A067698 -id:A067698 - cp's OEIS frontend

A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)log(log(r/p))) > sigma(r)/(rlog(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.

1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 5, 3, 3, 2, 3, 5, 3, 7, 2, 5, 5, 5, 3, 7, 7, 2

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Sep 29 2011

nonn

See comments, references, links and crossrefs in A067698.

			A067698(5) = 6 and sigma(6/2)/((6/2)*log(log(6/2))) = 14.17... > 3.42... = sigma(6)/(6*log(log(6))), so a(5) = 2.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33; see Table 1 and Lemma 11.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Cf. A067698.

A091902 Erroneous version of A067698.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360

Offset: 1

dead

A073004 Decimal expansion of exp(gamma).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

See references and additional links in A094644.
The Riemann hypothesis holds if and only if the inequality sigma(n)/(n*log(log(n))) < exp(gamma) is valid for all n >= 5041, (G. Robin, 1984). - Peter Luschny, Oct 18 2020
From Peter Bala, Aug 24 2025: (Start)
By definition, gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.
Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*exp(s(n+k)). Then it appears that E(n) converges rapidly to exp(gamma). For example, E(50) = 1.78107241799019798523650410310(43...) gives exp(gamma) correct to 29 decimal digits. Cf. A002389. (End)

			Exp(gamma) = 1.7810724179901979852365041031071795491696452143034302053...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.5.1 and 2.27.2, pp. 31, 187.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 166, 191, 208.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
Paul Erdős and S. K. Zaremba, The arithmetic function Sum_{d|n} log d/d, Demonstratio Mathematica, Vol. 6 (1973), pp. 575-579.
T. H. Gronwall, Some Asymptotic Expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
Simon Plouffe, The exp(gamma).
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Gronwall's Theorem.
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 2-3.
Eric Weisstein's World of Mathematics, Robin's Theorem.

Cf. A001620 (Euler-Mascheroni constant, gamma).

Cf. A001113, A002389, A067698, A080130, A091901, A094644 (continued fraction for exp(gamma)), A155969, A246499.

R:=RealField(100); Exp(EulerGamma(R)); // G. C. Greubel, Aug 27 2018

RealDigits[ E^(EulerGamma), 10, 110] [[1]]

PARI
```
exp(Euler)
```

By Mertens theorem, equals lim_{m->infinity}(1/log(prime(m))*Product_{k=1..m} 1/(1-1/prime(k))). - Stanislav Sykora, Nov 14 2014
Equals limsup_{n->oo} sigma(n)/(n*log(log(n))) (Gronwall, 1913). - Amiram Eldar, Nov 07 2020
Equals limsup_{n->oo} (Sum_{d|n} log(d)/d)/(log(log(n)))^2 (Erdős and Zaremba, 1973). - Amiram Eldar, Mar 03 2021
Equals Product_{k>=1} (1-1/(k+1))*exp(1/k). - Amiram Eldar, Mar 20 2022
Equals lim_{n->oo} n * Product_{prime p<=n} p^(1/(1-p)). - Thomas Ordowski, Jan 30 2023
Equals Product_{k>=1} (k/sqrt(2))^((-1)^k/(k*log(2))). - Antonio Graciá Llorente, Oct 11 2024
Equals lim_{n->oo} (1/log(n))*Product_{prime p<=n} p/(p - 1) [Mertens] (see Finch at p. 31). - Stefano Spezia, Oct 27 2024

A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 7, 2, 7, 5, 13, 0, 17, 9, 12, 8, 23, 5, 27, 8, 21, 20, 34, 1, 33, 25, 30, 17, 46, 7, 50, 22, 40, 37, 46, 6, 62, 43, 50, 19, 70, 19, 74, 37, 46, 55, 82, 9, 79, 46, 70, 47, 95, 32, 83, 38, 81, 74, 107, 2, 112, 81, 76, 56, 102, 45, 125, 70, 103, 58, 133, 14, 138, 101

Offset: 1

N. J. A. Sloane, Oct 12 2000

Theorem (Lagarias): a(n) is nonnegative for all n if and only if the Riemann Hypothesis is true.
Up to rank n=10^4, zeros occur only at n=1,2,4,6 and 12; ones occur at n=3 and n=24. The first occurrence of k = 0,1,2,3,... is at n = 1,3,8,-1,5,10,36,7,16,14,-1,-1,15,11,72,... where -1 means that k does not occur among the first 10^4 terms. - Robert G. Wilson v, Dec 06 2010, reformulated by M. F. Hasler, Sep 09 2011
Looking at the graph of this sequence, it appears that there is a slowly growing lower bound. It is even more apparent when larger ranges of points are computed. Numbers A176679(n+2) and A222761(n) give the (x,y) coordinates of the n-th point. - T. D. Noe, Mar 28 2013

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Peter Luschny, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.

Cf. A057640, A000203, A076633, A067698, A079526, A058209.

f[n_] := Block[{h = HarmonicNumber@n}, Floor[h + Exp@h*Log@h] - DivisorSigma[1, n]]; Array[f, 74] (* Robert G. Wilson v, Dec 06 2010 *)

a(n)={my(H=sum(k=1,n,1/k)); floor(exp(H)*log(H)+H) - sigma(n)}
list_A057641(Nmax,H=0,S=1)=for(n=S,Nmax, H+=1/n; print1(floor(exp(H)*log(H)+H) - sigma(n),","))  \\ M. F. Hasler, Sep 09 2011

a(n) = A057640(n) - A000203(n). - Omar E. Pol, Oct 25 2019

Five more terms from Robert G. Wilson v, Dec 06 2010

I deleted some unproved assertions by Robert G. Wilson v about the presence of 0's, 1's, ... in this sequence. - N. J. A. Sloane, Dec 07 2010

A091901 Values of n for which sigma(n) < e^gamma * n * log(log(n)).

