A065414 -id:A065414 - cp's OEIS frontend

A005596 Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).

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

Offset: 0

N. J. A. Sloane

On Simon Plouffe's web page (and in the book freely available at Gutenberg project) the value is given with an error of +1e-31, as "...651641..." instead of "...641641...". In the reference [Wrench, 1961] cited there, these digits are correct. They are also correct on the Plouffe's Inverter page, as computed by Oliveira e Silva, who comments it took 1 hour at 200 MHz with Mathematica. Using Amiram Eldar's PARI program, the same 500 digits are computed instantly (less than 0.1 sec). - M. F. Hasler, Apr 20 2021

Named after the Austrian mathematician Emil Artin (1898-1962). - Amiram Eldar, Jun 20 2021

			0.37395581361920228805472805434641641511162924860615...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 169.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..1000
Ivan Cherednik, A note on Artin's constant, arXiv:0810.2325 [math.NT], 2008.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2001; constant A_1^(1).
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Pieter Moree, The formal series Witt transform, Discr. Math., Vol. 295, No. 1-3 (2005), pp. 143-160. See p. 159.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
G. Niklasch, Artin's constant.
Simon Plouffe, The Artin's Constant=product(1-1/(p**2-p), p=prime) [backup on web.archive.org; chapter 8 of the free Gutenberg.org/ebooks/634]. [Warning: the value given in this reference is incorrect, cf. comment!]
Tomás Oliveira e Silva and Plouffe's Inverter, The first 500 digits of Artin's constant.
Eric Weisstein's World of Mathematics, Artin's constant.
Eric Weisstein's World of Mathematics, Full Reptend Prime.
R. G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
Index entries for sequences related to Artin's conjecture.

Cf. A048296, A065414, A001913, A001122.

a = Exp[-NSum[ (LucasL[n] - 1)/n PrimeZetaP[n], {n, 2, Infinity}, PrecisionGoal -> 500, WorkingPrecision -> 500, NSumTerms -> 100000]]; RealDigits[a, 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 taken from Mathematica's Help file on PrimeZetaP *)

prodinf(n=2,1/zeta(n)^(sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)) \\ Charles R Greathouse IV, Aug 27 2014

prodeulerrat(1-1/(p^2-p)) \\ Amiram Eldar, Mar 12 2021

Equals Product_{j>=2} 1/Zeta(j)^A006206(j), where Zeta = A013661, A002117 etc. is Riemann's zeta function. - R. J. Mathar, Feb 14 2009
Equals Sum_{k>=1} mu(k)/(k*phi(k)), where mu is the Moebius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Mar 11 2020
Equals 1/A065488. - Vaclav Kotesovec, Jul 17 2021

More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)

A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

Original entry on oeis.org

1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907

Offset: 1

N. J. A. Sloane

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.
The sequence defined by {a(n) - 1} plays a role for the computation of A065414, A146486, A146487, and A146488 equivalent to the role of A001610 for A005596, A146482, A146483 and A146484, see the variable a_{2,n} in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009
Except for n = 2, a(n) is the number of digits in n-th term of A049064. This can be derived form the T. Sillke link below. - Jianing Song, Apr 28 2019

			G.f. = x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 21*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..6012 (terms 1..500 from T. D. Noe)
Ignacio Cascudo, On squares of cyclic codes, arXiv:1703.01267 [cs.IT], 2017. See Theorem 6.1/Table 1.
Johann Cigler, Some remarks on generalized Fibonacci and Lucas polynomials, arXiv:1912.06651 [math.CO], 2019.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 91.
Matthew Macauley, Jon McCammond, Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217 (see Eq. 29).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Souvik Roy, Nazim Fatès, and Sukanta Das, Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence, hal-04456320 [nlin.CG], [cs], 2024. See p. 19.
T. Sillke, The binary form of Conway's sequence
Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390. - _N. J. A. Sloane_, Feb 12 2010
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A000930, A218439.

Cf. also A049064, A049194.

I:=[1,1,4]; [n le 3 select I[n] else Self(n-1)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Jun 28 2015

A001609:=-(1+3*z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation
f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-3), a(1)=1,a(2)=1,a(3)=4},a(n),remember):
map(f, [$1..100]); # Robert Israel, Jun 29 2015

Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
Table[ HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -(n/3)}, {1/2 - n/2, 1 - n/2}, -(27/4)], {n, 20}] (* Alexander R. Povolotsky, Nov 21 2008 *)
a[1] = a[2] = 1; a[3] = 4; m = 3; a[n_] := 1 + n*Sum [Binomial [n - 1 - (m - 1)*i, i - 1]/i, {i, n/m}] A001609 = Table[a[n], {n, 100}] (* Zak Seidov, Nov 21 2008 *)
LinearRecurrence[{1, 0, 1}, {1, 1, 4}, 50] (* Vincenzo Librandi, Jun 28 2015 *)

