A130291 -id:A130291 - cp's OEIS frontend

A006254 Numbers k such that 2k-1 is prime.

2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157

Offset: 1

Marc LeBrun

a(n) is the inverse of 2 modulo prime(n) for n >= 2. - Jean-François Alcover, May 02 2017
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
Positions of prime numbers among odd numbers. - Zak Seidov, Mar 26 2013
Also, the integers remaining after removing every third integer following 2, and, recursively, removing every p-th integer following the next remaining entry (where p runs through the primes, beginning with 5). - Pete Klimek, Feb 10 2014
Also, numbers k such that k^2 = m^2 + p, for some integers m and every prime p > 2. Applicable m values are m = k - 1 (giving p = 2k - 1).  Less obvious is: no solution exists if m equals any value in A047845, which is the complement of (A006254 - 1). - Richard R. Forberg, Apr 26 2014
If you define a different type of multiplication (*) where x (*) y = x * y + (x - 1) * (y - 1), (which has the commutative property) then this is the set of primes that follows. - Jason Atwood, Jun 16 2019

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A067076, A086801.
Equals A005097 + 1. A130291 is an essentially identical sequence.
Cf. A065091.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: this seq(k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [0..1000] | IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

Rest@Prime@Range@70/2 + 1/2 (* Robert G. Wilson v, Jun 16 2006 *)
Select[Range[200], PrimeQ[2#-1]&] (* Harvey P. Dale, Apr 06 2014 *)

a(n)=prime(n+1)\2+1 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime
def A006254(n): return prime(n+1)+1>>1 # Chai Wah Wu, Aug 02 2024

a(n) = (A000040(n+1) + 1)/2 = A067076(n-1) + 2 = A086801(n-1)/2 + 2.
a(n) = (1 + A065091(n))/2. - Omar E. Pol, Nov 10 2007
a(n) = sqrt((A065091^2 + 2*A065091+1)/4). - Eric Desbiaux, Jun 29 2009
a(n) = A111333(n+1). - Jonathan Sondow, Jan 20 2016

More terms from Erich Friedman

More terms from Omar E. Pol, Nov 10 2007

A130290 Number of nonzero quadratic residues modulo the n-th prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158

Offset: 1

M. F. Hasler, May 21 2007

Row lengths for formatting A063987 as a table: The number of nonzero quadratic residues modulo a prime p equals floor(p/2), or (p-1)/2 if p is odd. The number of squares including 0 is (p+1)/2, if p is odd (rows prime(i) of A096008 formatted as a table). In fields of characteristic 2, all elements are squares. For any m > 0, floor(m/2) is the number of even positive integers less than or equal to m, so a(n) also equals the number of even positive integers less than or equal to the n-th prime. For all n > 0, A130290(n+1) = A005097(n) = A102781(n+1) = A102781(n+1) = A130291(n+1)-1 = A111333(n+1)-1 = A006254(n)-1.
From Vladimir Shevelev, Jun 18 2016: (Start)
a(1)+2 and, for n >= 2, a(n)+1 is the smallest k such that there exists 0 < k_1 < k with the condition k_1^2 == k^2 (mod prime(n)).
Indeed, for n >= 2, if prime(n) = 4*t+1 then k = 2*t+1 = a(n)+1, since (2*t+1)^2 == (2*t)^2 (mod prime(n)) and there cannot be a smaller value of k; if prime(n) = 4*t-1, then k = 2*t = a(n)+1, since (2*t)^2 == (2*t-1)^2 (mod prime(n)). (End)
a(n) is the number of pairs (a,b) such that a + b = prime(n) with 1 <= a <= b. - Nicholas Leonard, Oct 02 2022

			a(1)=1 since the only nonzero element of Z/2Z equals its square.
a(3)=2 since 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are the only nonzero squares in Z/5Z.
a(1000000) = 7742931 = (prime(1000000)-1)/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue

Essentially the same as A005097.
Cf. A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130291.
Appears in A217983. - Johannes W. Meijer, Oct 25 2012

[Floor((NthPrime(n))/2): n in [1..60]]; // Vincenzo Librandi, Jan 16 2013

A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A000040 := proc(n): ithprime(n) end: seq(A130290(n), n=1..55); # Johannes W. Meijer, Oct 25 2012

Quotient[Prime[Range[66]], 2] (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)

PARI
```
A130290(n) = prime(n)>>1
```

from sympy import prime
def A130290(n): return prime(n)//2  # Chai Wah Wu, Jun 04 2022

a(n) = floor( A000040(n)/2 ) = #{ even positive integers <= A000040(n) }
a(n) = A055034(A000040(n)), n>=1. - Wolfdieter Lang, Sep 20 2012
a(n) = A005097(n-[n>1]) = A005097(max(n-1,1)). - M. F. Hasler, Dec 13 2019

