A136141 -id:A136141 - 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.

A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, -1, 0, 5, 2, -1, 0, 0, 0, -1, -1, 17, 0, 0, 0, 2, -1, -1, 0, 8, 4, -1, 14, 4, 0, -11, 0, 49, -1, -1, -1, 5, 0, -1, -1, 18, 0, -13, 0, 8, 6, -1, 0, 36, 6, 2, -1, 10, 0, 15, -1, 28, -1, -1, 0, -16, 0, -1, 10, 129, -1, -17, 0, 14, -1, -15, 0, 33

Offset: 1

Omar E. Pol, Dec 18 2012

sign

Observations: at least the first 50 indices of nonnegative terms are also the first 50 terms of A212165. Also at least the first 28 indices of negative terms are also the first 28 terms of A212168, since A212168 is the complement of A212165.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A001065, A003415.

Cf. A013661, A136141.

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Table[dn[n] - (DivisorSigma[1, n] - n), {n, 100}] (* T. D. Noe, Dec 27 2012 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A211991(n) = (A003415(n) - (sigma(n)-n)); \\ Antti Karttunen, Mar 08 2018

a(n) = A003415(n) - A001065(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (A136141 - A013661 + 1) / 2 = 0.0641113... . - Amiram Eldar, Mar 17 2024

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

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

Original entry on oeis.org

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68

Offset: 1

Antti Karttunen, Jun 05 2021

Coincides with A003415 only on perfect numbers (A000396).

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
Cf. also A051709, A344753 (analogous sequences).
Cf. A013661, A136141.

A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
a(n) = A001065(n) + A168036(n).
a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

A361205 a(n) = 2*omega(n) - bigomega(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, -1, 0, 2, 1, 1, 1, 2, 2, -2, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, -1, 1, 1, 3, 1, -3, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, -1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 2, 1, -4, 2, 3, 1, 1, 2, 3, 1, -1, 1, 2, 1, 1, 2, 3, 1, -1, -2, 2, 1, 2

Offset: 1

Gus Wiseman, Mar 16 2023

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

Without doubling omega we have -A046660.
Positions of 0's are A067801, counted by A239959.
Positions of negative terms are A360558, counted by A360254.
Positions of nonpositive terms are A361204, counted by A237363.
Positions of positive terms are A361393, counted by A237365.
Positions of nonnegative terms are A361395, counted by A361394.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
Cf. A061395, A067340, A111907, A324517, A324522, A326567/A326568.
Cf. A077761, A083342, A136141.

Table[2*PrimeNu[n]-PrimeOmega[n],{n,100}]

Additive with a(p^e) = 2 - e. - Amiram Eldar, Mar 26 2023

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = 2*A077761 - A083342 = A077761 - A136141 = -0.511659... . - Amiram Eldar, Oct 01 2023

A190121 Partial sums of the arithmetic derivative function A003415.

Original entry on oeis.org

0, 1, 2, 6, 7, 12, 13, 25, 31, 38, 39, 55, 56, 65, 73, 105, 106, 127, 128, 152, 162, 175, 176, 220, 230, 245, 272, 304, 305, 336, 337, 417, 431, 450, 462, 522, 523, 544, 560, 628, 629, 670, 671, 719, 758, 783, 784, 896, 910, 955, 975, 1031, 1032, 1113, 1129

Offset: 1

Giorgio Balzarotti, May 04 2011

See A229523 for a(10^n). - M. F. Hasler, Sep 25 2013

			1'+2'+3'+4'+5' = 0+1+1+4+1 = 7 -> a(5) = 7.

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., Vol. 4, No. 2 (May 1961), pp. 117-122.

Cf. A003415, A136141, A229523.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(der(i),i=1..j),j=1..100);

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; Table[d[n], {n, 1, 55}] // Accumulate (* Jean-François Alcover, Feb 21 2014 *)
A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]]; Table[Sum[A003415[k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Dec 29 2017 *)

s=0; A190121=vector(199,n,s+=A003415(n))

A190121(n)=sum(k=1,n,A003415(k)) \\ M. F. Hasler, Sep 26 2013

a(n)-> ~ 0.374*n^2 as n-> oo [Barbeau] (note: 1+2+3+4+5 ...-> ~ 1/2*n^2; the similarity stands also for higher power of the terms of sum). - Giorgio Balzarotti, Nov 14 2013

a(n) ~ c * n^2, where c = (1/2) * Sum_{p prime} 1/(p*(p-1)) = A136141 / 2 = 0.3865783345... . This constant was given by Barbeau (1961) but with the wrong value 0.374. - Amiram Eldar, Oct 06 2023

A275699 Excess of numbers that are not squarefree.

