A132233 -id:A132233 - cp's OEIS frontend

A030431 Primes of form 10n+3.

3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263, 283, 293, 313, 353, 373, 383, 433, 443, 463, 503, 523, 563, 593, 613, 643, 653, 673, 683, 733, 743, 773, 823, 853, 863, 883, 953, 983, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193

Offset: 1

Warut Roonguthai

nonn

Also primes of form 5n+3.
Union of A132233, A132235, {3}. - Ray Chandler, Apr 07 2009
Primes p such that arithmetic mean of divisors of p^4 is an integer. There are 2 such sequences of primes, this one and A030430. - Ctibor O. Zizka, Oct 20 2009
5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Intersection of A000040 and A017305. - 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. A045414, A049508, A053179, A102338.

Select[Prime@Range[200], Mod[ #, 10] == 3 &] (* Ray Chandler, Nov 07 2006 *)
Select[10 Range[0, 150] + 3, PrimeQ]  (* Harvey P. Dale, Apr 06 2011 *)

select(n->n%10==3, primes(500)) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = 10*A102338(n) + 3.

Extended by Ray Chandler, Nov 07 2006

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

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

A282322 Greater of twin primes congruent to 13 (mod 30).

Original entry on oeis.org

13, 43, 73, 103, 193, 283, 313, 433, 463, 523, 643, 823, 883, 1033, 1063, 1093, 1153, 1303, 1453, 1483, 1723, 1873, 1933, 2083, 2113, 2143, 2383, 2593, 2713, 2803, 3253, 3373, 3463, 3583, 3673, 3823, 3853, 4003, 4093, 4243, 4273, 4423, 4483, 4723, 4933, 5023, 5233, 5443, 5503, 5653, 5743

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A282321 and this sequence] is A132241.
The union of [{5, 7}, this sequence, A282324 and A282326] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
A181604 without the 3. [Proof: working mod 10 we see that each value here is in A181604. For the other direction: Except 3 all twin primes in A181604 are upper twin primes; they cannot be lower twin primes because the upper ones would be multiples of 5. The twin primes in A181604 could be == 3 (mod 30) or == 13 (mod 30) or == 23 (mod 30). The first case is excluded because they would be multiples of 3; the third case is excluded because the lower twin primes would be == 21 (mod 30) and also multiples of 3. So only the case == 13 (mod 30) remains.] - R. J. Mathar, Feb 14 2017
Number of terms < 10^k for k >= 1: 0, 3, 13, 67, 401, 2736, 19797, 146841, 1141217, 9137078, ..., . - Robert G. Wilson v, Jan 07 2018

Muniru A Asiru, Table of n, a(n) for n = 1..20000

Subset of A001097, A006512, A132233, A132241 and A132247.

Cf. A001359, A232880, A232881, A232882, A282321, A282323, A282324, A282326.

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 13 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

Select[13 + 30 Range[0, 200], PrimeQ[# - 2] && PrimeQ[#] &] (* Robert G. Wilson v, Jan 07 2018 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==13, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

Original entry on oeis.org

1, 12, 1, 6, 1, 3, 1, 6, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 2, 1, 2, 6, 1, 1, 1, 1, 1, 6, 2, 1, 2, 4, 3, 2, 2, 4, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 3, 2, 2, 4, 2, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

nonn

Primes can be classified according to their remainder modulo 30: remainder 1 (A136066), 7 (A132231), 11 (A132232), 13 (A132233), 17 (A039949), 19 (A132234), 23 (A132235), or 29 (A132236). In the sequence A007775 of all numbers (prime or nonprime) in any of these remainder classes, we look for runs of numbers that are successively prime or nonprime and place the lengths of these runs in this sequence.

			Groups of runs in A007775 are (1), (7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), (49), (53, 59, 61, 67, 71, 73), (77), (79, 83,...), which is 1 nonprime followed by 12 primes followed by 1 nonprime followed by 6 primes etc.

Cf. A136066, A132231, A132232, A132233, A039919, A132234, A132235, A132236.

A007775 := proc(n) option remember ; local a; if n = 1 then 1; else for a from A007775(n-1)+1 do if a mod 2 <>0 and a mod 3 <>0 and a mod 5 <> 0 then RETURN(a) ; fi ; od: fi ; end: A := proc() local al,isp,n; al := 0: isp := false ; n := 1: while n< 300 do a := A007775(n) ; if isprime(a) <> isp then printf("%d,",al) ; al := 1; isp := not isp ; else al := al+1 ; fi ; n := n+1: od: end: A() ; # R. J. Mathar, Jun 16 2008

Edited by R. J. Mathar, Jun 16 2008

A140387 Binary encoding of the location of primes in integer sets r+30*n with remainder r=1,7,11,..,29.

