A141164 -id:A141164 - cp's OEIS frontend

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1151

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Primes that are the sum of no fewer than four positive squares.
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Primes p such that x^4 = 2 has just two solutions mod p. Subsequence of A040098. Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 = 2 if and only if p - i is a solution mod p of x^4 = 2, so the sum of the two solutions is p. The solutions are given in A065907 and A065908. - Klaus Brockhaus, Nov 28 2001
As this is a subset of A001132, this is also a subset of the primes of form x^2 - 2y^2. And as this is also a subset of A038873, this is also a subset of the primes of form x^2 - 2y^2. - Tito Piezas III, Dec 28 2008
Subsequence of A141164. - Reinhard Zumkeller, Mar 26 2011
Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - Arkadiusz Wesolowski, Apr 05 2012
Primes p such that p XOR 6 = p - 6. - Brad Clardy, Jul 22 2012

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A004771.
Cf. A040098, A014754, A065907, A065908, A010051.
Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).

a007522 n = a007522_list !! (n-1)
a007522_list = filter ((== 1) . a010051) a004771_list
-- Reinhard Zumkeller, Jan 29 2013

[p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // Vincenzo Librandi, Jun 26 2014

select(isprime, [seq(i,i=7..10000,8)]); # Robert Israel, Nov 22 2016

Select[8Range[200] - 1, PrimeQ] (* Alonso del Arte, Nov 07 2016 *)

(A007522(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z == 2, print1(p, ", ")))); A007522(1400)  \\ Does not return a(m) but prints all terms <= m. - Edited to make it executable by M. F. Hasler, May 22 2025.

A007522_upto(N, start=1)=select(p->p%8==7, primes([start, N]))
#A7522=A007522_upto(10^5)
A007522(n)={while(#A7522A007522_upto(N*3\2, N+1))); A7522[n]} \\ M. F. Hasler, May 22 2025

Equals A000040 INTERSECT A004215. - R. J. Mathar, Nov 22 2006

a(n) = 7 + A139487(n)*8, n >= 1. - Wolfdieter Lang, Feb 18 2015

A188172 Number of divisors d of n of the form d == 7 (mod 8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

R. J. Mathar, Mar 23 2011

			a(A007522(i)) = 1, any i.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001842, A002325, A004771, A141164, A188169, A188170, A188171, A188226.

Cf. A001620, A016631, A354635 (psi(7/8)).

a188172 n = length $ filter ((== 0) . mod n) [7,15..n]
-- Reinhard Zumkeller, Mar 26 2011

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188172 := proc(n) sigmamr(n,8,7) ; end proc:

Table[Count[Divisors[n],?(Mod[#,8]==7&)],{n,90}] (* _Harvey P. Dale, Mar 08 2014 *)

a(n) = sumdiv(n, d, (d % 8) == 7); \\ Amiram Eldar, Nov 25 2023

A188170(n)+a(n) = A001842(n).
A188169(n)+A188170(n)-A188171(n)-a(n) = A002325(n).
a(A188226(n))=n and a(m)<>n for m<A188226(n), n>=0; a(A141164(n))=1. - Reinhard Zumkeller, Mar 26 2011
G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,8) - (1 - gamma)/8 = -0.212276..., gamma(7,8) = -(psi(7/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A343107 Numbers having exactly 1 divisor of the form 8k + 1, that is, numbers with no divisor of the form 8k + 1 other than 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 35, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 67, 69, 70, 71, 74, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 9, 17, 25, ...

			7 is a term since it has no divisor congruent to 1 modulo 8 other than 1.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: this sequence (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).

Indices of 1 in A188169.

Select[Range[100],NoneTrue[Rest[Divisors[#]],Mod[#,8]==1&]&] (* Harvey P. Dale, Jun 01 2022 *)

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343107(n) = (res(n,8,1) == 1)

A343108 Numbers having no divisor of the form 8*k + 3.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 20, 23, 25, 26, 28, 29, 31, 32, 34, 37, 40, 41, 46, 47, 49, 50, 52, 53, 56, 58, 61, 62, 64, 65, 68, 71, 73, 74, 79, 80, 82, 85, 89, 92, 94, 97, 98, 100, 101, 103, 104, 106, 109, 112, 113, 116, 119, 122, 124, 125, 127, 128

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 3, 11, 19, ...

			7 is a term since it has no divisor congruent to 3 modulo 8.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), this sequence (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).

Indices of 0 in A188170.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343108(n) = (res(n,8,3) == 0)

A343109 Numbers having no divisor of the form 8*k + 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 34, 36, 38, 41, 43, 44, 46, 47, 48, 49, 51, 54, 56, 57, 59, 62, 64, 66, 67, 68, 71, 72, 73, 76, 79, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 99, 102, 103, 107, 108, 112, 113

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 5, 13, 21, ...