{a(n) = if( n<1, n=-n; polcoeff( (3 + x^2) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( x * (1 + 3*x^2) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Aug 15 2016 */

G.f.: x*(1 + 3*x^2)/(1 - x - x^3).
a(n) = trace of successive powers of matrix ({{0,0,1},{1,0,0},{0,1,1}})^n. - Artur Jasinski, Jan 10 2007
a(n) = A000930(n) + 3*A000930(n-2). - R. J. Mathar, Nov 16 2007
Logarithmic derivative of Narayana's cows sequence A000930. - Paul D. Hanna, Oct 28 2012
a(n) = w1^n + w2^n + w3^n, where w1,w2,w3 are the roots of the cubic: (-1 - x^2 + x^3), see A092526. - Gerry Martens, Jun 27 2015

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
More terms from Michael Somos, Oct 03 2002
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A089189 Primes p such that p-1 is cubefree.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 373, 383

Offset: 1

Cino Hilliard, Dec 08 2003 and Reinhard Zumkeller, Aug 11 2004

The ratio of the count of primes p <= n such that p-1 is cubefree to the count of primes <= n converges to 0.69.. . This implies that roughly 70% of the primes less one are cubefree. This compares to about 0.37 of the primes less one are squarefree.

More accurately, the density of this sequence within the primes is Product_{p prime} (1-1/(p^2*(p-1))) = 0.697501... (A065414) (Mirsky, 1949). - Amiram Eldar, Feb 16 2021

			43 is included because 43-1 = 2*3*7.
41 is omitted because 41-1 = 2^3*5.
97 is omitted because 96 = 2^5*3 since higher powers are also tested for exclusion.

Reinhard Zumkeller and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A004709, A039787, A065414, A097380, A089194 (subsequence).

a097375 n = a097375_list !! (n-1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
-- Reinhard Zumkeller, May 27 2012

filter:= p -> isprime(p) and max(seq(t[2],t=ifactors(p-1)[2]))<=2:
select(filter, [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 11 2014

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]>2,a=1],{m,Length[FactorInteger[n]]}];a]; lst={};Do[p=Prime[n];If[f[p-1]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 15 2009 *)
Select[Prime[Range[100]],Max[Transpose[FactorInteger[#-1]][[2]]]<3&] (* Harvey P. Dale, Feb 05 2012 *)

lista(nn) = forprime(p=2, nn, f = factor(p-1)[,2]; if ((#f == 0) || vecmax(f) < 3, print1(p, ", "));) \\ Michel Marcus, Sep 11 2014

A212793(a(n) - 1) = 1. - Reinhard Zumkeller, May 27 2012

Corrected and extended by Harvey P. Dale, Feb 05 2012

A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.85654044485354217442616798413595388...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant A_1^(3) table 3.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065416.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 1, -1}, {0, 0, 0, 4, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-1/(p^4-p^3)) \\ Amiram Eldar, Mar 13 2021

A065416 Decimal expansion of Product_{p prime} (1-1/(p^5-p^4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.93126518416000433438923720555067698...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065415.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 0, 1, -1}, {0, 0, 0, 0, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 5, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-1/(p^5-p^4)) \\ Amiram Eldar, Mar 12 2021

A135177 a(n) = p^2*(p-1), where p = prime(n).

Original entry on oeis.org

4, 18, 100, 294, 1210, 2028, 4624, 6498, 11638, 23548, 28830, 49284, 67240, 77658, 101614, 146068, 201898, 223260, 296274, 352870, 383688, 486798, 564898, 697048, 903264, 1020100, 1082118, 1213594, 1283148, 1430128, 2032254, 2230930

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4) = 294 because the 4th prime number is 7, 7^2 = 49, 7-1 = 6 and 49 * 6 = 294.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Index to sequences related to prime powers.

Cf. A001248 (p^2), A030078 (p^3), A045991 (n^2 * (n-1)), A065414, A065483, A138416 (terms halved), A152441.

Column 4 of A379010.

[(p^3-p^2): p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

Table[Prime[n]^3-Prime[n]^2, {n, 1, 12}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[p^3-p^2,{p,Prime[Range[40]]}] (* Harvey P. Dale, Jan 15 2015 *)

forprime(p=2,1e3,print1(p^2*(p-1)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A135177(n) = eulerphi(prime(n)^3); \\ Antti Karttunen, Dec 14 2024

A135177(n) = ((p->(p-1)*p*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

a(n) = p^3 - p^2 = A030078(n) - A001248(n).
a(n) = A000010(prime(n)^3). - R. J. Mathar, Oct 15 2017
Sum_{n>=1} 1/a(n) = A152441. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065483.
Product_{n>=1} (1 - 1/a(n)) = A065414. (End)
a(n) = 2*A138416(n). - Antti Karttunen, Dec 14 2024

A089199 Primes p such that p+1 is divisible by a cube.