Original entry on oeis.org

1, 32, 16, 129, 73, 36, 194, 6, 42, 176, 225, 12, 21, 89, 18, 97, 25, 243, 44, 44, 196, 34, 166, 90, 149, 152, 109, 66, 135, 225, 89, 169, 169, 28, 82, 210, 33, 213, 179, 170, 38, 92, 15, 96, 252, 171, 94, 7, 209, 2, 187, 22, 153, 9, 236, 197, 71, 179, 212, 197, 186

Offset: 1

Juri-Stepan Gerasimov, Jun 10 2008

Classify all integers 30n+r with r= 1, 7, 11, 13, 17, 19, 23 or 29 as nonprime or prime and assign bit positions 0=LSB, 1, 2, 3, .., 7=MSB to the 8 remainders in the same order. Raise the bit if 30n+r is nonprime, erase it if 30n+r is prime.

The sequence interprets this as a number in base 2 and shows the decimal representation.

			For n=1, the 8 numbers 31 (r=1), 37 (r=7), 41 (r=11), 43 (r=17), 47 (r=17), 49 (r=19), 53 (r=23) and 59 (r=29) are prime, prime, prime, prime, prime, nonprime, prime, prime, prime, which is rendered into the binary 000100000 = 2^5=32=a(1).

Cf. A132236, A132235, A132234, A039919, A132233, A132232, A136066, A140378.

Cf. A105052 (analog in base 10, prime = bit 1, remainder 1 = MSB), A140891 (analog in base 14, prime = bit 0, remainder 1 = LSB).

Edited by R. J. Mathar, Jun 17 2008

A140533 Primes congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 103, 107, 137, 163, 167, 193, 197, 223, 227, 257, 283, 313, 317, 347, 373, 433, 463, 467, 523, 557, 587, 613, 617, 643, 647, 673, 677, 733, 797, 823, 827, 853, 857, 883, 887, 947, 977, 1033, 1063, 1093, 1097, 1123, 1153, 1187, 1213, 1217

Offset: 1

Juri-Stepan Gerasimov, Jul 28 2008

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

Cf. A132233, A039949.

Cf. A000040.

[p: p in PrimesUpTo(3000)|p mod 30 in {13,17}]; // Vincenzo Librandi, Dec 18 2010

A140533 := proc(n) local a; if n = 1 then 13; else a := nextprime(procname(n-1)) ; while not a mod 30 in {13,17} do a := nextprime(a) ; end do: return a; end if; end: seq(A140533(n),n=1..80) ; # R. J. Mathar, Oct 22 2009

Select[Prime[Range[300]],MemberQ[{13,17},Mod[#,30]]&] (* Vincenzo Librandi, Aug 15 2012 *)

A248474 Numbers congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 77, 103, 107, 133, 137, 163, 167, 193, 197, 223, 227, 253, 257, 283, 287, 313, 317, 343, 347, 373, 377, 403, 407, 433, 437, 463, 467, 493, 497, 523, 527, 553, 557, 583, 587, 613, 617, 643, 647, 673, 677, 703, 707, 733, 737, 763, 767, 793, 797

Offset: 1

Karl V. Keller, Jr., Oct 06 2014

The combination of A082369(30*n+13) and A128468(30*n+17) is the base sequence for A140533(Primes congruent to 13 or 17 mod 30).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A082369 (30*n+13), A128468 (30*n+17).

Cf. A039949 (Primes of the form 30n-13), A132233 (Primes congruent to 13 mod 30), A140533 (Primes congruent to 13 or 17 mod 30).

Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* Alonso del Arte, Oct 06 2014 *)

Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2014

for n in range(1,101):
  print (n*30-17),
  print (n*30-13),

From Colin Barker, Oct 07 2014: (Start)
a(n) = (-15-11*(-1)^n+30*n)/2.
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: x*(13*x^2+4*x+13) / ((x-1)^2*(x+1)). (End)
E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - David Lovler, Sep 10 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2*(5+sqrt(5)))+sqrt(3)-sqrt(15))*Pi / (30*(sqrt(6*(5+sqrt(5)))+sqrt(5)-1)). - Amiram Eldar, Jul 30 2024

cp's OEIS Frontend

A030431 Primes of form 10n+3.

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A132232 Primes congruent to 11 (mod 30).

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Magma

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A132235 Primes congruent to 23 (mod 30).

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Magma

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

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A282322 Greater of twin primes congruent to 13 (mod 30).

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A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

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A140387 Binary encoding of the location of primes in integer sets r+30*n with remainder r=1,7,11,..,29.

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A140533 Primes congruent to 13 or 17 mod 30.