A068468 -id:A068468 - cp's OEIS frontend

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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

Offset: 1

Benoit Cloitre, Apr 12 2003

Equals the Dirichlet zeta-function Sum_{n>=1} A001615(n)/n^s at s=3. - R. J. Mathar, Apr 02 2011
Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists. - Charles R Greathouse IV, Nov 26 2013
From Stanislav Sykora, Nov 14 2014: (Start)
Equals lim_{n->infinity} (Sum_{k=1..n} k/phi(k))/n, i.e., the limit mean value of k/phi(k), where phi(k) is Euler's totient function.
Also equals lim_{n->infinity} (Sum_{k=1..n} 1/phi(k))/log(n).
Proofs are trivial using the formulas for Sum_{k=1..n} k/phi(k) and Sum_{k=1..n} 1/phi(k) listed in the Wikipedia link.
For the limit mean value of phi(k)/k, see A059956. (End)
The asymptotic mean of A005361. - Amiram Eldar, Apr 13 2020

			1.94359643682075920505707036257476343718785850176780571602663568890 ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.7, p. 116.
Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329-345.
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.
Paul Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.
J. von zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
D. Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Eric Weisstein's World of Mathematics, Powerful Number.
Wikipedia, Euler's totient function.

Cf. A000010, A001615, A001694, A002117, A005117, A005361, A013661, A013664, A014197, A059956, A068468, A070243, A082696 (continued fraction), A157292.

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)

PARI
```
zeta(3)*315/2/Pi^4
```

Decimal expansion of Product_{p prime} (1+1/p/(p-1)) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707...
The sum of the reciprocals of the powerful numbers, A001694. - T. D. Noe, May 03 2006
Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.
Equals Sum_{k>=1} mu(k)^2/(k*phi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Euler totient function values, A000010). - Amiram Eldar, Aug 18 2020

New definition from Eric W. Weisstein, May 04 2006

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Original entry on oeis.org

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.

In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Mathematica
```
Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
```

a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022

A336591 Numbers whose exponents in their prime factorization are either 1, 3, or both.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is zeta(6)/(zeta(2) * zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) = 0.68428692418686231814196872579121808347231273672316377728461822629005... (Cohen, 1962).

First differs from A036537 at n = 89. A036537(89) = 128 = 2^7 is not a term of this sequence.

			1 is a term since it has no exponents, and thus it has no exponent that is not 1 or 3.
2 is a term since 2 = 2^1 has only the exponent 1 in its prime factorization.
24 is a term since 24 = 2^3 * 3^1 has the exponents 1 and 3 in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.

Intersection of A046100 and A036537.
Intersection of A046100 and A268335.
A005117 and A062838 are subsequences.
Cf. A068468.

seqQ[n_] := AllTrue[FactorInteger[n][[;;,2]], MemberQ[{1, 3}, #] &]; Select[Range[100], seqQ]

from itertools import count, islice
from sympy import factorint
def A336591_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e==1 or e==3 for e in factorint(n).values()),count(max(startvalue,1)))
A336591_list = list(islice(A336591_gen(),20)) # Chai Wah Wu, Jun 22 2023

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A333787 Fully multiplicative with a(2) = 2 and a(p) = p-1 for odd primes p.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 6, 8, 4, 8, 10, 8, 12, 12, 8, 16, 16, 8, 18, 16, 12, 20, 22, 16, 16, 24, 8, 24, 28, 16, 30, 32, 20, 32, 24, 16, 36, 36, 24, 32, 40, 24, 42, 40, 16, 44, 46, 32, 36, 32, 32, 48, 52, 16, 40, 48, 36, 56, 58, 32, 60, 60, 24, 64, 48, 40, 66, 64, 44, 48, 70, 32, 72, 72, 32, 72, 60, 48, 78, 64, 16, 80, 82, 48, 64

Offset: 1

Antti Karttunen, Apr 07 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000035, A003958, A006519, A068468.

Cf. also A329697.

