A179119 -id:A179119 - cp's OEIS frontend

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

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

Offset: 0

Jean-François Alcover, Apr 17 2016

This is the density of A060687, the numbers with one 2 and the rest 1s in the exponents of its prime factorization. - Charles R Greathouse IV, Aug 03 2016

			0.200755722019265986996250723114404765853535555352561916...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.

Cf. A059956, A060687, A085548, A136141, A179119, A222056.

digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ Charles R Greathouse IV, Aug 03 2016

sumeulerrat(1/(p*(p+1)))/zeta(2) \\ Amiram Eldar, Mar 18 2021

Equals (6/Pi^2)*A179119.

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

Offset: 1

Labos Elemer

nonn

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

			n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A048108.
A072357 is a subsequence.
Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

is(n)=n==1 || factorback(factor(n)[,2])<3 \\ Charles R Greathouse IV, Aug 25 2016

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

A028236 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 7, 1, 9, 8, 1, 1, 11, 1, 9, 10, 13, 1, 11, 1, 15, 1, 11, 1, 31, 1, 1, 14, 19, 12, 13, 1, 21, 16, 13, 1, 41, 1, 15, 14, 25, 1, 19, 1, 27, 20, 17, 1, 29, 16, 15, 22, 31, 1, 47, 1, 33, 16, 1, 18, 61, 1, 21, 26, 59, 1, 17, 1, 39, 28, 23, 18, 71, 1, 21, 1, 43

Offset: 1

N. J. A. Sloane

			Fractions begin with 1, 1/2, 1/3, 1/4, 1/5, 5/6, 1/7, 1/8, 1/9, 7/10, 1/11, 7/12, ...

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

Denominator is n (A000027).

Cf. A001221, A141809, A179119.

a028236 n = sum $ map (div n) $ a141809_row n
-- Reinhard Zumkeller, Nov 10 2013

a028236:=func< k | k eq 1 select 1 else Numerator(&+[ f[i, 1]^-f[i,2]: i in [1..#f] ]) where f is Factorization(k) >; [ a028236(n):n in [1..82] ]; // Klaus Brockhaus, Nov 06 2010

a[n_] := n * Total[1/Power @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Sep 29 2023 *)

Fraction is additive with a(p^e) = 1/p^e.
a(n) = Sum_{k=1..A001221(n)} n/A141809(n,k). - Reinhard Zumkeller, Nov 10 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/k = Sum_{p prime} 1/(p*(p+1)) = 0.330229... (A179119). - Amiram Eldar, Sep 29 2023

More terms from Erich Friedman

A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 17 2019

			0.12903892589780755649743486348177587763849321419920568830041270456398...

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
Index to constants which are prime zeta sums {0,2,2}

Cf. A154945 (eta_1), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

digits = 101; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta2 = Sum[n PrimeZetaP[2n + 2], {n, 1, m}], 10, digits][[1]]; rd[m0]; rd[m = 2m0]; While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[eta2, digits] ]; rd[m]

Sum_{p prime} 1/(p^2-1)^2.
Sum_{n>0} n P(2n+2) where P is the prime zeta P function.
Equals - A136141/4 + A086242/4 - A179119/4 + A382554/4. - Artur Jasinski, Mar 31 2025

A324834 Decimal expansion of eta_3, a constant related to the asymptotic density of certain sets of residues.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 17 2019

			0.03907240573557479188765950332042297638668483824477336035675406603269...

Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
Index to constants which are prime zeta sums {0,3,3}

Cf. A154945 (eta_1), A324833 (eta_2), A324835 (eta_4), A324836 (eta_5).

digits = 102; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta3 = Sum[n (n+1)/2 PrimeZetaP[2 n + 4], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2 m0]; While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[eta3, digits]]; rd[m]

Sum_{p prime} 1/(p^2-1)^3.
Sum_{n>0} (n(n+1)/2) P(2n+4) where P is the prime zeta P function.
Equals 3*A136141/16 - 3*A086242/16 + A380840/8 + 3*A179119/16 - 3*A382554/16 - A382555/8. - Artur Jasinski, Mar 31 2025

A036436 Numbers whose number of divisors is a square.

Original entry on oeis.org

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR (named after Hardy and Ramanujan) concept formation program.
Numbers in this sequence but not in A036455 are 1, 1260, 1440, 1800, 1980 etc. [From R. J. Mathar, Oct 20 2008]
tau(p^(n^2-1)) = n^2 so numbers of this form are in this sequence, and because tau is multiplicative: if a and b are in this sequence and (a,b)=1 then a*b is also in a(n). - Enrique Pérez Herrero, Jan 22 2013
What is the density of this sequence? It contains A030229 and thus has (lower) density at least 3/Pi^2 = 0.30396...; it does not contain any members of A030059 or A060687, and hence has (upper) density at most 1 - 3/Pi^2 - 6*A179119/Pi^2 = 0.49528.... - Charles R Greathouse IV, Jan 11 2025

			tau(6)=4, which is a square number, so 6 is in this sequence.

S. Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics (Unfortunately [403 Forbidden])
S. Colton and A. Bundy, Automatic Concept Formation in Pure Mathematics
S. Colton, A. Bundy and T. Walsh, Agent Based Cooperative Theory Formation in Pure Mathematics
S. Colton, R. McCasland and A. Bundy, Automated Theory Formation for Tutoring Tasks in Pure Mathematics, 2002.

Contains A030229 as a subsequence.

Cf. A009087, A033950, A057265, A049439.