A161148 Number of partitions of n such that each term of the partition is a squared divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 2, 6, 1, 9, 1, 8, 3, 6, 1, 16, 2, 7, 4, 12, 1, 21, 1, 15, 4, 9, 2, 39, 1, 10, 5, 25, 1, 35, 1, 24, 9, 12, 1, 76, 2, 21, 6, 32, 1, 61, 3, 38, 7, 15, 1, 174, 1, 16, 10, 46, 3, 93, 1, 50, 8, 42, 1, 231, 1, 19, 19, 60, 2, 135, 1

Offset: 1

R. J. Mathar, Jun 03 2009

nonn

			a(n=12)=5 counts these 5 partitions of 12: 1^2+1^2+..+1^2 = 1^2+1^2+...+1^2+2^2 = 1^2+1^2+..+1^2+2^2+2^2 = 1^2+1^2+1^2+3^2=2^2+2^2+2^2. Partitions with the divisors 4, 6 or 12 do not contribute to the count because 4^2, 6^2 and 12^2 are larger than n.

Cf. A000040, A130291, A018818, A001156, A002635.

a := proc(n) coeftayl(1/mul(1-x^(d^2),d=numtheory[divisors](n)),x=0,n) ; end:

a[n_] := SeriesCoefficient[1/Product[1-x^(d^2), {d, Divisors[n]}], {x, 0, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Apr 04 2024, after Maple code *)

a(p) = 1 if p a prime (A000040).
a(2p) = A130291(n) if p=A000040(n).
a(n) = [x^n] Product_{d|n} 1/( 1-x^(d^2) ).

A323703 Number of values of (X^3 + X) mod prime(n).

Original entry on oeis.org

1, 3, 3, 5, 7, 9, 11, 13, 15, 19, 21, 25, 27, 29, 31, 35, 39, 41, 45, 47, 49, 53, 55, 59, 65, 67, 69, 71, 73, 75, 85, 87, 91, 93, 99, 101, 105, 109, 111, 115, 119, 121, 127, 129, 131, 133, 141, 149, 151, 153, 155, 159, 161, 167, 171, 175, 179, 181, 185, 187, 189, 195

Offset: 1

Florian Severin, Jan 24 2019

nonn

a(n) is also the number of values of any other polynomial of degree 3, except X^3.

a(n) appears to approach (2/3)*prime(n) as n increases.

			a(1) = 1 since the only value X^3 + X takes mod 2 is 0.

R. Daublebsky von Sterneck, Über die Anzahl inkongruenter Werte, die eine ganze Funktion dritten Grades annimmt, Sitzungsber. Akad. Wiss. Wien (2A) 114 (1908), 711-717.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Thomas Brazelton, Joshua Harrington, Matthew Litman, and Tony W. H. Wong, Distinct residues of Lucas polynomials over Fp, arXiv:2103.09119 [math.NT], 2021. See p. 1.
Zhi-Hong Sun, On the theory of cubic residues and nonresidues, Acta Arithmetica 84.4 (1998): 291-335.
Zhi-Hong Sun, On the number of incongruent residues of x^4 +ax^2 +bx modulo p, Journal of Number Theory 119 (2006), 210-241. See p. 211.

Cf. A323704 (the number of values of X^3), A130291 (the number of values of X^2, which is also the number of values of any other polynomial of degree 2).

Array[Length@ Union@ Mod[Array[#^3 + # &, #], #] &@ Prime@ # &, 62] (* Michael De Vlieger, Jan 27 2019 *)

a(n) = #Set(vector(prime(n), k, Mod(k^3+k, prime(n)))); \\ Michel Marcus, Jan 25 2019

a(n) = prime(n) - 2*floor(prime(n)/6 + 1/2), for n >= 3. - Ridouane Oudra, Jun 13 2020

for n>=3, a(n) = (2*p + (p/3))/3 with p=prime(n) and where (p/3) is the Legendre symbol. See von Sterneck, Sun, and Brazelton et al. articles. - Michel Marcus, Mar 17 2021

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A006254 Numbers k such that 2k-1 is prime.

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A130290 Number of nonzero quadratic residues modulo the n-th prime.

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A161148 Number of partitions of n such that each term of the partition is a squared divisor of n.

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Maple

Mathematica

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A323703 Number of values of (X^3 + X) mod prime(n).

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Mathematica

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