Original entry on oeis.org

7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Eric W. Weisstein, Feb 09 2004

sigma(n) < e^gamma * n * log(log(n)) is "Robin's inequality" - see A067698. Sequence is cofinite if and only if the Riemann Hypothesis is true.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
Eric Weisstein's World of Mathematics, Robin's Theorem

Cf. A067698.

Select[Range[100], DivisorSigma[1, #] < E^EulerGamma*#*Log[Log[#]] &] (* Jean-François Alcover, Oct 30 2012 *)

a(n) = n + 27 for n > 5039, if and only if the Riemann Hypothesis is true. - Charles R Greathouse IV, May 31 2011

A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

Original entry on oeis.org

183783600, 367567200, 1396755360, 1745944200, 2327925600, 3491888400, 6983776800, 80313433200, 160626866400, 252706217563800, 288807105787200, 336941623418400, 404329948102080, 505412435127600, 673883246836800, 1010824870255200, 2021649740510400, 112201560598327200

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 03 2011

nonn

Infinitely many terms are superabundant (SA) A004394; the smallest is 183783600.
Infinitely many terms are colossally abundant (CA) A004490; the smallest is 367567200.
Infinitely many terms are odd (and hence neither SA nor CA); the smallest is 1058462574572984015114271643676625.
See Section 5 of "On SA, CA, and GA numbers".
For additional terms, in factored form, see "Table of proper GA1 numbers up to 10^60", where SA and CA numbers are starred * and **.

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the smallest proper GA1 number.

Amiram Eldar, Table of n, a(n) for n = 1..18408 (from the J.-L. Nicolas's table)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A000203, A001222, A067698, A197638, A197639, A201558, A216436.

Maple
```
See "Computation of GA1 numbers".
```

A197638(n) if A001222(A197638(n)) > 2

A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, May 30 2011

nonn

5040 is the last element in the sequence if and only if the Riemann Hypothesis is true. (See Akbary and Friggstad in A004394.)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A004394, A091901, A067698, A166981, A077006.

kmax = 10^4;
A004394 = Join[{1}, Reap[For[r = 1; k = 2, k <= kmax, k = k + 2, s = DivisorSigma[-1, k]; If[s > r, r = s; Sow[k]]]][[2, 1]]];
A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];
Intersection[A004394, A067698] (* Jean-François Alcover, Jan 28 2019 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)); \\ A067698
lista(nn) = my(r=1, t); forstep(n=2, nn, 2, t=sigma(n, -1); if(t>r && is(n), r=t; print1(n, ", "))); \\ Michel Marcus, Jan 28 2019; adapted from A004394

Equals A004394 intersect A067698.

Erroneous terms 1260 and 1680 removed by Jean-François Alcover, Jan 28 2019

A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 18, 24, 36, 48, 60, 72, 120, 180, 240, 360, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 02 2011

nonn

Subsequence of A067698.

A member > 5040 exists iff the Riemann Hypothesis is false, in which case the sequence is infinite. In any case, 3 and 5 are the only odd members. (See Sections 1 and 4 of "On SA, CA, and GA numbers".)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A000203, A067698, A197638, A201557.

nmax = 10^6; amax = 10;
G[k_] := DivisorSigma[1, k]/(k Log[Log[k]]);
okQ[n_] := DivisorSigma[1, n] > n Exp[EulerGamma] Log[Log[n]] && AllTrue[ Range[amax], Function[a, G[n] >= G[a*n]]];
Reap[For[n = 1, n <= nmax, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

A201558 Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 1, 1, 2, 4, 1, 2, 3, 7, 7, 7, 1, 4, 7

Offset: 3

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 03 2011

nonn

The number of GA1 numbers with one (resp., two) prime factors is zero (resp., infinity).
GA1 numbers with at least three prime factors are called "proper" - see A201557.
For a(n), see Section 6.2 of "On SA, CA, and GA numbers", and below "kmax" in "Table of proper GA1 numbers up to 10^60".

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the first of the a(13) = 2 GA1 numbers with 4 + 3 + 2 + 1 + 1 + 1 + 1 = 13 prime factors.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A067698, A197638, A197639, A201557.

Maple
```
See "Computation of GA1 numbers".
```

A192884 Non-superabundant numbers satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

3, 5, 8, 9, 10, 16, 18, 20, 30, 72, 84

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Jul 11 2011

nonn

If another term exists, it is > 5040 and the Riemann Hypothesis is false.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Cf. A004394 (superabundant), A091901 (Robin's inequality), A067698 (the reverse of Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality).

cp's OEIS Frontend

A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)*log(log(r/p))) > sigma(r)/(r*log(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.

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A091902 Erroneous version of A067698.

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A073004 Decimal expansion of exp(gamma).

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Magma

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PARI

Formula

A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

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References

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Mathematica

PARI

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Extensions

A091901 Values of n for which sigma(n) < e^gamma * n * log(log(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A197639 GA2 numbers: n with G(n) >= G(a*n) for all integers a > 0, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

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A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)log(log(r/p))) > sigma(r)/(rlog(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.

A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.