A136141 -id:A136141 - cp's OEIS frontend

A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

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

Offset: 0

R. J. Mathar, Jan 21 2013

			0.33022992626420324101.. = 1/(2*3) +1/(3*4) +1/(5*6) + 1/(7*8) +... = sum_{n>=1} 1/ (A000040(n)*A008864(n)).

Jason Kimberley, Table of n, a(n) for n = 0..683
Index to constants which are prime zeta sums {1,0,1}

Cf. A136141 for 1/(p(p-1)), A085548 for 1/p^2.
Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A307379.

R:=RealField(103);
ExhaustSum :=
  function(
    k_min, term
  : IZ := func)
    c:=R!0; k:=k_min;
    repeat
      t:=term(k); c+:=t; k+:=1;
    until IZ(t,k-1);
    return c;
  end function;
RealField(101)!
ExhaustSum(2,
  func
    : IZ:=func
    )>);
// Jason Kimberley, Jan 20 2017

interface(quiet=true):
read("transforms") ;
Digits := 300 ;
ZetaM := proc(s,M)
    local v,p;
    v := Zeta(s) ;
    p := 2;
    while p <= M do
        v := v*(1-1/p^s) ;
        p := nextprime(p) ;
    end do:
    v ;
end proc:
Hurw := proc(a)
        local T,p,x,L,i,Le,pre,preT,v,t,M ;
    T := 40 ;
    preT := 0.0 ;
    while true do
            1/p/(p+a) ;
            subs(p=1/x,%) ;
            exp(%) ;
            t := taylor(%,x=0,T) ;
            L := [] ;
            for i from 1 to T-1 do
                    L := [op(L),evalf(coeftayl(t,x=0,i))] ;
            end do:
            Le := EULERi(L) ;
        M := -a ;
            v := 1.0 ;
            pre := 0.0 ;
            for i from 2 to nops(Le) do
                    pre := log(v) ;
                    v := v*evalf(ZetaM(i,M))^op(i,Le) ;
                    v := evalf(v) ;
            end do:
        pre := (log(v)+pre)/2. ;
        printf("%.105f\n",%) ;
        if abs(1.0-preT/pre)  < 10^(-Digits/3) then
            break;
        end if;
        preT := pre ;
        T := T+10 ;
    end do:
        pre ;
end proc:
A179119 := proc()
    Hurw(1) ;
end proc:
A179119() ;

digits = 101; S = NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 5]; RealDigits[S, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

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)) \\ Charles R Greathouse IV, Aug 03 2016

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

P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - Charles R Greathouse IV, Aug 03 2016

A036689 Product of a prime and the previous number.

Original entry on oeis.org

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332, 1640, 1806, 2162, 2756, 3422, 3660, 4422, 4970, 5256, 6162, 6806, 7832, 9312, 10100, 10506, 11342, 11772, 12656, 16002, 17030, 18632, 19182, 22052, 22650, 24492, 26406, 27722, 29756, 31862, 32580, 36290, 37056, 38612, 39402, 44310

Offset: 1

Felice Russo

Records in A002618. - Artur Jasinski, Jan 23 2008

Also records in A174857. - Vladimir Shevelev, Mar 31 2010

			2*1, 3*2, 5*4, 7*6, 11*10, 13*12, 17*16, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to prime powers

Cf. A000040, A001248, A002618, A005596, A053650, A053192, A053193, A053650, A082695, A117495, A136141.
Twice the terms of A008837.
Subsequence of A002378 (oblong numbers).
Column 1 of A257251. (Row 1 of A257252.)
Column 2 of A379010.

a036689 n = a036689_list !! (n-1)
a036689_list = zipWith (*) a000040_list $ map pred a000040_list
-- Reinhard Zumkeller, Sep 17 2011

[ n*(n-1): n in PrimesUpTo(220) ]; // Bruno Berselli, Apr 11 2011

A036689 := proc(n) local p ; p := ithprime(n) ; p*(p-1) ; end proc: # R. J. Mathar, Apr 11 2011

Table[Prime[n] EulerPhi[Prime[n]], {n, 100}] (* Artur Jasinski, Jan 23 2008 *)
Table[Prime[n] (Prime[n] - 1), {n, 1, 50}] (* Bruno Berselli, Apr 22 2014 *)
#(#-1)&/@Prime[Range[50]] (* Harvey P. Dale, Sep 08 2019 *)

forprime(p=2,1e3,print1(p^2-p", ")) \\ Charles R Greathouse IV, Jun 10 2011

A036689(n) = ((p->(p-1)*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

(define (A036689 n) (* (A000040 n) (- (A000040 n) 1))) ;; Antti Karttunen, May 01 2015

a(n) = prime(n) * (prime(n) - 1).
a(n) = phi(prime(n)^2) = A000010(A001248(n)).
a(n) = prime(n) * phi(prime(n)). - Artur Jasinski, Jan 23 2008
From Reinhard Zumkeller, Sep 17 2011: (Start)
a(n) = A000040(n) * A006093(n) = A001248(n) - A000040(n).
A006530(a(n)) = A000040(n). (End)
a(n) = A009262(prime(n)). - Enrique Pérez Herrero, May 12 2012
a(n) = prime(n)! mod (prime(n)^2). - J. M. Bergot, Apr 10 2014
a(n) = 2*A008837(n). - Antti Karttunen, May 01 2015
Sum_{n>=1} 1/a(n) = A136141. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)*zeta(3)/zeta(6) (A082695).
Product_{n>=1} (1 - 1/a(n)) = A005596. (End)

Deleted two incorrect comments. - N. J. A. Sloane, May 07 2020

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

Original entry on oeis.org

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset: 1

Daniel Forgues, May 27 2008

nonn

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
From Michael De Vlieger, Aug 11 2025: (Start)
This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)

Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.

