A007519 -id:A007519 - cp's OEIS frontend

A095009 Number of 8k+1 primes (A007519) in range ]2^n,2^(n+1)].

0, 0, 0, 1, 1, 4, 4, 10, 18, 31, 64, 115, 216, 398, 752, 1413, 2692, 5092, 9642, 18355, 35089, 66907, 128431, 246479, 473201, 911650, 1756523, 3390156, 6551387, 12673576, 24545135, 47583812, 92329094, 179317195, 348545465, 678019818, 1319938243, 2571401536

Offset: 1

Antti Karttunen and Labos Elemer, 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. A007519, A091126, A095007, A095011, A095012, A095013.

a(n) = A095013(n) - A095012(n) = A095007(n) - A095011(n).

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

A096637 Smallest prime p == 1 mod 8 (A007519) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

Original entry on oeis.org

17, 73, 241, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 7921489, 3818929, 9257329, 22000801, 68204761, 48473881, 175244281, 1149374521, 427733329, 898716289

Offset: 0

Robert G. Wilson v, Jun 24 2004

nonn

Same as smallest prime p == 1 mod 8 with the property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2).

Cf. A094928, A002224, A096636.

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; If[Mod[p, 8] == 1, a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]]], {n, 10^9}]; t

Better name from Jonathan Sondow, Mar 07 2013

A185377 Product of exactly two distinct primes congruent to 1 mod 8 (A007519).

Original entry on oeis.org

697, 1241, 1513, 1649, 1921, 2329, 2993, 3281, 3649, 3961, 3977, 4097, 4369, 4633, 4777, 5321, 5617, 5729, 6001, 6497, 6817, 6953, 7081, 7361, 7633, 7769, 7913, 8249, 8633, 8857, 9553, 9673, 9809, 9881, 10001, 10057, 10081, 10217, 10489, 10537

Offset: 1

Jonathan Vos Post, Feb 20 2011

Subset of semiprimes A001358. Subset of {d = p_1 * p_2 * ... * p_m where p_i == 1 (mod 8), 1 <= i <= m are distinct primes} as occurs in Wei, p. 2.

			10001 is in this sequence because 10001 = 73 * 137 = A007519(3) * A007519(7).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dasheng Wei, On the equation x^2-Dy^2=n, Feb 18, 2011.

Cf. A000040, A001358, A007519, A017077.

p = Select[Prime[Range[200]], Mod[#, 8] == 1 &]; Sort[Reap[Do[n=p[[i]] p[[j]]; If[n <= p[[1]]p[[-1]], Sow[n]], {i, 2, Length[p]}, {j, i - 1}]][[2,1]]]

list(lim)=my(v=List(),P=List(),t); forprime(p=2,lim\17, if(p%8==1, listput(P,p))); for(i=2,#P, my(p=P[i]); for(j=1,i-1, t=p*P[j]; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jul 03 2016

{A007519(i) * A007519(j) for i < j}.

{A000040(i) * A000040(j) for i < j, and A000040(i) in A017077 and A000040(j) in A017077}.

A186293 (A007519(n)-1)/2.

Original entry on oeis.org

8, 20, 36, 44, 48, 56, 68, 96, 116, 120, 128, 140, 156, 168, 176, 200, 204, 216, 224, 228, 260, 284, 288, 296, 300, 308, 320, 336, 380, 384, 404, 428, 440, 464, 468, 476, 488, 504, 516, 524, 548, 564, 576, 596, 600, 608

Offset: 1

Marco Matosic, Feb 17 2011

Cf. A186305.

(Select[1 + 8 Range@ 155, PrimeQ] -1)/2 (* Robert G. Wilson v *)

a(n) = A186294(n)-1.

a(n) = 4*A005123(n).

A186294 (A007519(n)+1)/2.

