A132236 -id:A132236 - cp's OEIS frontend

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

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

A282324 Greater of twin primes congruent to 19 (mod 30).

Original entry on oeis.org

19, 109, 139, 199, 229, 349, 619, 829, 859, 1279, 1429, 1489, 1609, 1669, 1699, 1789, 1879, 1999, 2029, 2089, 2239, 2269, 2659, 2689, 3169, 3259, 3469, 3529, 3559, 3769, 3919, 4129, 4159, 4219, 4339, 4519, 4549, 4639, 4789, 4969, 5419, 5479, 5659, 5869, 6199

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A282323 and this sequence] is A132242.
The union of [{5, 7}, A282322, this sequence 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.
Number of terms less than 10^k, k=2,3,4,...: 1, 9, 64, 414, 2734, 19674, 146953, ... - Muniru A Asiru, Feb 09 2018

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Subset of A001097, A006512, A132234, A132242 and A132247.

Cf. A001359, A132235, A132236, A232880, A232881, A232882, A282321, A282322, A282323, A060229, A282326.

Filtered(List([1..220], k -> 30*k-11), n -> IsPrime(n) and IsPrime(n-2));  # Muniru A Asiru, Feb 02 2018

[p: p in PrimesUpTo(7000) | IsPrime(p-2) and p mod 30 eq 19 ]; // Vincenzo Librandi, Feb 13 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 19 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;
# More efficient
select(n -> isprime(n-2) and isprime(n), [seq(30*k+19, k=0..220)]); # Muniru A Asiru, Jan 30 2018

Select[Prime[Range[1000]], PrimeQ[# - 2] && Mod[#, 30] == 19 &] (* Vincenzo Librandi, Feb 13 2017 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==19, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A138905 a(n) is n-th prime == -1 (mod 6n).

Original entry on oeis.org

5, 23, 71, 167, 179, 431, 461, 863, 863, 839, 1583, 1511, 1949, 2099, 2339, 4127, 4283, 4751, 4673, 4919, 5669, 6599, 8693, 10079, 7349, 10607, 12149, 11087, 12527, 11159, 15809, 19583, 16829, 19583, 13859, 25703, 24197, 25307, 23633, 21839, 34439

Offset: 1

Zak Seidov, Apr 03 2008

nonn

			a(1) = 1st term in A007528 (Primes of form 6n-1)
a(2) = 2nd term in A068231 (Primes congruent to 11 (mod 12))
a(3) = 3rd term in A061242 (Primes of form 18n-1)
a(4) = 4th term in A134517 (Primes of form 24n-1)
a(5) = 5th term in A132236 (Primes congruent to 29 (mod 30))

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A007528, A061242, A068231, A093871, A132236, A134517, A138906, A138907.

a[n_]:=Module[{p=1,cnt=0},Until[cnt==n,If[Mod[Prime[p],6n]==6n-1,cnt++];p++];Prime[p-1]];Array[a,41] (* James C. McMahon, Jun 22 2025 *)

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

A215850 Primes p such that 2*p + 1 divides Lucas(p).

Original entry on oeis.org

5, 29, 89, 179, 239, 359, 419, 509, 659, 719, 809, 1019, 1049, 1229, 1289, 1409, 1439, 1499, 1559, 1889, 2039, 2069, 2129, 2339, 2399, 2459, 2549, 2699, 2819, 2939, 2969, 3299, 3329, 3359, 3389, 3449, 3539, 3779, 4019, 4349, 4409, 4919, 5039, 5279, 5399, 5639

Offset: 1

Arkadiusz Wesolowski, Aug 24 2012

nonn

An equivalent definition of this sequence: 5 together with primes p such that p == -1 (mod 30) and 2*p + 1 is also prime.
Sequence without the initial 5 is the intersection of A005384 and A132236.
These numbers do not occur in A137715.
From Arkadiusz Wesolowski, Aug 25 2012: (Start)
The sequence contains numbers like 1409 which are in A053027.
a(n) is in A002515 if and only if a(n) is congruent to -1 mod 60. (End)

			29 is in the sequence since it is prime and 59 is a factor of Lucas(29) = 1149851.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
C. K. Caldwell, "Top Twenty" page, Lucas cofactor
Eric Weisstein's World of Mathematics, Lucas Number

Supersequence of A230809. Cf. A000032, A132236.

