A271780 -id:A271780 - cp's OEIS frontend

A008837 a(n) = p*(p-1)/2 for p = prime(n).

1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306

Offset: 1

N. J. A. Sloane, Ken Levasseur

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014
From Jianing Song, Apr 13 2019: (Start)
a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

[ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008

a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Table[Prime[n] * (Prime[n] - 1)/2, {n, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[Binomial[Prime[n], 2], {n, 40}] (* Alonso del Arte, Aug 22 2014, based on the formula from Enrique Pérez Herrero *)
(#(#-1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Oct 02 2019 *)

{ n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009

(define (A008837 n) (/ (A036689 n) 2)) ;; Antti Karttunen, May 01 2015

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree.

Original entry on oeis.org

5, 11, 13, 29, 43, 59, 61, 67, 83, 131, 139, 157, 173, 211, 227, 229, 277, 283, 317, 331, 347, 373, 389, 419, 421, 443, 461, 509, 547, 563, 571, 619, 643, 653, 659, 661, 691, 709, 733, 787, 797, 821, 853, 859, 877, 907, 941, 947, 997, 1019, 1021, 1069, 1091, 1093, 1109, 1123, 1163, 1181, 1213

Offset: 1

Bill McEachen, Mar 06 2016

The associated prime factors will always include 2 and 3.
Will every prime number be encountered as a prime factor from the sequence entries?
The sequence appears to share many of it terms with A001122.
What is the asymptotic behavior?
Conjecture: sequence has density A271780/2 = A005597*4/Pi^2 = 0.2675535... in the primes. - Charles R Greathouse IV, Jan 24 2018
The prime terms of A179017 (except 3). - Bill McEachen, Oct 21 2021

			277 = 138 + 139 = 2*3*23 + 139 is in the sequence since both terms are squarefree.
281 = 140 + 141 = 2^2*5*7 + 3*47 is not in the sequence since the former term is divisible by 2^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Bill McEachen, A269844_vs_A001122

Cf. A001122 (primes with primitive root 2), A179017.

Cf. A005597, A271780.

Select[Prime@ Range[3, 200], PrimeOmega@ # == PrimeNu@ # &[# (# + 1)] &@ Floor[#/2] &] (* Michael De Vlieger, Mar 07 2016 *)

genit(maxx)={for(i5=3,maxx,n=prime(i5);a=factor(floor(n/2.));b=factor(ceil(n/2.));clear=1;for(j5=1,omega(floor(n/2.)),if(a[j5,2]<>1,clear=0));
for(j7=1,omega(ceil(n/2.)),if(b[j7,2]<>1,clear=0));if(clear>0,print1(n,",")));}

is(n)=isprime(n) && issquarefree(n\2) && issquarefree(n\2+1) \\ Charles R Greathouse IV, Jan 24 2018

list(lim)=my(v=List(),t=1); forfactored(k=3,(lim+1)\2, if(vecmax(k[2][,2])>1, t=0, ; if(t && isprime(t=2*k[1]-1), listput(v,t)); t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 24 2018

A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

0, 6, 0, 8, 2, 1, 6, 5, 5, 1, 2, 0, 3, 0, 5, 0, 8, 6, 0, 0, 5, 6, 3, 2, 2, 7, 5, 4, 6, 1, 9, 2, 0, 8, 5, 5, 4, 3, 1, 3, 3, 7, 3, 7, 3, 4, 7, 5, 7, 6, 7, 9, 4, 1, 9, 8, 2, 6, 4, 3, 4, 0, 3, 1, 5, 0, 4, 0, 8, 0, 4, 3, 5, 0, 7, 2, 1, 2, 5, 6, 1, 6, 9, 5, 8, 6, 1, 8, 8, 8, 7, 3, 4, 8, 5, 8, 6, 6, 2, 4, 6, 8, 7, 3, 4, 0

Offset: 0

Jean-François Alcover, Apr 16 2016

			0.0608216551203050860056322754619208554313373734757679419826434...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Cf. A005596, A065414, A065415, A065416, A271780, A271798.

digits = 70; $MaxExtraPrecision = 1000; m0 = 2000; dm = 200; Clear[s]; LR =
LinearRecurrence[{2, 2, -6, 4, -1}, {0, 6, 0, 22, 5}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> 2 m0, WorkingPrecision -> digits+10] // Exp; s[m0]; s[m = m0+dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[ s[m-dm], 10, digits][[1]], Print[m]; m = m + dm]; Join[{0}, RealDigits[ s[m], 10, digits][[1]]]

prodeulerrat(1 - (p^3 - (p - 1)^3)/(p^3*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_3 = Product_{p prime} 1 - (p^3 - (p - 1)^3)/(p^3*(p - 1)).

More digits from Vaclav Kotesovec, Jun 19 2020

A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

0, 2, 6, 1, 0, 7, 4, 4, 6, 3, 1, 4, 9, 1, 7, 7, 0, 8, 0, 8, 3, 2, 4, 9, 3, 9, 4, 3, 1, 3, 8, 2, 1, 4, 6, 7, 2, 5, 5, 6, 2, 6, 6, 7, 3, 6, 4, 0, 5, 5, 3, 8, 0, 4, 5, 2, 7, 6, 1, 1, 7, 3, 3, 7, 1, 0, 2, 4, 9, 8, 2, 0, 0, 5, 6, 5, 8, 7, 0, 1, 4, 0, 9, 9, 6, 8, 4, 7, 0, 4, 8, 1, 5, 1, 1, 5, 2, 2, 6, 0, 3, 8, 6, 9, 4, 0

Offset: 0

Jean-François Alcover, Apr 16 2016

			0.026107446314917708083...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Cf. A005596, A065414, A065415, A065416, A271780, A271798, A271869.

$MaxExtraPrecision = 2000; LR = LinearRecurrence[{2, 3, -10, 10, -5, 1}, {0, -8, 6, -40, 35, -194}, 10^4]; r[n_Integer] := LR[[n]]; NSum[r[n] PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2000, WorkingPrecision -> 300, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, 20]& // First // Prepend[#, 0]&
$MaxExtraPrecision = 1000; Clear[f]; f[p_] := 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)); Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 105]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)

prodeulerrat(1 - (p^4 - (p - 1)^4)/(p^4*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_4 = Product_{p prime} 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)).

More digits from Vaclav Kotesovec, Jun 19 2020

cp's OEIS Frontend

A008837 a(n) = p*(p-1)/2 for p = prime(n).

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A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree.

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A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

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A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.

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Mathematica

PARI

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