129, 1249, 1279, 1399, 1429, 1459, 1489, 1549, 1579, 1609, 1669, 1699, 1759, 1789, 1879, 1999, 2029, 2089, 2179, 2239, 2269, 2389, 2539, 2659, 2689

Offset: 1

Omar E. Pol, Aug 15 2007

Also primes congruent to 4 (mod 15). - N. J. A. Sloane, Jul 11 2008

Primes ending in 9 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches

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, A039949, A132230-A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 19 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{19},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[19,2700,30],PrimeQ] (* Harvey P. Dale, Dec 26 2014 *)

is(n)=isprime(n) && n%30==19 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A158806(n)*30 + 19. - Chandler

Intersection of A030433 and A002476. - Chandler

Extended by Ray Chandler, Apr 07 2009

cp's OEIS Frontend

A030430 Primes of the form 10*n+1.

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A039949 Primes of the form 30n - 13.

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A132236 Primes congruent to 29 (mod 30).

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A073102 Primes of the form 210n + 1.

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A132231 Primes congruent to 7 (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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