A030431 -id:A030431 - cp's OEIS frontend

A095023 Number of 5k+3 primes (A030431) in range ]2^n,2^(n+1)].

1, 0, 1, 1, 2, 4, 5, 11, 18, 36, 63, 117, 220, 407, 760, 1435, 2682, 5074, 9683, 18392, 35113, 67054, 128503, 246433, 473717, 911310, 1756933, 3390711, 6551833, 12673497, 24546404, 47584981, 92331321, 179315178, 348550417, 678028865, 1319937828, 2571418193

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A030431, A036378.

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A129078 Prime numbers that are the sum of consecutive prime numbers with the final digit 3 (primes in A030431).

Original entry on oeis.org

3, 1259, 51241, 81749, 230149, 245621, 253567, 269879, 286801, 331301, 482731, 540041, 551917, 564013, 625943, 638669, 746777, 975427, 1093129, 1145537, 1181149, 1272679, 1528187, 1569479, 1675679, 1741517, 1970867, 2066951

Offset: 1

Tomas Xordan, May 11 2007

			a(2)=1259 because 1259=A030431(1)+ A030431(2)+A030431(3)+A030431(4)+ A030431(5)+A030431(6)+A030431(7)+ A030431(8)+A030431(9)+A030431(10)+A030431(11)+A030431(12)+A030431(13)= 3+ 13+ 23+ 43+ 53+ 73+ 83+ 103+ 113+ 163+ 173+ 193+ 223 and 1259 is a prime number.

Cf. A030431; A000040.

Select[Accumulate[Select[10Range[0,1500]+3,PrimeQ]],PrimeQ]  (* Harvey P. Dale, Apr 06 2011 *)

a(n)=A030431(1)+A030431(2)+...+A030431(x); a is a prime number.

A007528 Primes of the form 6k-1.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A030432 Primes of form 10n+7.

Original entry on oeis.org

7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277, 307, 317, 337, 347, 367, 397, 457, 467, 487, 547, 557, 577, 587, 607, 617, 647, 677, 727, 757, 787, 797, 827, 857, 877, 887, 907, 937, 947, 967, 977, 997, 1087, 1097, 1117, 1187, 1217, 1237

Offset: 1

Warut Roonguthai

Union of A132231 and A039949. - Ray Chandler, Apr 07 2009
5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Also primes of the form 5n+2 with positive n. - Danny Rorabaugh, Feb 20 2016
Intersection of A000040 and A017353. - 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. A030430 (10n+1), A030431 (10n+3), A030433 (10n+9).

Cf. A045357, A045380, A049509, A102342.

[n: n in [7..1240 by 10] | IsPrime(n)]; // Bruno Berselli, Apr 06 2011

Select[Prime@Range[210], Mod[ #, 10] == 7 &] (* Ray Chandler, Nov 07 2006 *)

is(n)=n%10==7 && isprime(n) \\ Charles R Greathouse IV, Jul 01 2013

lista(nn) = forprime(p=7, nn, if(p%10==7, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

[10*n+7 for n in range(124) if is_prime(10*n+7)] # Danny Rorabaugh, Feb 20 2016

a(n) = 10*A102342(n) + 7.

a(n) ~ 4n log n. - Charles R Greathouse IV, Jul 01 2013

Extended by Ray Chandler, Nov 07 2006

A244763 Prime numbers ending in the prime number 13.

Original entry on oeis.org

13, 113, 313, 613, 1013, 1213, 1613, 1913, 2113, 2213, 2713, 3313, 3413, 3613, 4013, 4513, 4813, 5113, 5413, 5813, 6113, 7013, 7213, 8513, 8713, 9013, 9413, 9613, 10313, 10513, 10613, 11113, 11213, 11813, 12113, 12413, 12613, 12713, 13313, 13513, 13613, 13913

Offset: 1

Vincenzo Librandi, Jul 06 2014

Also primes of the form 100*n+13. Subsequence of A141885, A141937, A166573.

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

Cf. A141885, A141937, A166573.

Cf. Prime numbers ending in the prime number k: A030431 (k=3), A030432 (k=7), A167442 (k=11), this sequence (k=13), A244764 (k=17), A244765 (k=19), A244766 (k=23), A244767 (k=29), A167388 (k=31), A244768 (k=37), A167443 (k=41), A244769 (k=43), A244770 (k=47), A244771 (k=53), A244772 (k=59), A167445 (k=61), A244773 (k=67), A167441 (k=71), A244774 (k=73), A244775 (k=79), A244776 (k=83), A244777 (k=89), A244778 (k=97), A167626 (k=101), A167627 (k=163).

[n: n in PrimesUpTo(14000) | n mod 100 eq 13];

select(isprime, [13+100*n $ n=0..1000]); # Robert Israel, Jul 06 2014

Select[Prime[Range[5, 2000]], Take[IntegerDigits[#], -2]=={1, 3}&]

select(x->(x % 100)==13, primes(2000)) \\ Michel Marcus, Jul 06 2014

[p for p in primes(14000) if mod(p,100) == 13] # Bruno Berselli, Jul 07 2014

A105854 Primes of the form 20*k + 3.

Original entry on oeis.org

3, 23, 43, 83, 103, 163, 223, 263, 283, 383, 443, 463, 503, 523, 563, 643, 683, 743, 823, 863, 883, 983, 1063, 1103, 1123, 1163, 1223, 1283, 1303, 1423, 1483, 1523, 1543, 1583, 1663, 1723, 1783, 1823, 2003, 2063, 2083, 2143, 2203, 2243, 2383, 2423, 2503, 2543

Offset: 1

Zak Seidov, May 05 2005

Cf. A030431: Primes of the form 10*k + 3.

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

Cf. A000040, A030431.

[n: n in PrimesUpTo(2600) | IsDivisibleBy(n-3,20)];  // Bruno Berselli, Apr 05 2011

[ a: n in [0..250] | IsPrime(a) where a is 20*n+3 ]; // Vincenzo Librandi, Apr 06 2011

Select[Range[3, 2003, 20], PrimeQ[ # ]&]

More terms from N. J. A. Sloane, Jul 11 2008

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

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A385800 Primes having only {6, 8, 9} as digits.

Original entry on oeis.org

89, 6689, 6869, 6899, 8669, 8689, 8699, 8969, 8999, 9689, 66889, 68669, 68699, 68899, 69899, 86689, 86869, 86969, 88969, 89669, 89689, 89899, 89989, 96989, 98669, 98689, 98869, 98899, 98999, 99689, 99989, 666889, 666989, 668699, 668869, 668989, 668999, 669689, 669869

Offset: 1

Jason Bard, Jul 14 2025

Subsequence of A030431, A106111.
Supersequence of A020472.
Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [6, 8, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{6, 8, 9}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [6, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("689"), 41))) # uses function/imports in A385776

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

cp's OEIS Frontend

A095023 Number of 5k+3 primes (A030431) in range ]2^n,2^(n+1)].

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A129078 Prime numbers that are the sum of consecutive prime numbers with the final digit 3 (primes in A030431).

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Mathematica

Formula

A007528 Primes of the form 6k-1.

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GAP

Haskell

Maple

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PARI

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A030432 Primes of form 10n+7.

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Magma

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PARI

Sage

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A244763 Prime numbers ending in the prime number 13.

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Magma

Maple

Mathematica

PARI

Sage

A105854 Primes of the form 20*k + 3.

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Magma

Magma

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A132235 Primes congruent to 23 (mod 30).

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