Original entry on oeis.org

9, 21, 37, 45, 49, 57, 69, 97, 117, 121, 129, 141, 157, 169, 177, 201, 205, 217, 225, 229, 261, 285, 289, 297, 301, 309, 321, 337, 381, 385, 405, 429, 441, 465, 469, 477, 489, 505, 517, 525, 549, 565, 577, 597, 601, 609, 625, 645, 649, 661, 681

Offset: 1

Marco Matosic, Feb 17 2011

nonn

isok(n) = (n % 4 == 1) && isprime(2*n-1) \\ Michel Marcus, Jul 16 2013

More terms from Michel Marcus, Jul 16 2013

A157115 Alternate terms of A007519, A007520, A007521, A007522.

Original entry on oeis.org

17, 3, 5, 7, 41, 11, 13, 23, 73, 19, 29, 31, 89, 43, 37, 47, 97, 59, 53, 71, 113, 67, 61, 79, 137, 83, 101, 103, 193, 107, 109, 127, 233, 131, 149, 151, 241, 139, 157, 167, 257, 163, 173, 191, 281, 179, 181, 199, 313, 211, 197, 223, 337, 227, 229, 239, 353, 251, 269

Offset: 1

Zak Seidov and N. J. A. Sloane, Feb 23 2009

Or, read the following table by columns:
17,41,73,89,97,113,137,193,233,241,257,281,313,337,353,401,409,... (primes = = 1 mod 8)
3,11,19,43,59,67,83,107,131,139,163,179,211,227,251,283,307,331,... (primes == 3 mod 8)
5,13,29,37,53,61,101,109,149,157,173,181,197,229,269,277,293,317,... (primes == 5 mod 8)
7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,... (primes == 7 mod 8)

			The first four primes congruent to (1,3,5,7) mod 8 are 17,3,5,7, hence a(1..4)=17,3,5,7;
The next four primes congruent to (1,3,5,7) mod 8 are 41,11,13,23, hence a(5..8)=41,11,13,23, etc.

Zak Seidov, Table of n, a(n) for n = 1..4000

Cf. A007519 - A007522, A042987.

s[i_]:=(c=0;a=2*i-1;Reap[Do[If[PrimeQ[a],c++;Sow[a]];If[c>99,Break[],a = a+8],{10^8}]][[2,1]]);Flatten[Transpose[Table[s[i],{i,4}]]]; (* Zak Seidov, Jan 16 2013 *)

A185379 Product of exactly three distinct primes congruent to 1 mod 8 (A007519).

Original entry on oeis.org

50881, 62033, 67609, 78761, 95489, 110449, 120377, 134521, 140233, 146761, 162401, 167977, 170017, 170969, 179129, 186337, 195857, 207281, 218161, 225913, 234889, 239513, 246041, 263177, 266377, 279497, 285073, 289153, 290321, 292009, 299081, 301801, 312953

Offset: 1

Jonathan Vos Post, Feb 20 2011

Subset of numbers that are divisible by exactly 3 primes (counted with multiplicity), also known as triprimes or 3-almost primes, A014612. Subset of {d = p_1 * p_2 * ... * p_m where p_i == 1 (mod 8), 1 <= i <= m are distinct primes} as occurs in Wei, p.2.

			a(12) = 170017 = 17 * 73 * 137 = A007519(1) * A007519(3) * A007519(7).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Dasheng Wei, On the equation x^2-Dy^2=n, Feb 18, 2011.

Cf. A007519, A014612, A017077, A185377.

p = Select[Prime[Range[100]], Mod[#, 8] == 1 &]; Sort[Reap[Do[n = p[[i]] p[[j]] p[[k]]; If[n <= p[[1]] p[[2]] p[[-1]], Sow[n]], {i, 2, Length[p]}, {j, i - 1}, {k, j - 1}]][[2, 1]]]

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\697, if(p%8==1, listput(u,p))); for(i=1,#u-2, for(j=i+1, #u-1, if(u[i]*u[j]*u[j+1]>lim, break); for(k=j+1,#u, t=u[i]*u[j]*u[k]; if(t>lim, break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

{A007519(i) * A007519(j) * A007519(k) for i < j < k}. {A000040(i) * A000040(j) * A000040(k) for i < j < k, and A000040(i) in A017077 and A000040(j) in A017077 and A000040(k) in A017077}.

A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).