With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Aug 11 2025 *)

isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - Amiram Eldar, Aug 15 2025

A088529 Numerator of Bigomega(n)/Omega(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 1, 1, 5, 4, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 3, 1, 1, 1, 3, 1, 3, 3, 2

Offset: 2

Cino Hilliard, Nov 16 2003

			bigomega(24) / omega(24) = 4/2 = 2, so a(24) = 2.

H. Z. Cao, On the average of exponents, Northeast. Math. J., Vol. 10 (1994), pp. 291-296.

Antti Karttunen, Table of n, a(n) for n = 2..10000
R. L. Duncan, On the factorization of integers, Proc. Amer. Math. Soc. 25 (1970), 191-192.
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001222, A070012, A070013, A070014, A088530 (gives the denominator).

Cf. A136141, A272531.

Table[Numerator[PrimeOmega[n]/PrimeNu[n]], {n, 2, 100}] (* Michael De Vlieger, Jul 12 2017 *)

for(x=2,100,y=bigomega(x)/omega(x);print1(numerator(y)","))

from sympy import primefactors, Integer
def bigomega(n): return 0 if n==1 else bigomega(Integer(n)/primefactors(n)[0]) + 1
def omega(n): return Integer(len(primefactors(n)))
def a(n): return (bigomega(n)/omega(n)).numerator
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Jul 13 2017

Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = numerator of B/O.
a(n) = A136565(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010
Sum_{k=2..n} a(k)/A088530(k) ~ n + O(n/log(log(n))) (Duncan, 1970). - Amiram Eldar, Oct 14 2022
Sum_{k=2..n} a(k)/A088530(k) = n + c_1 * n/log(log(n)) + c_2 * n/log(log(n))^2 + O(n/log(log(n))^3), where c_1 = A136141 and c_2 = A272531 (Cao, 1994; Finch, 2020). - Amiram Eldar, Dec 15 2022

A129283 a(n) = (arithmetic derivative of n) + n.

Original entry on oeis.org

0, 1, 3, 4, 8, 6, 11, 8, 20, 15, 17, 12, 28, 14, 23, 23, 48, 18, 39, 20, 44, 31, 35, 24, 68, 35, 41, 54, 60, 30, 61, 32, 112, 47, 53, 47, 96, 38, 59, 55, 108, 42, 83, 44, 92, 84, 71, 48, 160, 63, 95, 71, 108, 54, 135, 71, 148, 79, 89, 60, 152, 62, 95, 114, 256, 83, 127, 68

Offset: 0

Reinhard Zumkeller, Apr 07 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A003415, A051674, A129284, A129285, A129286, A136141.

a129283 n = a003415 n + n  -- Reinhard Zumkeller, Nov 01 2013

A129283 := proc(n)
    n+A003415(n) ;
end proc:
seq(A129283(n),n=0..40) ; # R. J. Mathar, Feb 04 2022

ad[n_] := Switch[n, 0|1, 0, ?PrimeQ, 1, , Sum[Module[{p, e}, {p, e} = pe; n*e/p], {pe, FactorInteger[n]}]];
a[n_] := ad[n] + n;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 23 2023 *)

a(n) = A003415(n) + n.
a(n) = A003415(n*A051674(k)) / A051674(k).
a(A129284(n)) > 1, a(A129285(n)) > 1, a(A129286(n)) > 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + Sum_{p prime} 1/(p*(p-1)) = 1 + A136141 = 1.773156... . - Amiram Eldar, May 14 2025

A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

By geometric series expansion, the same as the sum over the prime zeta function at even arguments, P(2i), i=1,2,....

(Pi^2/6)*density of A190641, the numbers divisible by exactly one prime with exponent greater than 1. - Charles R Greathouse IV, Aug 02 2016

			0.551693297656999184439731023971343578131500377778628252230...