[5] cat [n: n in [29..5639 by 30] | IsPrime(n) and IsPrime(2*n+1)];

Select[Prime@Range[740], Divisible[LucasL[#], 2*# + 1] &]
Prepend[Select[Range[29, 5639, 30], PrimeQ[#] && PrimeQ[2*# + 1] &], 5]

is_A215850(n)=isprime(n)&!real((Mod(2,2*n+1)+quadgen(5))*quadgen(5)^n) \\ - M. F. Hasler, Aug 25 2012

A229947 Primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

Original entry on oeis.org

11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 71, 73, 79, 89, 101, 103, 107, 109, 131, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 257, 269, 271, 281, 283, 311, 313, 317, 331, 347, 349, 359, 373, 379, 389

Offset: 1

Omar E. Pol, Oct 27 2013

For twin primes congruent to {1, 11, 13, 17, 19, 29} mod 30 see A132247.

Complement of A132237, primes congruent to 7 or 23 (mod 30), in the set of primes > 5. - M. F. Hasler, Nov 02 2013

Omar E. Pol, Prime numbers in a sieve with 30 columns

Cf. A000040, A001097, A039949, A132230, A132232-A132234, A132236, A132238-A132259, A230462.

[p: p in PrimesUpTo(500) | p mod 30 in {1,11,13,17, 19,29} ]; // Vincenzo Librandi, Apr 05 2015

Select[Flatten[Table[30n + {1, 11, 13, 17, 19, 29}, {n, 0, 11}]], PrimeQ] (* Alonso del Arte, Nov 01 2013 *)
Select[Prime@Range[100], MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

is(n)=isprime(n) && setsearch([1,11,13,17,19,29], n%30) \\ Charles R Greathouse IV, Mar 08 2015

a(n) ~ 4/3 n log n. - Charles R Greathouse IV, Mar 08 2015

A263769 Smallest prime q such that q == -1 (mod prime(n)-1).

Original entry on oeis.org

2, 3, 3, 5, 19, 11, 31, 17, 43, 83, 29, 71, 79, 41, 137, 103, 173, 59, 131, 139, 71, 233, 163, 263, 191, 199, 101, 211, 107, 223, 251, 389, 271, 137, 443, 149, 311, 647, 331, 859, 1423, 179, 379, 191, 587, 197, 419, 443

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

a(n): A000040(1), A065091(1), A002145(1), A007528(1), A030433(1), A068231(1), A127576(1), A061242(1), A141857(1), A141976(1), A132236(1), A142111(1), A142198(1), A141898(1), A141926(1), A142531(1), A142004(1), A142799(1), A142068(1), A142099(1), ...

Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

			a(4) = 5 because 5 == -1 (mod prime(4)-1) and is prime.

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A263730, A263770.

Cf. A000040, A065091, A002145, A007528, A030433, A068231, A127576, A061242, A141857, A141976, A132236, A142111, A142198, A141898, A141926, A142531, A142004, A142799, A142068, A142099.

for n from 1 to 100 do
  k:= ithprime(n)-1;
  q:= 2;
  while (1 + q) mod k <> 0 do
    q:= nextprime(q)
  od;
  A[n]:= q;
od:
seq(A[i],i=1..1000); # Robert Israel, Oct 26 2015

Table[q = 2; z = Prime@ n - 1; While[Mod[q, z] != z - 1, q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

Corrected and edited by Robert Israel, Oct 26 2015,

A293425 Primes of the form 2^a * 3^b * 5^c - 1 for positive a, b, c.