Original entry on oeis.org

2, 3, 10, 12, 33, 18, 10, 9, 12, 8, 4, 60, 5, 85, 70, 45, 31, 79, 92, 170, 43, 76, 152, 59, 59, 139, 256, 64, 62, 40, 44, 188, 177, 18, 14, 156, 227, 192, 231, 223, 79, 31, 75, 362, 7, 239, 338, 402, 6, 235, 114, 72, 342, 511, 15, 483, 310, 355, 104, 292, 232

Offset: 1

Michel Lagneau, Oct 22 2012

nonn

A007519(n) : primes of form 8n+1.

			a(5) = 33 because 33^4+1 = 1185922 = 2 * 97 * 6113 with A007519(5) = 97.

Robert Israel, Table of n, a(n) for n = 1..10000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.

Cf. A017077, A007519.

V:= Vector(100): count:= 0:
for p from 9 by 8 while count < 100 do
  if isprime(p) then
      count:= count+1; V[count]:=min(map(rhs@op,[msolve(k^4+1,p)]))
    fi
od:
convert(V,list); # Robert Israel, Mar 13 2018

aa = {}; Do[p = Prime[n]; If[Mod[p, 8] == 1, k = 1; While[ ! Mod[k^4 + 1, p] == 0, k++ ]; AppendTo[aa, k]], {n, 300}]; aa

A064496 a(n) is the least k such that k * A007519(n) + 1 = 0 (mod 12).

Original entry on oeis.org

7, 7, 11, 7, 11, 7, 7, 11, 7, 11, 7, 7, 11, 11, 7, 7, 11, 11, 7, 11, 7, 7, 11, 7, 11, 7, 7, 11, 7, 11, 7, 7, 7, 7, 11, 7, 7, 11, 11, 7, 7, 11, 11, 7, 11, 7, 11, 7, 11, 11, 7, 7, 7, 7, 11, 7, 7, 11, 11, 7, 7, 11, 11, 11, 11, 7, 7, 11, 11, 7, 11, 11, 7, 11, 7

Offset: 1

Jon Perry, Oct 05 2001

nonn

Original name: Values of n such that 4j = np+1 where p = 8x+1, x integer, p prime and j mod 3 = 0.

			For example, A007519(1)=17, 4j=17p+1 implies k is 7 and j is 30. The values that give 7 form the basis for solutions for the Erdős-Straus conjecture: 4/n=1/a+1/b+1/c for n >= 2, a,b,c>0 and integers.

H. Mishima, Overview of "Mathematician's Secret Room"

Entry revised by Sean A. Irvine, Jul 14 2023

A186295 A007519(n)-2.

Original entry on oeis.org

15, 39, 71, 87, 95, 111, 135, 191, 231, 239, 255, 279, 311, 335, 351, 399, 407, 431, 447, 455, 519, 567, 575, 591, 599, 615, 639, 671, 759, 767, 807, 855, 879, 927, 935, 951, 975, 1007, 1031, 1047, 1095, 1127, 1151, 1191, 1199, 1215

Offset: 1

Marco Matosic, Feb 17 2011

nonn

isok(n) = isprime(p=n+2) && (p % 8 == 1) \\ Michel Marcus, Jul 16 2013

a(n) = 2*A186294(n) - 3. - Michel Marcus, Jul 16 2013

cp's OEIS Frontend

A095009 Number of 8k+1 primes (A007519) in range ]2^n,2^(n+1)].

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A096637 Smallest prime p == 1 mod 8 (A007519) and p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

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A185377 Product of exactly two distinct primes congruent to 1 mod 8 (A007519).

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A186293 (A007519(n)-1)/2.

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A186294 (A007519(n)+1)/2.

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A157115 Alternate terms of A007519, A007520, A007521, A007522.

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A185379 Product of exactly three distinct primes congruent to 1 mod 8 (A007519).

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A218028 a(n) is the smallest positive integer k such that k^4 + 1 == 0 mod p, where p is the n-th prime of the form p = 1 + 8*b (see A007519).

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A064496 a(n) is the least k such that k * A007519(n) + 1 = 0 (mod 12).

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