Jacques Grah, Comportement moyen du cardinal de certains ensembles de facteurs premiers, Monatsh. Math., Vol. 118 (1994), pp. 91-109, Corollary 6.1.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.
Index to constants which are prime zeta sums {0,1,1}

Cf. A000040, A000961, A056798, A084920, A085548, A085964, A085966, A085968, A152447, A154932, A190641.

digits = 105; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[delta1 = Sum[PrimeZetaP[2n], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2m0];
While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[delta1, digits]]; rd[m] (* Jean-François Alcover, Sep 11 2015, updated Mar 16 2019 *)

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))))
sumpos(n=1,primezeta(2*n)) \\ Charles R Greathouse IV, Aug 02 2016

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

Equals Sum_{k>=1} 1/A084920(k) = Sum_{i>=1} P(2i) = A085548+A085964+A085966+A085968+... = A152447+A085548-A154932.
Equals Sum_{k>=2} 1/A000961(k)^2 = Sum_{k>=2} 1/A056798(k). - Amiram Eldar, Sep 21 2020
Equals (A136141 + A179119)/2. - Artur Jasinski, Mar 31 2025

More digits from Jean-François Alcover, Sep 11 2015

A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

Original entry on oeis.org

0, -1, -1, -2, 0, -4, -1, -6, 4, -3, -3, -10, 4, -12, -5, -7, 16, -16, 3, -18, 4, -11, -9, -22, 20, -15, -11, 0, 4, -28, 1, -30, 48, -19, -15, -23, 24, -36, -17, -23, 28, -40, -1, -42, 4, -6, -21, -46, 64, -35, -5, -31, 4, -52, 27, -39, 36, -35, -27, -58, 32, -60, -29

Offset: 0

Paolo P. Lava, Nov 17 2009

Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A007850, A051674, A054377, A136141.

Cf. A003415, A051674, A083347, A083348.

a168036 n = a003415 n - n  -- Reinhard Zumkeller, May 22 2015

with(numtheory);
A168036:=proc(q)
local n,p;
for n from 0 to q do
  print(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])-n); od; end:
A168036(1000); # Paolo P. Lava, Nov 05 2012

np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
Table[np[n] - n, {n, 0, 100}] (* Robert Price, Mar 14 2020 *)

a(A083347(n)) < 0; a(A051674(n)) = 0; a(A083348(n)) > 0. - Reinhard Zumkeller, May 22 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - Amiram Eldar, Dec 08 2023

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 3

Offset: 1

Antti Karttunen, Aug 11 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.
Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
Cf. A085548, A136141.

Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k,2] > 1, f[k,2])); \\ Michel Marcus, Jul 19 2017

sub a275812 { vecsum( grep {$> 1} map {$->[1]} factor_exp(shift) ); } # Dana Jacobsen, Aug 15 2016

from sympy import factorint, primefactors
def a001222(n):
    return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
def a(n):
    return a001222(n) - a056169(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, and for n > 1, if A067029(n)=1 [when n is one of the terms of A247180], a(n) = a(A028234(n)), otherwise a(n) = A067029(n)+a(A028234(n)).
a(n) = A001222(n) - A056169(n).
a(n) = A001222(A057521(n)). - Antti Karttunen, Jul 19 2017
From Amiram Eldar, Sep 28 2023: (Start)
Additive with a(p) = 0, and a(p^e) = e for e >= 2.
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 + 1/(p*(p-1))) = A085548 + A136141 = 1.22540408909086062637... . (End)
a(n) = A046660(n) + A056170(n). - Amiram Eldar, Jan 09 2024

A380840 Decimal expansion of Sum_{p prime} 1/(p-1)^3.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 19 2025

nonn

			1.1475290977585800469332838..,

Cf. A085541, A086242, A085548, A136141, A152441, A154945, A179119, A369632, A324833.

PARI
```
sumeulerrat(1/(p-1)^3)
```

A083342 Decimal expansion of average deviation of the total number of prime factors.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 25 2003

Or, decimal expansion of constant B2 from the summatory function of the restricted divisor function.

The constant A in the asymptotic formula Sum_{prime p <= n} 1/(p-1) = log(log(n)) + A + O(1/log(n)) (Jakimczuk, 2017). - Amiram Eldar, Mar 18 2024

			1.03465388189743791161979429846463825467030798434438525450307...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.
József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, p. 155, Chapter V, 1) b).

Rafael Jakimczuk, On Sums of Powers of the p-adic Valuation of n!, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit upper bound for Siegel zeros of imaginary quadratic fields, arXiv:2001.05782 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Mertens Constant.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A000010, A001620, A007434, A077761, A086242, A136141.

digits = 102; Mp = EulerGamma - NSum[PrimeZetaP[n]/n - PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[Mp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)

Equals A077761 + A136141. - Jean-François Alcover, Sep 02 2015
Equals gamma + Sum_{p prime} (log(1-1/p) + 1/(p-1)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021
From Amiram Eldar, Mar 18 2024: (Start)
Equals gamma + Sum_{k>=2} phi(k) * log(zeta(k)) / k, where phi = A000010.
Equals gamma - Sum_{p prime} 1/(p-1)^2 + Sum_{k>=2} J_2(k) * log(zeta(k)) / k, where J_2 = A007434.
Both formulas are from Jakimczuk (2017). (End)

cp's OEIS Frontend

A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

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A036689 Product of a prime and the previous number.

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A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

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A088529 Numerator of Bigomega(n)/Omega(n).

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A129283 a(n) = (arithmetic derivative of n) + n.

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A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

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