A132235 -id:A132235 - 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

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

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

A106859 Primes of the form 2x^2 + xy + 2y^2.

Original entry on oeis.org

2, 3, 5, 17, 23, 47, 53, 83, 107, 113, 137, 167, 173, 197, 227, 233, 257, 263, 293, 317, 347, 353, 383, 443, 467, 503, 557, 563, 587, 593, 617, 647, 653, 677, 683, 743, 773, 797, 827, 857, 863, 887, 947, 953, 977, 983, 1013, 1097, 1103, 1163, 1187, 1193

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-15.
If p is a prime >= 17 in this sequence then k==0 (mod 4) for all k satisfying "B(2k)(p^k-1) is an integer" where B are the Bernoulli numbers. - Benoit Cloitre, Nov 14 2005
Equals {2, 3, 5 and primes congruent to 17, 23 (mod 30)}; see A039949 and A132235. Except for 2, the same as primes of the form 3x^2 + 5y^2, which has discriminant -60. - T. D. Noe, May 02 2008
Equals {3, 5 and primes congruent to 2, 8 (mod 15)} sorted; see A033212. This form is in the only non-principal class (respectively, genus) for fundamental discriminant -15. - Rick L. Shepherd, Jul 25 2014 [See A343241 for the 2, 8 (mod 15) primes.]
From Wolfdieter Lang, Jun 08 2021: (Start)
Regarding the above comment of T. D. Noe on the form [3, 0, 5]: the class number h(-60) = 2 = A000003(15), and [1, 0, 15] is the principal reduced form, representing the primes given in A033212.
The form [3, 0, 5] represents the proper equivalence class of the second genus of forms of discriminant Disc = -60. The Legendre symbol for the odd primes, not 3 or 5, satisfy L(-3|p) = -1 and L(5|p) = -1, leading to primes p == {17, 23, 47, 53} (mod 60). See the Buell reference, p. 52, for the two characters L(p|3) and L(p|5). The prime 2 is represented by the imprimitive reduced form [2, 2, 8] of Disc = -60. (End)

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 51-52.

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi, next 691 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A000003, A139827, A039949, A132235, A033212, A343241.

QuadPrimes2[2, 1, 2, 100000] (* see A106856 *)

{ fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();
for(n=1,M, p = prime(n);
t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));
print(t1);
}
fc(2,1,2,1000); \\ N. J. A. Sloane, Jun 06 2014

Removed defective Mma program and extended the b-file using the PARI program fc. - N. J. A. Sloane, Jun 06 2014

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

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

A214360 Primes congruent to 23 modulo 3120613860.

Original entry on oeis.org

23, 3120613883, 6241227743, 9361841603, 12482455463, 15603069323, 18723683183, 21844297043, 24964910903, 28085524763, 34326752483, 43688594063, 62412277223, 115462712843, 124824554423, 156030693023, 159151306883, 171633762323, 180995603903, 196598673203

Offset: 1

Reinhard Zumkeller, Jul 13 2012

nonn

A211889(9) = 3120613860;

the first 10 terms constitute row 9 of triangle A211890, an arithmetic progression of 10 primes.

Cf. A010051.

Sequences of numbers congruent 23 modulo m: A134517 m=24, A141945 m=25, A140375 m=26, A141963 m=27, A141974 m=28, A141999 m=29, A132235 m=30, A142027 m=31, A142044 m=32, A142062 m=33, A142091 m=35, A142107 m=36, A142132 m=37, A142173 m=39, A142192 m=40, A142220 m=41, A142244 m=42, A142272 m=43, A142302 m=44, A142324 m=45, A142374 m=47, A142405 m=48, A142433 m=49, A142490 m=51, A142518 m=52, A142553 m=53, A142617 m=55, A142650 m=56, A142679 m=57, A142750 m=59, A142790 m=60, A142821 m=61, A142902 m=63, A142935 m=64, A140844 m=210.

a214360 n = a214360_list !! (n-1)
a214360_list = [x | k <- [0..], let x = 3120613860*k+23, a010051' x == 1]

select(isprime,[seq(23+i*3120613860,i=0..1000)]); # Robert Israel, Jun 07 2015

Select[Range[23, 2 10^11, 3120613860], PrimeQ] (* Vincenzo Librandi, Jun 07 2015 *)

is(n)=isprime(n) && n%3120613860==23 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 658414080n log n. - Charles R Greathouse IV, Jul 02 2016

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

A177966 Indices m for which A177961(m) = 2 + m.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 20, 23, 26, 27, 29, 35, 41, 42, 44, 50, 53, 56, 57, 65, 68, 74, 83, 86, 87, 89, 95, 98, 113, 116, 117, 119, 125, 128, 131, 132, 134, 140, 146, 147, 155, 158, 173, 176, 177, 179, 191, 192, 194, 200, 209, 215, 221, 222, 224, 230, 233, 239, 245, 251, 252, 254

Offset: 1

Vladimir Shevelev, May 16 2010

nonn

All m for which 2*m+1 is in A003627 are in the sequence:
This concerns m=2, 5, 8, 11, 14, 20, 23, 26, 29, 35,...
Union of (A003627-1)/2 and (A132235+1)/2. - Robert Israel, Jul 31 2015

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

Cf. A003627, A132235, A177961, A177964, A177965.

A090368 := proc(n) A020639(2*n-1) ; end proc:
A177961 := proc(n) (A090368(n)+A090368(n+1)) /2 ; end proc:
isA177966 := proc(n) A177961(m) = m+2 ; end proc:
for m from 1 to 800 do if isA177966(m) then printf("%d,",m) ; end if; end do:
# R. J. Mathar, Oct 25 2010
N:= 1000: # to get all terms <= N
A1:= map(t -> (t-1)/2, select(isprime, {seq(6*i-1, i=1..(N+1)/3)})):
A2:= map(t -> (t+1)/2, select(isprime, {seq(23+30*i,i=0..(N-12)/15)})):
sort(convert(A1 union A2,list));
# Robert Israel, Jul 31 2015

M = 1000; (* to get all terms <= M *)
A1 = (Select[Table[6 i - 1, {i, 1, (M + 1)/3}], PrimeQ] - 1)/2;
A2 = (Select[Table[23 + 30 i, {i, 0, (M - 12)/15}], PrimeQ] + 1)/2;
Union[A1, A2] (* Jean-François Alcover, Jul 17 2020, after Robert Israel *)

Corrected (11, 23, 27, etc. inserted) and extended by R. J. Mathar, Oct 25 2010

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

cp's OEIS Frontend

A030431 Primes of form 10n+3.

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A132236 Primes congruent to 29 (mod 30).

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Magma

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A132232 Primes congruent to 11 (mod 30).

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Magma

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A106859 Primes of the form 2x^2 + xy + 2y^2.

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Mathematica

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

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

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GAP

Magma

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A214360 Primes congruent to 23 modulo 3120613860.

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