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A005596
Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).
(Formerly M2608)
+10
89
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
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
FORMULA
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
EXAMPLE
0.37395581361920228805472805434641641511162924860615...
MATHEMATICA
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 *)
PROG
(PARI) 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
(PARI) prodeulerrat(1-1/(p^2-p)) \\ Amiram Eldar, Mar 12 2021
CROSSREFS
Cf. A048296, A065414, A001913, A001122.
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)
STATUS
approved
A003959
If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.
+10
86
1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Completely multiplicative.
Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. - Franklin T. Adams-Watters, Jan 25 2010
LINKS
Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Index to divisibility sequences
FORMULA
Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson, Aug 01 2001
Sum_{n>0} a(n)/n^s = Product_{p prime} 1/(1-p^(-s)-p^(1-s)) (conjectured). - Ralf Stephan, Jul 07 2013
This follows from the absolute convergence of the sum (compare with a(n) = n^2) and the Euler product for completely multiplicative functions. Convergence occurs for at least Re(s)>3. - Thomas Anton, Jul 15 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - Vaclav Kotesovec, Jul 17 2021
From Thomas Scheuerle, Jul 19 2021: (Start)
a(n) = gcd(A166642(n), A166643(n)).
a(n) = A166642(n)/A061142(n).
a(n) = A166643(n)/A165824(n).
a(n) = A166644(n)/A165825(n).
a(n) = A166645(n)/A165826(n).
a(n) = A166646(n)/A165827(n).
a(n) = A166647(n)/A165828(n).
a(n) = A166649(n)/A165830(n).
a(n) = A166650(n)/A165831(n).
a(n) = A167351(n)/A166590(n). (End)
Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - Vaclav Kotesovec, Aug 22 2021
MAPLE
a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80); # Alois P. Heinz, Sep 13 2017
MATHEMATICA
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* Jean-François Alcover, Apr 22 2011 *)
PROG
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X-p*X))[n]) /* Ralf Stephan */
(Haskell)
a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
CROSSREFS
Apart from initial terms, same as A064478.
Cf. A003958, A027746, A036987, A061142, A163407.
Cf. A063441, A165824, A165825, A165826, A165827.
Cf. A165828, A165829, A165831, A167350, A166590.
Cf. A166642, A166643, A166644, A166645, A166646.
Cf. A166647, A166649, A166650, A166586, A165830.
Cf. A167351, A168065, A168066.
KEYWORD
nonn,easy,nice,mult
AUTHOR
Marc LeBrun
EXTENSIONS
Definition reedited (with formula) by Daniel Forgues, Nov 17 2009
STATUS
approved
A351219
Multiplicative with a(p^e) = Fibonacci(e+1).
+10
7
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 13, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
These numbers were called Zetanacci numbers by Bruckman (1983).
The distinct values of the terms are in A065108.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul S. Bruckman, Problem H-359, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 21, No. 3 (1983), p. 238; Zetanacci, Solution to Problem H-359 by C. Georghiou, ibid., Vol. 23, No. 1 (1985), pp. 91-92.
FORMULA
Dirichlet g.f.: Product_{p prime} 1/(1 - p^(-s) - p^(-2*s)).
a(n) = 1 if and only if n is a squarefree number (A005117).
Sum_{k=1..n} a(k) ~ c * n, where c = A065488 = Product_{p primes} (1 + 1/(p^2 - p - 1)) = 2.67411272557... - Vaclav Kotesovec, Feb 10 2022
MATHEMATICA
f[p_, e_] := Fibonacci[e + 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1] = fibonacci(f[k, 2]+1); f[k, 2]=1); factorback(f); \\ Michel Marcus, Feb 05 2022
(Python)
from math import prod
from sympy import factorint, fibonacci
def a(n): return prod(fibonacci(e+1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 05 2022
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022
CROSSREFS
Cf. A000045, A005117, A065108, A065488.
KEYWORD
nonn,mult,easy
AUTHOR
Amiram Eldar, Feb 05 2022
STATUS
approved
A079579
Totally multiplicative with p -> (p-1)*p, p prime.
