A111175 -id:A111175 - cp's OEIS frontend

A132230 Primes congruent to 1 (mod 30).

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

A158806 Numbers n such that 30*n + 19 is prime.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 11, 12, 13, 14, 16, 20, 23, 24, 25, 27, 28, 30, 33, 34, 35, 37, 41, 42, 46, 47, 48, 49, 51, 52, 53, 55, 56, 58, 59, 62, 66, 67, 69, 72, 74, 75, 79, 84, 88, 89, 90, 91, 100, 101, 102, 103, 105, 107, 108, 110, 115, 116, 117, 118, 123, 124, 125, 129, 130

Offset: 1

Ki Punches, Mar 27 2009

nonn

Encoded primes with LSD 9, (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: 3019, with LSD 9, (SOD-1)/3 integer; Then 3019/30 = 100.633, or 100, which is in the sequence, thus 3019 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A111175, A158573, A158614, A158648, A158746, A158791, A158850.

Select[Range[0,150],PrimeQ[30#+19]&] (* Harvey P. Dale, Apr 05 2025 *)

is(n)=isprime(30*n+19) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132234(n) - 19)/30 = Floor[A132234(n)/30]. - Chandler

Edited 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

A158614 Numbers n such that 30*n + 11 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 13, 14, 15, 16, 17, 21, 23, 25, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 43, 45, 48, 49, 50, 52, 53, 57, 60, 62, 63, 64, 69, 70, 71, 78, 79, 80, 81, 84, 86, 87, 90, 91, 93, 95, 100, 101, 106, 107, 108, 112, 115, 116, 119, 122, 123, 125, 127, 128

Offset: 1

Ki Punches, Mar 22 2009, Mar 29 2009

nonn

Encoded primes with LSD 1 and (SOD-1)/3 non-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: 3191, with LSD 1 and (SOD-1)/3 = 4.33 (non-integer); Then 3191/30=106.367, or 106 which is in the sequence, thus 3191 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A111175, A158573, A158648, A158746, A158791, A158806, A158850.

Select[Range[0,130],PrimeQ[30#+11]&] (* Harvey P. Dale, Jul 26 2011 *)

is(n)=isprime(30*n+11) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132232(n) - 11)/30 = floor(A132232(n)/30). - Ray Chandler, Apr 07 2009

Edited by Ray Chandler, Apr 07 2009

A158648 Numbers n such that 30*n + 17 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 8, 10, 11, 15, 18, 19, 20, 21, 22, 26, 27, 28, 29, 31, 32, 36, 39, 40, 42, 43, 45, 47, 49, 53, 54, 55, 56, 59, 61, 62, 63, 66, 67, 69, 73, 74, 75, 76, 78, 80, 81, 82, 88, 89, 92, 94, 96, 97, 98, 104, 105, 108, 111, 113, 115, 117, 118, 120, 122, 125, 126

Offset: 1

Ki Punches, Mar 23 2009

nonn

Encoded primes with LSD 7 and (SOD-1)/3 non-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: 3557, with LSD 7 and (SOD-1)/3 = 6.333 (non-integer); Then 7557/30 = 118.566, or 118 which is in the sequence, and thus 3557 is prime.

Cf. A111175, A158573, A158614, A158746, A158791, A158806, A158850.

Select[Range[0,200],PrimeQ[30#+17]&] (* Harvey P. Dale, Mar 21 2018 *)

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

a(n) = (A039949(n) - 17)/30 = Floor[A039949(n)/30]. - Chandler

Edited by Ray Chandler, Apr 07 2009

A158746 Numbers n such that 30*n + 13 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 12, 14, 15, 17, 20, 21, 22, 24, 27, 28, 29, 34, 35, 36, 37, 38, 40, 43, 47, 48, 49, 51, 55, 56, 57, 58, 59, 62, 64, 66, 68, 69, 70, 71, 73, 76, 79, 82, 83, 86, 89, 90, 93, 94, 98, 105, 108, 110, 111, 112, 114, 115, 119, 120, 121, 122, 124, 126, 127

Offset: 1

Ki Punches, Mar 25 2009

nonn

Encoded primes with LSD 3, (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: 3163, with LSD 3 and (SOD-1)/3 = 4 (integer); Then 3163/30 = 105.433, or 105 which is in the sequence, thus 3163 is prime.

Cf. A111175, A158573, A158614, A158648, A158791, A158806, A158850.

Select[Range[0,200],PrimeQ[30#+13]&] (* Harvey P. Dale, Oct 16 2018 *)

is(n)=isprime(30*n+13) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132233(n) - 13)/30 = Floor[A132233(n)/30]. - Chandler

Edited by Ray Chandler, Apr 07 2009

A158791 Numbers n such that 30*n + 23 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 25, 28, 31, 32, 33, 36, 38, 39, 40, 42, 45, 47, 49, 50, 51, 52, 53, 57, 60, 63, 65, 66, 68, 71, 73, 74, 75, 77, 79, 80, 84, 87, 88, 89, 91, 94, 96, 98, 100, 102, 106, 110, 113, 117, 119, 120, 126, 127, 128, 130, 133

Offset: 1

Ki Punches, Mar 26 2009

nonn

Encoded primes with LSD 3 and (SOD-1)/3 non-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: 3623, with LSD 3 and (SOD-1)/3 non-integer; Then 3623/30 = 120.766, or 120, which is in the sequence, thus 3623 is prime.