Original entry on oeis.org

29, 59, 89, 149, 179, 239, 269, 359, 449, 479, 599, 719, 809, 1439, 1499, 1619, 2399, 2699, 2879, 2999, 4049, 4799, 5399, 7499, 8999, 9719, 10799, 11519, 12149, 12959, 13499, 15359, 18749, 20249, 21599, 23039, 25919, 33749, 35999, 40499, 51839, 56249, 59999, 65609, 67499, 69119, 71999

Offset: 1

Muniru A Asiru, Oct 09 2017

nonn

a(n) is congruent to 29 (mod 30).

			a(1) = 29 = 2^1 * 3^1 * 5^1 - 1.
a(2) = 59 = 2^2 * 3^1 * 5^1 - 1.
a(3) = 89 = 2^1 * 3^2 * 5^1 - 1.
a(4) = 149 = 2^1 * 3^1 * 5^2 - 1.
a(5) = 179 = 2^2 * 3^2 * 5^1 - 1.
list of (a, b, c): (1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2), (2, 2, 1), (4, 1, 1), (1, 3, 1), (3, 2, 1), (1, 2, 2), (5, 1, 1), (3, 1, 2), (4, 2, 1), (1, 4, 1), (5, 2, 1), (2, 1, 3), (2, 4, 1), ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A005105, A030433, A132236, A293194.

K:=10^5+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
A293425:=List(Positions(List(A,i->Elements(Factors(i+1))),[2,3,5]),i->A[i]);

N:= 10^6: # to get all terms < N
R:= {}:
for c from 1 to floor(log[5]((N+1)/6)) do
for b from 1 to floor(log[3]((N+1)/2/5^c)) do
     R:= R union select(isprime, {seq(2^a*3^b*5^c-1,
             a=1..ilog2((N+1)/(3^b*5^c)))})
od od:
sort(convert(R,list)); # Robert Israel, Oct 15 2017

With[{n = 10^5}, Sort@ Select[Flatten@ Table[2^a*3^b*5^c - 1, {a, Log2@ n}, {b, Log[3, n/(2^a)]}, {c, Log[5, n/(2^a*3^b)]}], PrimeQ]] (* Michael De Vlieger, Oct 11 2017 *)

lista(nn) = {forprime(p=2,nn, f = factor(p+1); if ((vecmax(f[,1]) <= 5) && (#f~==3), print1(p, ", ")););} \\ Michel Marcus, Oct 09 2017

from itertools import count, islice
from sympy import integer_log, isprime
def A293425_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = x
        for i in range(1,integer_log(x,5)[0]+1):
            for j in range(1,integer_log(m:=x//5**i,3)[0]+1):
                c -= (m//3**j).bit_length()-1
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A293425_list = list(islice(A293425_gen(),30)) # Chai Wah Wu, Mar 31 2025

A141866 Primes of the form 2357k+89, k >= 0.

Original entry on oeis.org

89, 509, 719, 929, 1559, 1979, 2399, 2609, 2819, 3449, 3659, 4079, 4289, 4919, 6389, 6599, 7019, 7229, 7649, 8069, 8699, 9539, 9749, 10169, 10589, 10799, 12269, 12479, 12689, 12899, 13109, 14159, 14369, 15629, 16889, 17099, 17519, 17729, 17939, 18149

Offset: 1

Juri-Stepan Gerasimov, Sep 15 2008

nonn

Subsequence of A132236. - R. J. Mathar, Sep 19 2008

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

Cf A000040.

Select[210 Range[0, 200] + 89, PrimeQ] (* Harvey P. Dale, Nov 11 2013 *)

Definition edited, 7019 inserted and extended by R. J. Mathar, Sep 19 2008

cp's OEIS Frontend

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

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A282324 Greater of twin primes congruent to 19 (mod 30).

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A138905 a(n) is n-th prime == -1 (mod 6n).

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

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Maple

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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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A215850 Primes p such that 2*p + 1 divides Lucas(p).

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Magma

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A229947 Primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

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Magma

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A263769 Smallest prime q such that q == -1 (mod prime(n)-1).

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A141866 Primes of the form 2357k+89, k >= 0.