+10
2
1, 2, 6, 4, 20, 12, 42, 8, 36, 40, 110, 24, 156, 84, 120, 16, 272, 72, 342, 80, 252, 220, 506, 48, 400, 312, 216, 168, 812, 240, 930, 32, 660, 544, 840, 144, 1332, 684, 936, 160, 1640, 504, 1806, 440, 720, 1012, 2162, 96, 1764, 800, 1632, 624, 2756, 432, 2200, 336
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The Dirichlet inverse is 1, -2, -6, 0, -20, 12, -42, 0, 0, 40, -110, 0, -156, 84, 120, 0, -272, ..., i.e., the sequence defined by mu(n)*a(n). - R. J. Mathar, Dec 20 2011
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
a(n) <= n^2.
a(n) = n iff n = 2^k.
a(n) = n*A003958(n).
Multiplicative sequence with a(p^e) = p^e*(p-1)^e for prime p. - Jaroslav Krizek, Nov 01 2009
Dirichlet g.f.: sum_{n>=1} a(n)/n^s = Product_{primes p} 1/(1+p^(1-s)-p^(2-s)). - R. J. Mathar, Dec 20 2011
From Amiram Eldar, Oct 23 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(6)/(3*zeta(2)*zeta(3)) = 2*Pi^4/(945*zeta(3)) = A068468 / 3 = 0.171503... .
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2-p-1)) (A065488). (End)
MATHEMATICA
f[p_, e_] := ((p - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Oct 23 2022 *)
PROG
(Haskell)
a079579 1 = 1
a079579 n = product $ zipWith (*) pfs $ map (subtract 1) pfs
where pfs = a027746_row n
-- Reinhard Zumkeller, Jan 05 2012
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]*(f[i, 1]-1))^f[i, 2]); } \\ Amiram Eldar, Oct 23 2022
CROSSREFS
Cf. A003958, A008683, A027746, A065488, A068468.
KEYWORD
nonn,mult
AUTHOR
Reinhard Zumkeller, Jan 24 2003
STATUS
approved
A167338
Totally multiplicative sequence with a(p) = p*(p+1) = p^2+p for prime p.
+10
2
1, 6, 12, 36, 30, 72, 56, 216, 144, 180, 132, 432, 182, 336, 360, 1296, 306, 864, 380, 1080, 672, 792, 552, 2592, 900, 1092, 1728, 2016, 870, 2160, 992, 7776, 1584, 1836, 1680, 5184, 1406, 2280, 2184, 6480, 1722, 4032, 1892, 4752, 4320, 3312, 2256, 15552
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
Multiplicative with a(p^e) = (p*(p+1))^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+1))^e(k).
a(n) = n * A003959(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 1)) = A065489 = 1.419562880505485919317235861789735359166071586305122542698983695564330971... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.8913709085... . - Amiram Eldar, Dec 15 2022, c = A065488/3. - Vaclav Kotesovec, Apr 05 2023
Dirichlet g.f.: zeta(s-2) * Product_{p prime} (1 + 1/(p^(s-1) - p - 1)). - Vaclav Kotesovec, Apr 05 2023
MATHEMATICA
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 1/(1/X/p - p - 1))/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023
CROSSREFS
Cf. A003959, A065489, A133205.
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Nov 01 2009
STATUS
approved
A306190
a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.
+10
2
1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
EXAMPLE
a(3) = 19 because 5^2 - 5 - 1 = 19.
MAPLE
map(p -> p^2-p-1, [seq(ithprime(i), i=1..100)]); # Robert Israel, Mar 11 2019
MATHEMATICA
Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)
PROG
(PARI) a(n) = {p=prime(n); p^2-p-1; } \\ Jinyuan Wang, Feb 02 2019
CROSSREFS
Supersequence of A091568.
Subsequence of A028387 or A165900.
Cf. A000040, A030431, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.
KEYWORD
nonn
AUTHOR
Kritsada Moomuang, Jan 28 2019
STATUS
approved
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