Array[If[# == 1, 1, Apply[Times, FactorInteger[#] /. {p_Integer, e_Integer} :> If[p == 2, 2, p - 1]^e]] &, 85]  (* Michael De Vlieger, Apr 15 2020 *)

A333787(n) = { my(f=factor(n)); for(i=1,#f~,f[i,1] -= (f[i,1]%2)); factorback(f); };

Multiplicative with a(p^e) = (p-A000035(p))^e.
a(n) = A003958(n) * A006519(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/(210*zeta(3)) = (3/4) * A068468 = 0.385882... . - Amiram Eldar, Nov 10 2022

A074467 Least k such that Sum_{i=1..k} 1/phi(i) >= n.

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 38, 63, 105, 177, 296, 495, 828, 1386, 2318, 3879, 6489, 10854, 18158, 30375, 50811, 84998, 142187, 237853, 397885, 665589, 1113411, 1862534, 3115683, 5211973, 8718687, 14584783, 24397699, 40812930, 68272636, 114207749, 191048868, 319590137

Offset: 1

Labos Elemer, Aug 29 2002

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 177, p. 55, Ellipses, Paris 2008.
E. Landau, Uber die Zahlentheoretische Function ϕ(n) und ihre Beziehung zum Goldbachschen satz, Nachrichten der Koniglichten Gesel lschaft der Wissenschaften zu Göttingen mathematisch Physikalische klasse, Jahrgang (1900), pp. 177-186.

H. L. Montgomery, Primes in arithmetic progressions, Michigan Math. J. 17:1 (1970), pp. 33-39. [alternate link]

Cf. A002387, A004080, A046024, A074631, A074633.

{s=0, s1=0}; Do[s=s+(1/EulerPhi[n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

a(n)=my(s,k);while(sCharles R Greathouse IV, Jan 29 2013

a(n) ~ k exp(cn) for c = zeta(6)/zeta(2)/zeta(3) = A068468 and k = exp(-gamma + A085609) = 1.0316567993311528...; see Montgomery or Koninck. - Charles R Greathouse IV, Jan 29 2013

More terms from Ryan Propper, Jul 09 2005

a(32)-a(38) from Donovan Johnson, Aug 21 2011

A079579 Totally multiplicative with p -> (p-1)*p, p prime.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 24 2003

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003958, A008683, A027746, A065488, A068468.

a079579 1 = 1
a079579 n = product $ zipWith (*) pfs $ map (subtract 1) pfs
   where pfs = a027746_row n
-- Reinhard Zumkeller, Jan 05 2012

f[p_, e_] := ((p - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Oct 23 2022 *)

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

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)

A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

Original entry on oeis.org

72, 108, 200, 360, 392, 500, 504, 540, 600, 675, 756, 792, 936, 968, 1125, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1500, 1656, 1800, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2312, 2484, 2520, 2600, 2646, 2664, 2700, 2888, 2904, 2952, 3087

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers whose powerful part (A057521) is a term of A375073.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) + 1/zeta(2) - zeta(6)/(zeta(2) * zeta(3)) * c = A215267 - A088453 + A059956 - A068468 * c = 0.0156712080080470088619..., where c = Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5).

Equals A046100 \ (A004709 UNION A336591).
Disjoint union of A375073 and A375075.
Cf. A051903, A057521.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A215267.

Select[Range[3000], Union[Select[FactorInteger[#][[;; , 2]], # > 1 &]] == {2, 3} &]

is(k) = Set(select(x -> x > 1, factor(k)[,2])) == [2, 3];

A051903(a(n)) = 3.

A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A163569 at n = 25: a(25) = 2520 = 2^3 * 3^2 * 5 * 7 is not a term of A163569.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 2, 3}.
The asymptotic densities of this sequence and A375074 are equal (0.0156712..., see A375074 for a formula), since the terms in A375074 that are not in this sequence (A375073) have a density 0.

Intersection of A375072 and A317090.
Equals A375074 \ A375073.
Subsequence of A046100 and A176297.
A163569 is a subsequence.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A136568, A215267.

Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]

is(k) = Set(factor(k)[,2]) == [1, 2, 3];

cp's OEIS Frontend

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

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A336591 Numbers whose exponents in their prime factorization are either 1, 3, or both.

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A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

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A333787 Fully multiplicative with a(2) = 2 and a(p) = p-1 for odd primes p.

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A074467 Least k such that Sum_{i=1..k} 1/phi(i) >= n.

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A079579 Totally multiplicative with p -> (p-1)*p, p prime.

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