A045323 -id:A045323 - cp's OEIS frontend

A040103 Duplicate of A045323.

2, 3, 7, 11, 17, 19, 23, 31, 41, 43, 47, 59, 67, 71, 73, 79, 83, 89, 97, 103, 107, 113, 127

Offset: 0

dead

A007521 Primes of the form 8k + 5.

5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 541, 557, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 877, 941, 997, 1013, 1021, 1061, 1069, 1093, 1109, 1117, 1181, 1213

Offset: 1

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

Primes of the form 4x^2 - 4xy + 5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

Prime numbers 2k+1 with k even such that 2k+1 | 2^k+1. - Hilko Koning, Jan 21 2022

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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].

Cf. A010051.

Subsequence of A004770; see also A045323.

a007521 n = a007521_list !! (n-1)
a007521_list = filter ((== 1). a010051) a004770_list
-- Reinhard Zumkeller, Aug 17 2012

[p: p in PrimesUpTo(2000) | p mod 8 eq 5]; // Vincenzo Librandi, Jun 26 2014

Select[8Range[0, 160] + 5, PrimeQ]  (* Harvey P. Dale, Apr 22 2011 *)

list(lim)=select(n->n%8==5,primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

from sympy import isprime
print(list(filter(isprime, range(5, 1214, 8)))) # Michael S. Branicky, May 13 2021

A004776 Numbers not congruent to 5 (mod 8).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78

Offset: 1

N. J. A. Sloane

Also, numbers whose binary expansion does not end in 101.

Numbers that are congruent to {0, 1, 2, 3, 4, 6, 7} mod 8. - Wesley Ivan Hurt, Jul 22 2016

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A004770 (complement), A045323 (primes).

a004776 n = a004776_list !! (n-1)
a004776_list = filter ((/= 5) . (`mod` 8)) [0..]
-- Reinhard Zumkeller, Aug 17 2012

[n : n in [0..100] | n mod 8 in [0, 1, 2, 3, 4, 6, 7]]; // Wesley Ivan Hurt, Jul 22 2016

A004776:=n->8*floor(n/7)+[0, 1, 2, 3, 4, 6, 7][(n mod 7)+1]: seq(A004776(n), n=0..100); # Wesley Ivan Hurt, Jul 22 2016

DeleteCases[Range[0,80],?(Mod[#,8]==5&)] (* _Harvey P. Dale, Apr 28 2014 *)

is(n)=n%8!=5 \\ Charles R Greathouse IV, Mar 07 2013

A004776(n)=n+(n-6)\7  \\ M. F. Hasler, Nov 02 2013

def A004776(n): return n-1+(n+1)//7 # Chai Wah Wu, Feb 24 2025

Numbers that are congruent to {0, 1, 2, 3, 4, 6, 7} mod 8.
G.f.: x^2*(1+x+x^2+x^3+2*x^4+x^5+x^6) / ((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2). - R. J. Mathar, Oct 25 2011
a(n) = n + floor((n-6)/7). - M. F. Hasler, Nov 02 2013
From Wesley Ivan Hurt, Jul 22 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; a(n) = a(n-7) + 8 for n>7.
a(n) = (56*n - 63 + (n mod 7) - 6*((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) + ((n+6) mod 7))/49.
a(7k) = 8k-1, a(7k-1) = 8k-2, a(7k-2) = 8k-4, a(7k-3) = 8k-5, a(7k-4) = 8k-6, a(7k-5) = 8k-7, a(7k-6) = 8k-8. (End)

Edited by M. F. Hasler, Nov 02 2013

A120347 Numerator of Sum_{k=1..n-1} 1/k^n.

Original entry on oeis.org

1, 9, 1393, 257875, 47463376609, 940908897061, 972213062238348973121, 7727182467755471289426059, 10338014371627802833957102351534201, 26038773205374138944970092886340352227, 205885410277133543091182509665217407908365393153956577

Offset: 2

Alexander Adamchuk, Aug 16 2006, Oct 31 2006

Prime p>2 divides a(p). p^3 divides a(p) for prime p>3. p divides a((p+1)/2) for prime p = {7,11,17,19,23,31,41,43,47,59,67,71,73,79,83,89,97,103,...} = all primes excluding 2 and 3 from A045323[n] Primes congruent to {1, 2, 3, 7} mod 8.

a(n) = Numerator( H(n-1,n) ), where H(k,r) = Sum_{i=1..k} 1/i^r is the generalized harmonic number.

Vincenzo Librandi, Table of n, a(n) for n = 2..49
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A045323, A120289, A120352 (a(prime(n))), A119722 (a(prime(n))/prime(n)^3).

Table[Numerator[Sum[1/k^n,{k,1,n-1}]],{n,2,15}]

a(n) = Numerator(Sum_{k=1..n-1} 1/k^n). a(n) = Numerator[Zeta[n] - Zeta[n,n]].

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A040103 Duplicate of A045323.

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A007521 Primes of the form 8k + 5.

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A004776 Numbers not congruent to 5 (mod 8).

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A120347 Numerator of Sum_{k=1..n-1} 1/k^n.

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