A124175 -id:A124175 - cp's OEIS frontend

A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

1, 1, 1, 2, 4, 16, 32, 192, 768, 4608, 18432, 184320, 737280, 8847360, 53084160, 424673280, 3397386240, 54358179840, 326149079040, 5870683422720, 46965467381760, 563585608581120, 5635856085811200, 123988833887846400, 991910671102771200, 19838213422055424000

Offset: 0

Simon Plouffe

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n [Smith and Mansion]. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001

The matrix M(i,j) = gcd(i,j) is sequence A003989. - Michael Somos, Jun 25 2012

			a(2) = 1 because the matrix M is: [1,1; 1,2] and det(A) = 1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.
M. Petkovsek et al., A=B, Peters, 1996, p. 21.

Antoine Mathys, Table of n, a(n) for n = 0..496 (first 100 terms by T. D. Noe)
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
E. C. Catalan, Théorème de MM. Smith et Mansion, Nouvelle correspondance mathématique, 4 (1878) 103-112. [_Philippe Deléham_, Dec 22 2003]
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, 76 (2003), 392-394.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A000010, A060238, A060239, A059381, A059382, A059383, A059384, A002088.

Cf. A003989.

List([1..30],n->Product([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018

a001088 n = a001088_list !! (n-1)
a001088_list = scanl1 (*) a000010_list
-- Reinhard Zumkeller, Mar 04 2012

with(numtheory,phi); A001088 := proc(n) local i; mul(phi(i),i=1..n); end;
seq(A001088(n), n=0..30);

A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* Enrique Pérez Herrero, Sep 19 2010 *)
Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* Harvey P. Dale, Dec 09 2011 *)

a(n)=prod(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Mar 04 2012

a(n) = phi(1) * phi(2) * ... * phi(n).

Limit_{n->infinity} a(n)^(1/n) / n = exp(-1) * A124175 = 0.205963050288186353879675428232497466485878059342058515016427881513657493... (see Mathoverflow link). - Vaclav Kotesovec, Jun 09 2021

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023

A126908 Numbers k such that 1 + k^2 + k^4 + k^6 + k^7 is prime.

Original entry on oeis.org

1, 4, 13, 15, 24, 30, 37, 40, 55, 93, 138, 139, 148, 153, 154, 159, 160, 165, 184, 195, 204, 223, 258, 303, 355, 360, 373, 459, 472, 475, 510, 519, 534, 577, 594, 607, 615, 627, 658, 672, 688, 723, 735, 739, 795, 805, 807, 817, 819, 820, 847, 874, 879, 904

Offset: 1

Artur Jasinski, Dec 31 2006

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^7], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[1000],PrimeQ[1+#^2+#^4+#^6+#^7]&] (* Harvey P. Dale, Jan 15 2016 *)

is(n)=isprime(1+n^2+n^4+n^6+n^7) \\ Charles R Greathouse IV, Feb 17 2017

A126916 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23 is prime.

Original entry on oeis.org

1, 2, 11, 23, 47, 64, 77, 80, 103, 251, 290, 321, 331, 335, 375, 382, 387, 403, 507, 568, 590, 594, 649, 801, 805, 828, 830, 840, 847, 854, 905, 925, 926, 959, 982, 986, 1034, 1086, 1094, 1102, 1122, 1129, 1147, 1160, 1391

Offset: 1

Artur Jasinski, Dec 31 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[1400],PrimeQ[Total[#^Range[2,22,2]]+1+#^23]&] (* Harvey P. Dale, Oct 04 2018 *)

is(n)=isprime(1+n^2+n^4+n^6+n^8+n^10+n^12+n^14+n^16+n^18+n^20+n^22+n^23) \\ Charles R Greathouse IV, Feb 17 2017

A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 09 2021

			1.561596846931024164326967889144555644364737646822232169945866457...

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20), constant c.

Cf. A001088, A066843, A066780, A124175.

$MaxExtraPrecision = 1000; m = 500; prod = 1; Do[Clear[f]; f[p_] := ((p^(k + 1) - 1)/(p^(k + 1) - p))^(1/p^k); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; prod *= f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 100]]; Print[prod], {k, 1, 200}]

Equals exp(1) * lim_{n->infinity} (A066780(n)^(1/n)) / n.

A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 11, 1, 1, 4, 1, 9, 2, 2, 1, 1, 4, 4, 2, 2, 2, 1, 14, 1, 2, 2, 2, 7, 2, 2, 1, 1, 4, 2, 4, 1, 11, 7, 2, 8, 32, 2, 1, 293, 2, 145, 1, 2, 1, 21, 1, 1, 3, 1, 1, 8, 8, 5, 2, 3, 4, 3, 1, 3, 1, 1, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 19, 3, 2, 1, 15, 1, 2, 1, 2, 5, 3, 1, 1, 1, 38, 1, 10, 1, 2, 1, 80, 1

Offset: 0

Martin Fuller, Dec 20 2006

This might be interpreted as the expected value of phi(n)/n for very large n. - David W. Wilson, Dec 05 2006

			0.55986561693237348...

Cf. A124175, A085548, A085541, A085964, A085965, A085966, A085967, A085968, A085969.

contfrac(exp(-suminf(m=2,log(zeta(m))*sumdiv(m,k,if(k

A126906 Smallest k such that 1 + k^(2n+1) + Sum_{j=1..n} k^(2j) is prime.