Original entry on oeis.org

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

Offset: 1

Felix Fröhlich, Aug 05 2016

nonn

The "excess" of a number is the number of prime divisors with multiplicity (the Omega function, A001222) minus the number of distinct prime divisors (the omega function, A001221). A046660(n) gives the excess of n.

Since squarefree numbers have no excess, this sequence is essentially A046660 with the 0's removed.

			Since 16 = 2^4, 16 has four prime divisors, but only one distinct divisor. Hence Omega(16) - omega(16) = 4 - 1 = 3. As 16 is the fifth number that is not squarefree, its corresponding 3 is a(5) in this sequence.
17 is prime and thus has no excess and no corresponding term in this sequence.
18 = 2 * 3^2, Omega(18) - omega(18) = 3 - 2 = 1, thus a(6) = 1.

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

Cf. A001221, A001222, A013929, A046660, A136141, A229099.

DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)

for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))

a(n) = A046660(A013929(n)).

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) / (1-6/Pi^2) = A136141/A229099 = 1.9719717... - Amiram Eldar, Feb 10 2021

A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

Original entry on oeis.org

9, 18, 25, 27, 36, 49, 50, 54, 72, 81, 98, 100, 108, 121, 125, 144, 162, 169, 196, 200, 216, 242, 243, 250, 288, 289, 324, 338, 343, 361, 392, 400, 432, 484, 486, 500, 529, 576, 578, 625, 648, 676, 686, 722, 729, 784, 800, 841, 864, 961, 968, 972, 1000, 1058

Offset: 1

Gus Wiseman, May 02 2019

nonn

Also Heinz numbers of integer partitions that are not hooks but whose augmented differences are hooks, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k), and a hook is a partition of the form (n,1,1,...,1). The enumeration of these partitions by sum is given by A325459.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    18: {1,2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   169: {6,6}
   196: {1,1,4,4}
   200: {1,1,1,3,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 2's in A325355.
Numbers n such that n does not belong to A093641 but A325351(n) does.
Cf. A056239, A112798, A136141, A325366, A325389, A325394, A325395, A325396.

N:= 1000: # to get terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):
B:= map(proc(p) local y;  seq(p^y, y=2..floor(log[p](N))) end proc, P):
sort(map(proc(t) local z;  seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # Robert Israel, May 03 2019

Select[Range[1000],MatchQ[FactorInteger[2*#],{{2,},{?(#>2&),_?(#>1&)}}]&]

Sum_{n>=1} 1/a(n) = 2 * Sum_{p prime} 1/(p*(p-1)) - 1 = 2 * A136141 - 1 = 0.54631333809959025572... - Amiram Eldar, Sep 30 2020

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

A152441 Decimal expansion of Sum_{primes p} 1/(p^2*(p-1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 04 2008

Generally, sum_p 1/(p^s*(p-1)) equals A136141 minus the sum over all prime zeta functions with index 2 to s (see A085964 to A085969).

			0.320909249008729629357824095023694461445509992843293626574587137005544001125... = 1/(4*1) + 1/(9*2) + 1/(25*4) + 1/(49*6) + ...

Index to constants which are prime zeta sums {2,1,0}

Cf. A085548, A136141, A246549.

digits = 104; sp = NSum[PrimeZetaP[n], {n, 3, Infinity},  WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

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

Equals A136141 minus A085548 .

Equals Sum_{n>=1} 1/A246549(n). - Amiram Eldar, Oct 27 2020

More digits from Jean-François Alcover, Sep 11 2015

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.

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A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.

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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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A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

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A361205 a(n) = 2*omega(n) - bigomega(n).

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A190121 Partial sums of the arithmetic derivative function A003415.

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A275699 Excess of numbers that are not squarefree.

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