			9 is a term since it has no divisor congruent to 5 modulo 8.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), this sequence (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).

Indices of 0 in A188171.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343109(n) = (res(n,8,5) == 0)

A343110 Numbers having no divisor of the form 8*k + 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 29, 32, 33, 34, 36, 37, 38, 40, 41, 43, 44, 48, 50, 51, 52, 53, 54, 57, 58, 59, 61, 64, 65, 66, 67, 68, 72, 73, 74, 76, 80, 81, 82, 83, 85, 86, 88, 89, 96, 97, 99, 100, 101, 102

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 7, 15, 23, ...

			9 is a term since it has no divisor congruent to 7 modulo 8.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), this sequence (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).

Indices of 0 in A188172.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343110(n) = (res(n,8,7) == 0)

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.

Original entry on oeis.org

9, 17, 18, 25, 27, 33, 34, 36, 41, 45, 49, 50, 51, 54, 57, 63, 65, 66, 68, 72, 73, 75, 82, 85, 89, 90, 97, 98, 100, 102, 105, 108, 113, 114, 117, 119, 121, 123, 125, 126, 129, 130, 132, 135, 136, 137, 144, 145, 146, 147, 150, 161, 164, 165, 169, 170, 175

Offset: 1

Jianing Song, Apr 05 2021

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisors congruent to 1 modulo 8 are 1 and 9.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), this sequence (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 2 in A188169.
A007519 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343111(n) = (res(n,8,1) == 2)

A343112 Numbers having exactly 1 divisor of the form 8*k + 3.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 18, 19, 21, 22, 24, 30, 35, 36, 38, 39, 42, 43, 44, 45, 48, 55, 59, 60, 63, 67, 69, 70, 72, 76, 77, 78, 83, 84, 86, 87, 88, 90, 91, 93, 95, 96, 107, 110, 111, 115, 117, 118, 120, 121, 126, 131, 133, 134, 138, 139, 140, 141, 143, 144

Offset: 1

Jianing Song, Apr 05 2021

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisor congruent to 3 modulo 8 is 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), this sequence (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188170.
A007520 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343112(n) = (res(n,8,3) == 1)

A343113 Numbers having exactly 1 divisor of the form 8*k + 5.

Original entry on oeis.org

5, 10, 13, 15, 20, 21, 25, 26, 29, 30, 35, 37, 39, 40, 42, 50, 52, 53, 55, 58, 60, 61, 63, 69, 70, 74, 75, 77, 78, 80, 84, 87, 91, 93, 95, 100, 101, 104, 106, 109, 110, 111, 115, 116, 120, 122, 126, 133, 138, 140, 141, 143, 147, 148, 149, 150, 154, 155

Offset: 1

Jianing Song, Apr 05 2021

			52 is a term since among the divisors of 52 (namely 1, 2, 4, 13, 26 and 52), the only divisor congruent to 5 modulo 8 is 13.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), this sequence (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188171.
A007521 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343113(n) = (res(n,8,5) == 1)

A188226 Smallest number having exactly n divisors of the form 8*k + 7.

Original entry on oeis.org

1, 7, 63, 315, 945, 1575, 3465, 19845, 10395, 17325, 26775, 127575, 45045, 266805, 190575, 155925, 135135, 2480625, 225225, 130203045, 405405, 1289925, 2168775, 1715175, 675675, 3898125, 3468465, 1576575, 3239775, 67798585575, 2027025, 16769025, 2297295, 20539575, 42170625, 27286875, 3828825, 117661005

Offset: 0

Reinhard Zumkeller, Mar 26 2011

nonn

A188172(a(n)) = n and A188172(m) <> n for m < a(n).

Bert Dobbelaere, Table of n, a(n) for n = 0..200

Cf. A004771, A141164, A188172.

Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), A343106 (i=5), this sequence (i=7).

import Data.List  (elemIndex)
import Data.Maybe (fromJust)
a188226 n = a188226_list !! n
a188226_list =
   map (succ . fromJust . (`elemIndex` (map a188172 [1..]))) [0..]

a(19)-a(35) from Nathaniel Johnston, Apr 06 2011

More terms from Bert Dobbelaere, Apr 09 2021

cp's OEIS Frontend

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A188172 Number of divisors d of n of the form d == 7 (mod 8).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A343107 Numbers having exactly 1 divisor of the form 8*k + 1, that is, numbers with no divisor of the form 8*k + 1 other than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A343108 Numbers having no divisor of the form 8*k + 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A343109 Numbers having no divisor of the form 8*k + 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A343110 Numbers having no divisor of the form 8*k + 7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A343111 Numbers having exactly 2 divisors of the form 8*k + 1, that is, numbers with exactly 1 divisor of the form 8*k + 1 other than 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A343107 Numbers having exactly 1 divisor of the form 8k + 1, that is, numbers with no divisor of the form 8k + 1 other than 1.

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.