A086242 -id:A086242 - cp's OEIS frontend

A382554 Decimal expansion of Sum_{p prime} 1/(p + 1)^2.

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.24447730415659330695194837134735096763737...

Index to constants which are prime zeta sums {0,0,2}.

Cf. A085548, A086242.

sumeulerrat(1/(p+1)^2) \\ Amiram Eldar, Apr 01 2025

Equals Sum_{k>=2} (-1)^k * (k-1) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

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)

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

A119686 Numerator of Sum_{k=1..n} 1/(prime(k) - 1)^2.

Original entry on oeis.org

1, 5, 21, 193, 4861, 2443, 78401, 707209, 85701889, 4203312961, 841345613, 841819933, 4211020661, 4212763061, 2229320057669, 376856710434461, 317005189060740101, 317069381268836117, 317122432680485717

Offset: 1

Alexander Adamchuk, Jun 08 2006

Lim_{n -> infinity} a(n)/A334746(n) = A086242.

			The first few fractions are 1, 5/4, 21/16, 193/144, 4861/3600, 2443/1800, 78401/57600, 707209/518400, ... = A119686/A334746.

Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A000040, A006093, A086242, A334746 (denominators).

(* First program *)
Numerator[Table[Sum[1/(Prime[i]-1)^2,{i,1, n}], {n,1,30}]]
(* Second program *)
Numerator[Accumulate[1/(Prime[Range[20]]-1)^2]] (* Harvey P. Dale, Jun 28 2017 *)

a(n)=numerator(sum(k=1,n,1/(prime(k)-1)^2)) \\ Charles R Greathouse IV, Apr 24 2015

a(n) = numerator(Sum_{k=1..n} 1/(Prime(k) - 1)^2).

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

A005722 a(n) = (prime(n) - 1)^2.

Original entry on oeis.org

1, 4, 16, 36, 100, 144, 256, 324, 484, 784, 900, 1296, 1600, 1764, 2116, 2704, 3364, 3600, 4356, 4900, 5184, 6084, 6724, 7744, 9216, 10000, 10404, 11236, 11664, 12544, 15876, 16900, 18496, 19044, 21904, 22500, 24336, 26244, 27556, 29584, 31684, 32400, 36100

Offset: 1

Scorpion(AT)aol.com

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Richard K. Guy, Letter to N. J. A. Sloane, 1987.

Cf. A001248, A005597, A006093, A065485, A086242, A095874, A192134.

[(p-1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

A005722:=n->(ithprime(n)-1)^2; seq(A005722(n), n=1..40); # Wesley Ivan Hurt, Mar 27 2014

Table[(Prime[n] - 1)^2, {n, 40}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = (prime(n) - 1)^2; \\ Michel Marcus, Nov 09 2020

a(n) = A192134(A095874(A001248(n))) - 1. - Reinhard Zumkeller, Jun 26 2011
a(n) = A006093(n)^2. - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = A086242. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065485.
Product_{n>=2} (1 - 1/a(n)) = A005597. (End)

A334746 Denominator of Sum_{k=1..n} 1/(prime(k) - 1)^2.

Original entry on oeis.org

1, 4, 16, 144, 3600, 1800, 57600, 518400, 62726400, 3073593600, 614718720, 614718720, 3073593600, 3073593600, 1625931014400, 274782341433600, 231091949145657600, 231091949145657600, 231091949145657600, 231091949145657600, 77030649715219200

Offset: 1

Petros Hadjicostas, May 11 2020

Lim_{n -> infinity} A119686(n)/a(n) = A086242.

			The first few fractions are 1, 5/4, 21/16, 193/144, 4861/3600, 2443/1800, 78401/57600, 707209/518400, ... = A119686/A334746.

Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A000040, A006093, A086242, A119686 (numerators).

Denominator @ Accumulate @ Table[1/(Prime[k] - 1)^2, {k, 1, 21}] (* Amiram Eldar, May 12 2020 *)

a(n) = denominator(sum(k=1, n, 1/(prime(k) - 1)^2)); \\ Michel Marcus, May 12 2020

A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

Value computed and communicated by Bill Allombert and confirmed by Pascal Sebah.

			1.9069738480349544...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533.

/* author Bill Allombert */
\p150
pz(x, ex=0)=
{
my(s=bitprecision(x));
my(B=s/real(polcoef(x, 0))+ex);
sum(n=1, B, my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
}
my(P=primes([2, 61])); intnum(x=1, [oo, log(67)], (pz(x)-vecsum([p^-x|p<-P]))*intnum(s=0, [oo, 1], (x-1)^s/gamma(1+s))) + vecsum([1/p/log(p)/log(log(p))|p<-P])

A053198 Totients of consecutive pure powers of primes.

Original entry on oeis.org

2, 4, 6, 8, 20, 18, 16, 42, 32, 54, 110, 100, 64, 156, 162, 128, 272, 294, 342, 256, 506, 500, 486, 812, 930, 512, 1210, 1332, 1640, 1806, 1024, 1458, 2028, 2162, 2058, 2756, 2500, 3422, 3660, 2048, 4422, 4624, 4970, 5256, 6162, 4374, 6498, 6806, 7832, 4096

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Totients of prime powers are prime powers only for powers of 2.

			The 10th pure power of prime (but not a prime) is 81, so a(10) = EulerPhi(81) = 54.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A086242.

EulerPhi[Select[Range[2^13], CompositeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Dec 21 2020 *)

a(n) = A000010(A025475(n+1)).
Numbers of the form phi(p^k) = (p-1)*p^(k-1), where p is prime and k > 1.
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p-1)^2 = A086242 = 1.3750649947... - Amiram Eldar, Dec 21 2020

cp's OEIS Frontend

A382554 Decimal expansion of Sum_{p prime} 1/(p + 1)^2.

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

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

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

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A119686 Numerator of Sum_{k=1..n} 1/(prime(k) - 1)^2.

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

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A005722 a(n) = (prime(n) - 1)^2.

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A334746 Denominator of Sum_{k=1..n} 1/(prime(k) - 1)^2.

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A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).