Original entry on oeis.org

1, 2, 1, 2, 1, 10, 17, 2, 1, 2, 1, 94, 122, 22, 1, 80, 1, 4, 6, 2, 1, 242, 3, 6, 5, 80, 1, 12, 1, 82, 96, 2, 7, 188, 1, 136, 69, 158, 1, 2, 1, 954, 50, 118, 1, 570, 14, 90, 45, 6, 1, 228, 38, 4, 6, 22, 1, 12, 1, 580, 86, 336, 24, 768, 1, 1170, 408, 340, 1, 896

Offset: 1

Artur Jasinski, Dec 31 2006

nonn

1 is a term if and only if number of terms in polynomial is prime.

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A124151, A119863, A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a[n_]: = Module[{k = 1}, While[!PrimeQ[1 + k^(2*n+1) + Sum[k^(2*j), {j, 1, n}]], k++]; k]; Array[a, 30] (* Amiram Eldar, Mar 13 2020 *)

a(n) = my(k = 1); while(! isprime(1 + k^(2*n+1) + sum(j=1, n, k^(2*j))), k++); k; \\ Michel Marcus, Mar 13 2020

More terms from Amiram Eldar, Mar 13 2020

A126907 Numbers n such that 1 + n^2 + n^4 + n^5 is prime.

Original entry on oeis.org

2, 4, 6, 8, 12, 18, 32, 34, 68, 70, 78, 88, 110, 114, 116, 118, 120, 122, 132, 134, 142, 150, 172, 180, 186, 190, 210, 216, 238, 246, 254, 272, 294, 322, 362, 376, 380, 386, 388, 408, 476, 500, 502, 506, 508, 520, 530, 542, 564, 584, 588, 590, 616, 620, 632

Offset: 1

Artur Jasinski, Dec 31 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^5], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[700],PrimeQ[1+#^2+#^4+#^5]&] (* Harvey P. Dale, Jun 24 2018 *)

is(n)=isprime(1+n^2+n^4+n^5) \\ Charles R Greathouse IV, Jun 13 2017

A126909 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^9 is prime.

Original entry on oeis.org

2, 18, 48, 56, 116, 120, 128, 146, 194, 198, 200, 230, 266, 278, 282, 288, 324, 362, 372, 390, 396, 420, 434, 458, 488, 576, 594, 708, 714, 728, 740, 774, 818, 830, 860, 888, 896, 912, 914, 990, 996, 1002, 1008, 1010, 1016, 1044, 1124, 1128, 1140, 1146, 1260

Offset: 1

Artur Jasinski, Dec 31 2006

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^8 + n^9], AppendTo[a, n]], {n, 1, 1400}]; a
Select[Range[1300],PrimeQ[1+#^2+#^4+#^6+#^8+#^9]&] (* Harvey P. Dale, Apr 25 2020 *)

is(n)=isprime(1+n^2+n^4+n^6+n^8+n^9) \\ Charles R Greathouse IV, Jun 13 2017

A126910 Numbers k such that 1 + k^2 + k^4 + k^6 + k^8 + k^10 + k^11 is prime.

Original entry on oeis.org

1, 2, 3, 35, 48, 77, 97, 105, 111, 112, 122, 128, 161, 168, 175, 216, 231, 255, 271, 276, 297, 338, 361, 370, 378, 422, 485, 513, 525, 558, 622, 658, 661, 662, 667, 675, 700, 718, 725, 742, 753, 766, 770, 795, 796, 833, 875, 886, 921, 993, 1027, 1066, 1078

Offset: 1

Artur Jasinski, Dec 31 2006

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^11], AppendTo[a, n]], {n, 1, 1400}]; a

is(n)=isprime(1+n^2+n^4+n^6+n^8+n^10+n^11) \\ Charles R Greathouse IV, Jun 13 2017

A126911 Numbers k such that 1 + k^2 + k^4 + k^6 + k^8 + k^10 + k^12 + k^13 is prime.

Original entry on oeis.org

10, 24, 60, 148, 174, 180, 268, 274, 280, 294, 346, 472, 484, 516, 522, 598, 654, 804, 834, 856, 858, 898, 994, 1012, 1036, 1054, 1066, 1102, 1168, 1272, 1294, 1338, 1342, 1368, 1420, 1462, 1500, 1536, 1564, 1588, 1608, 1624, 1710, 1746, 1786, 1792, 1822, 1992

Offset: 1

Artur Jasinski, Dec 31 2006

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049407, A124175 A124176 A124177, A124178, A124179, A124180, A124181, A126908-A126916.

a = {}; Do[If[PrimeQ[1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^13], AppendTo[a, n]], {n, 1, 1400}]; a

is(n)=isprime(1+n^2+n^4+n^6+n^8+n^10+n^12+n^13) \\ Charles R Greathouse IV, Jun 13 2017

from sympy import isprime
def ok(k): return isprime(1+sum(k**i for i in [2, 4, 6, 8, 10, 12, 13]))
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Oct 24 2021

cp's OEIS Frontend

A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

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A126908 Numbers k such that 1 + k^2 + k^4 + k^6 + k^7 is prime.

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A126916 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^10 + n^12 + n^14 + n^16 + n^18 + n^20 + n^22 + n^23 is prime.

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A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

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A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

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A126906 Smallest k such that 1 + k^(2*n+1) + Sum_{j=1..n} k^(2*j) is prime.

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A126907 Numbers n such that 1 + n^2 + n^4 + n^5 is prime.

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A126909 Numbers n such that 1 + n^2 + n^4 + n^6 + n^8 + n^9 is prime.

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A126906 Smallest k such that 1 + k^(2n+1) + Sum_{j=1..n} k^(2j) is prime.