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

is(n)=issquare(numdiv(n)) \\ Charles R Greathouse IV, Jan 22 2013

Links corrected and edited by Daniel Forgues, Jun 30 2010

A068050 Number of values of k, 1<=k<=n, for which floor(n/k) is prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 2, 2, 4, 5, 3, 4, 5, 6, 5, 6, 5, 6, 6, 7, 9, 10, 6, 7, 9, 9, 9, 10, 10, 11, 9, 10, 12, 14, 11, 12, 13, 14, 13, 14, 13, 14, 14, 15, 17, 18, 13, 14, 16, 17, 18, 19, 17, 19, 18, 19, 21, 22, 18, 19, 20, 21, 19, 21, 22, 23, 23, 24, 26, 27, 21, 22, 23, 24, 24, 26, 27

Offset: 1

Amarnath Murthy, Feb 12 2002

			a(10) = 4 as floor(10/k) for k = 1 to 10 is 10,5,3,2,2,1,1,1,1,1, respectively; this is prime for k = 2,3,4,5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Randell Heyman, Primes in floor function sets, arXiv:2111.00408 [math.NT], 2021.

Cf. A067514.

Cf. A010051, A179119, A205745.

a068050 n = length [k | k <- [1..n], a010051 (n `div` k) == 1]
-- Reinhard Zumkeller, Jan 31 2012

a[n_] := Length[Select[Table[Floor[n/i], {i, 1, n}], PrimeQ]]
Table[Count[Table[Floor[n/k],{k,n}],?PrimeQ],{n,80}] (* _Harvey P. Dale, Nov 19 2022 *)

a(n) = sum(k=1, n, isprime(n\k)); \\ Michel Marcus, Jun 03 2024

If p is a prime other than 3, a(p) = a(p-1) + 1. - Franklin T. Adams-Watters, Apr 27 2020
a(n) = A179119*n + O(n^(1/2)). - Randell Heyman, Oct 06 2022
a(n) = Sum_{p prime and p<=n} (floor(n/p) - floor(n/(p+1))). - Ridouane Oudra, Jun 03 2024

Edited by Dean Hickerson, Feb 12 2002

A369632 Decimal expansion of Sum_{primes p} 1/(p*(p^2 - 1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 28 2024

			0.22146337139279594342463643588459881748724095830455...

Index to constants which are prime zeta sums {1,1,1}

Cf. A127917, A136141, A179119.

RealDigits[NSum[PrimeZetaP[2*k + 1], {k, 1, Infinity}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Jan 28 2024 *)

sumeulerrat(1/(p*(p^2-1))) \\ Amiram Eldar, Jan 28 2024

Equals Sum_{i>=1} 1/A127917(i) = (A136141 - A179119)/2.

Equals Sum_{k>=1} P(2*k+1), where P(s) is the prime zeta function. - Amiram Eldar, Jan 28 2024

More terms from Amiram Eldar, Jan 28 2024

A185380 Decimal expansion of sum 1/(p*(p+2)) over the primes p.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

If we omit the first term 1/(2*4)=0.125 from the sum, 0.138672... remains, which is an upper limit of A209329 in the sense that we "fake" prime gaps of 2 here [which are actually larger on average].

			0.263672061761153178749842188233776 .. = 1/(2*4) +1/(3*5) + 1/(5*7) + 1/(7*9) + 1/(11*13)+ ...

Cf. A136141 (1/(p(p-1))), A179119 (1/(p(p+1))).

read("transforms") ;
Digits := 300 ;
# insert coding of ZetaM(s,M) and Hurw(a) from A179119 here...
A185380 := proc()
        Hurw(2) ;
end proc:
A185380() ;

sumeulerrat(1/(p*(p+2))) \\ Amiram Eldar, Mar 19 2021

Equals -1/8 + Sum_{k>=2} (-1)^k * 2^(k-2) * P(k), where P is the prime zeta function. - Vaclav Kotesovec, Jan 13 2021

More digits from Vaclav Kotesovec, Jan 13 2021

A307379 Decimal expansion of Sum_{n >= 1} 2/(k(n)*(k(n) + 1)), with k = A018252 (nonprime numbers).

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Apr 06 2019

We know that Sum_{n >= 1} 2/(n^2 + n) = 2 and Sum_{n >= 1} 2/(p(n)*(p(n) + 1)) = 2*A179119, where p = A000040. Therefore, the present decimal expansion 1/1 + 1/10 + 1/21 + 1/36 + ... = 2*(1 - A179119).

			1.3395401474715935179... = 2 - (1/3 + 1/(3*2) + 1/(5*3) + 1/(7*4) + 1/(11*6) + ...) = 2*(1 - A179119).

Cf. A000040, A008864, A018252, A179119.

digits = 87;
S = 2 - 2 NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5];
RealDigits[S, 10, digits][[1]] (* Jean-François Alcover, Jun 20 2019 *) [From A179119]

2*(1 - sumeulerrat(1/(p*(p+1)))) \\ Amiram Eldar, Mar 18 2021

Equals 2*(1 - A179119) = 2*(1 - Sum_{n>=1} 1/(A000040(n)*A008864(n))).

Edited by Wolfdieter Lang, Jul 10 2019

cp's OEIS Frontend

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A028236 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Formula

Extensions

A324833 Decimal expansion of eta_2, a constant related to the asymptotic density of certain sets of residues.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A324834 Decimal expansion of eta_3, a constant related to the asymptotic density of certain sets of residues.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A036436 Numbers whose number of divisors is a square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A068050 Number of values of k, 1<=k<=n, for which floor(n/k) is prime.

Views

Author

Keywords

Examples

Links

Crossrefs