Original entry on oeis.org

7, 23, 31, 47, 53, 71, 79, 103, 107, 127, 151, 167, 191, 199, 223, 239, 263, 269, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 499, 503, 593, 599, 607, 631, 647, 701, 719, 727, 743, 751, 809, 823, 839, 863, 887, 911, 919, 967, 971, 983, 991

Offset: 1

Cino Hilliard, Dec 08 2003

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p^2*(p-1))) = 1 - A065414 = 0.302498... (Mirsky, 1949). - Amiram Eldar, Apr 07 2021

Robert Israel, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Includes A007522 and A141965.

Cf. A049098, A065414.

filter:= proc(p)
  isprime(p) and ormap(t -> t[2]>=3, ifactors(p+1)[2])
end proc:
select(filter, [seq(i,i=3..2000,2)]); # Robert Israel, Jan 11 2019

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)

ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }

A089200 Primes p such that p-1 is divisible by a cube.

Original entry on oeis.org

17, 41, 73, 89, 97, 109, 113, 137, 163, 193, 233, 241, 251, 257, 271, 281, 313, 337, 353, 379, 401, 409, 433, 449, 457, 487, 521, 541, 569, 577, 593, 601, 617, 641, 673, 751, 757, 761, 769, 809, 811, 857, 881, 919, 929, 937, 953, 977

Offset: 1

Cino Hilliard, Dec 08 2003

This sequence is infinite and its relative density in the sequence of primes is 1 - Product_{p prime} (1-1/(p^2*(p-1))) = 1 - A065414 = 0.30249864150363409671... (Jakimczuk, 2024). - Amiram Eldar, Jul 20 2024

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Numbers of the form p-1 where p is prime, ResearchGate preprint, 2024.

Cf. A039787, A046099, A065414.

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
Select[Prime[Range[200]],Count[Transpose[FactorInteger[#-1]][[2]], ?(#>2&)]>0&] (* _Harvey P. Dale, Jan 01 2012 *)

ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }

A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 11, 14, 20, 27, 39, 52, 75, 102, 145, 201, 286, 397, 565, 791, 1123, 1581, 2248, 3173, 4517, 6399, 9112, 12945, 18457, 26270, 37502, 53478, 76416, 109146, 156135, 223301, 319764, 457884, 656288, 940795, 1349671, 1936620

Offset: 1

N. J. A. Sloane, Nov 15 2001

nonn

Inverse Euler transform of A078012. (The inverse of 1-1/(p^3-p^2) is p^2(p-1)/(p^3-p^2-1) = 1-1/(1+p^2-p^3). Setting 1/p=x gives (1-x)/(1-x-x^3), the g.f. of A078012.) - R. J. Mathar, Jul 26 2010

			x^3 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + 7*x^12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, sequence gamma_{2,j}^(A).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Index entries for sequences related to Artin's conjecture

Cf. A065414.

read("transforms") ;
A078012 := proc(n) option remember; if n <3 then op(n+1,[1,0,0]) ; else procname(n-1)+procname(n-3) ; end if; end proc:
a078012 := [seq(A078012(n),n=1..80)] ; EULERi(%) ;
# R. J. Mathar, Jul 26 2010

A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];
s = {}; For[i = 1, i <= m, i++, AppendTo[s, i*a078012[[i]] - Sum[s[[d]] * a078012[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d ], 0]*s[[d]], {d, 1, i}]/i, {i, m}] (* Jean-François Alcover, Apr 15 2016, after R. J. Mathar *)

a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - Vaclav Kotesovec, Jun 13 2020

More terms from R. J. Mathar, Jul 26 2010

A089191 Primes p such that p+1 is cubefree.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 59, 61, 67, 73, 83, 89, 97, 101, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 233, 241, 251, 257, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 373, 379, 389, 397, 401, 409

Offset: 1

Cino Hilliard, Dec 08 2003

The ratio of the count of primes p <= n such that p+1 is cubefree to the count of primes <= n converges to 0.69+ slightly higher than the p-1 variety.

More accurately, the density of this sequence within the primes is Product_{p prime} (1-1/(p^2*(p-1))) = 0.697501... (A065414) (Mirsky, 1949). - Amiram Eldar, Feb 16 2021

			43 is included because 43+1 = 2^2*11.
71 is omitted because 71+1 = 2^3*3^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A004709, A049097, A065414, A089189.

filter:= t -> isprime(t) and max(map(s -> s[2], ifactors(t+1)[2]))<3:
select(filter, [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 18 2018

Select[Prime[Range[100]],Max[Transpose[FactorInteger[#+1]][[2]]]<3&] (* Harvey P. Dale, Jun 06 2013 *)

is(n) = isprime(n) && vecmax(factor(n+1)[,2]) < 3 \\ Amiram Eldar, Feb 16 2021

cp's OEIS Frontend

A005596 Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).

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A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

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A089189 Primes p such that p-1 is cubefree.

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A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

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A065416 Decimal expansion of Product_{p prime} (1-1/(p^5-p^4)).

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A135177 a(n) = p^2*(p-1), where p = prime(n).

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