A045429 -id:A045429 - cp's OEIS frontend

A216057 a(n) = A045429(n) - A045356(n).

1, 6, 6, 6, 12, 12, 6, 6, 2, 4, -6, -6, 4, -4, 4, 4, 14, 24, 24, 24, 24, 14, 14, 24, 6, 12, 12, 6, 2, 4, -24, -24, -26, -34, -18, -6, 6, 4, 12, 24, 22, 14, 4, 12, 6, 24, 24, 34, 24, 32, 16, 14, 24, 24, 26, 32, 34, 26, 34, 14, 14, 6, -18, -18, 6, 4, -6, -8, -14

Offset: 1

Zak Seidov, Aug 31 2012

sign

Zak Seidov, Table of n, a(n) for n = 1..1000
Zak Seidov, Plot of terms up to a(1000) (gif image)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319v1 [math.NT]
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A007350, A007351, A038691, A038698, A045356, A045429, A156749.

A039703 a(n) = n-th prime modulo 5.

Original entry on oeis.org

2, 3, 0, 2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 3, 2, 1, 2, 4, 4, 1, 2, 3, 2, 3, 4, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 4, 1, 1, 2, 3, 4, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 4, 3, 4, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 2, 1, 4, 3, 4, 1, 3, 1, 2, 2, 3, 4, 1

Offset: 1

Clark Kimberling

a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A010874, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.

[p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)

primes(105) % 5 \\ Zak Seidov, Apr 09 2013

Sum_k={1..n} a(k) ~ (5/2)*n. - Amiram Eldar, Dec 11 2024

A020451 Primes that contain digits 1 and 3 only.

Original entry on oeis.org

3, 11, 13, 31, 113, 131, 311, 313, 331, 3313, 3331, 11113, 11131, 11311, 13313, 13331, 31333, 33113, 33311, 33331, 113111, 113131, 131111, 131113, 131311, 311111, 313133, 313331, 313333, 331333, 333131, 333331, 1111333, 1131113, 1131131, 1131133, 1131331

Offset: 1

David W. Wilson

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Subsequence of A030096, A045429, and A032917.

[p: p in PrimesUpTo(1131331) | Set(Intseq(p)) subset [1,3]]; // Bruno Berselli, Jul 27 2012

N:= 8: # to get all a(n) with at most N digits
S:= {}:
for d from 1 to N do
  r:= (10^d-1)/9;
  S:= S union select(isprime,map(`+`,map(convert,combinat[powerset]
      ({seq(2*10^i,i=0..d-1)}),`+`),r));
od:
S; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list)); # Robert Israel, May 04 2015

Flatten[Table[Select[FromDigits/@Tuples[{1,3},n],PrimeQ],{n,7}]] (* Vincenzo Librandi, Jul 27 2012 *)

from sympy import primerange
def checkd(a, c):
    b =  set(int(i) for i in set(str(a)))
    return b.issubset(c)
for n in primerange(2, 2000000):
    if checkd(n, [1, 3]):
        print(n)
# Abhiram R Devesh, May 04 2015

A045356 Primes congruent to {0, 2, 4} mod 5.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 37, 47, 59, 67, 79, 89, 97, 107, 109, 127, 137, 139, 149, 157, 167, 179, 197, 199, 227, 229, 239, 257, 269, 277, 307, 317, 337, 347, 349, 359, 367, 379, 389, 397, 409, 419, 439, 449, 457, 467

Offset: 1

N. J. A. Sloane

A039703(A049084(a(n))) = even; complement of A045429. - Reinhard Zumkeller, Feb 25 2008

R. Zumkeller, Table of n, a(n) for n = 1..1001

Cf. A045370, A045378, A137977.

[ p: p in PrimesUpTo(1000) | p mod 5 in {0, 2, 4} ]; // Vincenzo Librandi, Aug 07 2012

Select[Prime[Range[400]], MemberQ[{0, 2, 4}, Mod[#, 5]] &] (* Vincenzo Librandi, Aug 07 2012 *)
Select[Prime[Range[100]], EvenQ[Mod[#, 5]] &]  (* Zak Seidov, Aug 31 2012 *)
Select[Flatten[#+{0,2,4}&/@(5*Range[0,100])],PrimeQ] (* Harvey P. Dale, Sep 14 2019 *)

A045415 Primes congruent to {1, 3, 5} mod 7.

Original entry on oeis.org

3, 5, 17, 19, 29, 31, 43, 47, 59, 61, 71, 73, 89, 101, 103, 113, 127, 131, 157, 173, 197, 199, 211, 227, 229, 239, 241, 257, 269, 271, 281, 283, 311, 313, 337, 353, 367, 379, 383, 397, 409, 421, 439, 449, 463, 467

Offset: 1

N. J. A. Sloane

A039705(A049084(a(n))) = odd; complement of A045370. - Reinhard Zumkeller, Feb 25 2008

R. Zumkeller, Table of n, a(n) for n = 1..1001

Cf. A000040, A045429, A137978.