Cf. A111175, A158573, A158614, A158648, A158746, A158806, A158850.

Select[Range[0,200],PrimeQ[30#+23]&] (* Harvey P. Dale, Jul 31 2014 *)

is(n)=isprime(30*n+23) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132235(n) - 23)/30 = Floor[A132235(n)/30]. - Chandler

Edited by Ray Chandler, Apr 07 2009

A158850 Numbers n such that 30*n + 29 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 26, 27, 30, 33, 34, 36, 40, 41, 42, 43, 46, 47, 49, 51, 53, 56, 62, 64, 65, 67, 68, 69, 70, 76, 77, 79, 81, 84, 85, 86, 89, 90, 92, 93, 95, 96, 97, 98, 99, 102, 103, 106, 109, 110, 111, 112, 114, 117, 121, 123, 125

Offset: 1

Ki Punches, Mar 28 2009

nonn

Encoded primes with LSD 9 and (SOD-1)/3 non-integer, (LSD, least significant digit; SOD, sum of digits). Divide any such number by 30, if the whole number portion from the quotient is in the sequence, the number is prime.

			Example: 3209 with LSD 9 and (SOD-1)/3 non-integer; Then 3209/30 = 106.966, or 106, which is in the sequence, thus 3209 is prime.

Cf. A111175, A158573, A158614, A158648, A158746, A158791, A158806.

is(n)=isprime(30*n+29) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132236(n) - 29)/30 = Floor[A132236(n)/30]. - Chandler

Edited by Ray Chandler, Apr 07 2009

A138220 Numbers k such that 900*k^2 + 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 14, 19, 23, 25, 26, 31, 32, 36, 38, 43, 44, 45, 47, 48, 62, 64, 66, 77, 81, 82, 85, 90, 91, 92, 95, 108, 112, 113, 116, 122, 129, 130, 136, 138, 139, 142, 147, 151, 153, 155, 164, 178, 179, 181, 183, 185, 190, 192, 195, 196, 199, 201, 202, 204

Offset: 1

Zak Seidov, May 05 2008

Includes all terms from A125251.

Conjecture: (30*k)^(2^m)+1 is prime for some numbers k at all values of m >= 0. - Richard R. Forberg, Feb 06 2021

Cf. A125251, A111175, A341183.

[n: n in [0..10000]| IsPrime(1+900*n^2)] // Vincenzo Librandi, Dec 13 2010

Select[Range[250],PrimeQ[900#^2+1]&] (* Harvey P. Dale, May 20 2012 *)

is(n)=isprime(1+900*n^2) \\ Charles R Greathouse IV, Jun 13 2017

A284659 Numbers n such that numbers 30(n+k) + 1 are prime for k=0..5.

Original entry on oeis.org

18, 74, 4386, 4505, 9314, 10357, 21095, 29621, 38784, 102463, 105200, 116134, 163300, 179967, 186918, 210515, 252830, 348709, 354022, 362345, 396820, 400915, 431568, 438862, 457748, 464118, 470852, 477341, 493070

Offset: 1

Zak Seidov, Mar 31 2017

nonn

Numbers n through n+5 are terms in A111175. There are no cases of 7 consecutive numbers in A111175.

All terms are congruent to 4 mod 7.

			a(1)=18 because 1 + 30*k for k=18..23 are 541, 571, 601, 631, 661, 691 all primes: A000040(k) for k={100, 105, 110, 115, 121, 125}.

Robert Israel, Table of n, a(n) for n = 1..1000
Index entries for primes in AP

Cf. A111175, A132230.

filter:= t -> andmap(isprime, [seq(30*(t+k)+1, k=0..5)]):
select(filter, [seq(seq(77*k + i,i=[18,39,53,60,74]),k=0..10000)]); # Robert Israel, Apr 04 2017

Select[Range[18, 1000000, 7], PrimeQ[1 + 30*#] && PrimeQ[1 + 30*(# + 1)] && PrimeQ[1 + 30*(# + 2)] && PrimeQ[1 + 30*(# + 3)] && PrimeQ[1 + 30*(# + 4)] && PrimeQ[1 + 30*(# + 5)] &]
Select[Range[4,10^6,7],AllTrue[30(#+Range[0,5])+1,PrimeQ]&] (* Harvey P. Dale, Dec 03 2023 *)

cp's OEIS Frontend

A132230 Primes congruent to 1 (mod 30).

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A158806 Numbers n such that 30*n + 19 is prime.

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

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A158614 Numbers n such that 30*n + 11 is prime.

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A158648 Numbers n such that 30*n + 17 is prime.

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A158746 Numbers n such that 30*n + 13 is prime.

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A158791 Numbers n such that 30*n + 23 is prime.

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