[ p: p in PrimesUpTo(600) | p mod 7 in {1,3,5} ]; // Vincenzo Librandi, Aug 12 2012

Select[Prime[Range[100]],MemberQ[{1,3,5},Mod[#,7]]&] (* Harvey P. Dale, May 15 2011 *)

A137978 Primes congruent to {1, 3, 5, 7, 9} modulo 11.

Original entry on oeis.org

3, 5, 7, 23, 29, 31, 47, 53, 67, 71, 73, 89, 97, 113, 137, 139, 157, 163, 179, 181, 199, 223, 227, 229, 251, 269, 271, 293, 311, 313, 317, 331, 337, 353, 359, 379, 383, 397, 401, 419, 421, 443, 449, 463, 467, 487, 491, 509, 557, 577, 599, 601, 617, 619, 641

Offset: 1

Reinhard Zumkeller, Feb 25 2008

nonn

A039709(A049084(a(n))) = odd; complement of A137977.

R. Zumkeller, Table of n, a(n) for n = 1..1001

Cf. A045415, A045429.

[ p: p in PrimesUpTo(740) | p mod 11 in {1, 3, 5, 7, 9} ] // Vincenzo Librandi, Jan 25 2011

Select[Flatten[# + {1, 3, 5, 7, 9}&/@(11Range[0, 80])], PrimeQ] (* Harvey P. Dale, Jan 15 2011 *)

A259932 Primes whose anti-divisors sum to a prime.

Original entry on oeis.org

3, 5, 13, 41, 113, 761, 1201, 1741, 1861, 2113, 9661, 9941, 12641, 13613, 15313, 21841, 23981, 30013, 34061, 47741, 49613, 60901, 70313, 83641, 101701, 237361, 241513, 252761, 303421, 335381, 377581, 413141, 489061, 491041, 525313, 529421, 637321, 695021, 718801

Offset: 1

Paolo P. Lava, Jul 09 2015

nonn

See A066272 for definition of anti-divisor.
Subsequence of A109350.
Apparently, apart from 5, all terms are congruent to {1, 3} mod 5 (see A045429).

			The anti-divisor of 3 is 2, which is prime.
The anti-divisors of 41 are 2, 3, 9, and 27, whose sum is 41, which is prime.
The anti-divisors of 9941 are 2, 3, 9, 47, 59, 141, 337, 423, 2209, and 6627, whose sum is 9857, which is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A000040, A045429, A066272, A066417, A109350.

with(numtheory): P:=proc(q) local a,i,j,n;
for n from 3 to q do if isprime(n) then
i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od;
if isprime(sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^i)*2^(i+1)-6*n-2)
then print(n); fi; fi; od; end: P(10^9);

ad[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Prime@ Range@ 1250, PrimeQ[Total@ ad@ #] &] (* _Michael De Vlieger, Jul 10 2015 *)

A269789 Primes p such that 2*p + 59 is a square.

Original entry on oeis.org

11, 31, 83, 151, 191, 283, 811, 983, 1171, 1483, 1831, 2083, 2351, 3251, 3583, 3931, 4111, 4483, 4871, 5483, 6131, 8291, 9631, 11071, 12611, 14251, 14591, 15991, 18211, 20983, 24391, 27583, 29983, 30971, 34031, 35083, 36151, 36691, 37783, 38891, 39451, 40583

Offset: 1

Vincenzo Librandi, Mar 24 2016

nonn

Primes of the form 2*k^2 + 2*k - 29.

			a(1) = 11 because 2*11 + 59 = 81, which is a square.

Cf. A000040.
Subsequence of A002144, A045429.
Cf. similar sequences listed in A269784.

[p: p in PrimesUpTo(50000) | IsSquare(2*p+59)];

Select[Prime[Range[4500]], IntegerQ[Sqrt[2 # + 59]] &]

lista(nn) = {forprime(p=2, nn, if(issquare(2*p + 59), print1(p, ", "))); } \\ Altug Alkan, Mar 24 2016

A212719 Numbers n with property that the subset A039703(1..n) contains equal numbers of odd and even terms.

Original entry on oeis.org

2, 6, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 44, 48, 54, 56, 58, 60, 70, 72, 74, 76, 84, 86, 90, 100, 122, 124, 128, 130, 132, 134, 136, 138, 142, 144, 192, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 234, 236, 238, 240, 254, 256, 258, 282, 284, 286

Offset: 1

Zak Seidov, Aug 29 2012

nonn

It appears that the sequence is infinite and unbound.

Cf. A039703, A096629.
Cf. A045356 (primes congruent to {0,2,4} mod 5).
Cf. A045429 (primes congruent to {1,3} mod 5).

cp's OEIS Frontend

A216057 a(n) = A045429(n) - A045356(n).

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A039703 a(n) = n-th prime modulo 5.

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A020451 Primes that contain digits 1 and 3 only.

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A045356 Primes congruent to {0, 2, 4} mod 5.

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Magma

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A045415 Primes congruent to {1, 3, 5} mod 7.

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Magma

Mathematica

A137978 Primes congruent to {1, 3, 5, 7, 9} modulo 11.

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Magma

Mathematica

A259932 Primes whose anti-divisors sum to a prime.

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Maple

Mathematica

A269789 Primes p such that 2*p + 59 is a square.

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