A185393 -id:A185393 - cp's OEIS frontend

A349117 a(n) = Sum_{m=1..n} (Sum_{k=1..m} (Sum_{j=1..k} j^k)).

1, 7, 49, 445, 5266, 77258, 1349554, 27306462, 627568355, 16142172173, 459332766227, 14324480721391, 485783513552956, 17798331858727376, 700589353757045796, 29484907446960975744, 1321168518044435497005, 62795290373559355285155, 3155553461189975793914005

Offset: 1

Vaclav Kotesovec, Nov 08 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Cf. A086787, A349116.

Table[Sum[Sum[Sum[j^k, {j, 1, k}], {k, 1, m}], {m, 1, n}], {n, 1, 20}]

a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393 = 1/A068996 = 1.581976706...

A349964 a(n) = Sum_{k=0..n} (k*n)^n.

Original entry on oeis.org

1, 1, 20, 972, 90624, 13828125, 3133930176, 988501957072, 414139067400192, 222497518123837665, 149143419250000000000, 122020951254446884154196, 119671520043865789861724160, 138593796657903100873209121453

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Cf. A031971, A155956, A185393, A349966, A349969,

a[n_] := Sum[If[k == n == 0, 1, (k*n)^n], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Dec 07 2021 *)

PARI
```
a(n) = sum(k=0, n, (k*n)^n);
```

a(n) = n^n * [x^n] Sum_{k>=0} (k * x)^k/(1 - k * x) = n^n * A031971(n).

a(n) ~ c * n^(2*n), where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A052824 A simple grammar: cycles of pairs of cycles.

Original entry on oeis.org

0, 0, 2, 6, 34, 220, 1808, 17388, 194724, 2478096, 35418192, 561533280, 9785418432, 185921027136, 3825633439392, 84756646285920, 2011657535668128, 50924796197369088, 1369659967551038976, 39003791158314816768, 1172394903935534452992, 37094744191300029964800

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 789

Cf. A185393.

spec := [S,{B=Cycle(Z),C=Prod(B,B),S=Cycle(C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[-Log[1-(Log[1-x])^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: log(-1/(-1+log(-1/(-1+x))^2)).
E.g.f.: -log(1-(log(1-x))^2). - Vaclav Kotesovec, Oct 01 2013
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - Vaclav Kotesovec, Oct 01 2013

More terms from Alois P. Heinz, Mar 16 2016

A366342 a(n) = Product_{k=1..n} Sum_{j=1..k} j^k.

Original entry on oeis.org

1, 5, 180, 63720, 281961000, 18939602331000, 22733280436308624000, 561162207057469095693888000, 322278252906706683140441912431680000, 4806568058842248598039183477606983722184000000, 2055653754202086984879290521714456895014175320595424000000

Offset: 1

Vaclav Kotesovec, Oct 07 2023

nonn

Cf. A002109, A031971, A074962, A185393, A366329.

Table[Product[Sum[j^k, {j, 1, k}], {k, 1, n}], {n, 1, 12}]
Table[Product[HarmonicNumber[k, -k], {k, 1, n}], {n, 1, 12}] // FunctionExpand

a(n) = A002109(n) * Product_{k=1..n} Sum_{j=1..k} (j/k)^k.
a(n) ~ A002109(n) * c * d^n / n^f, where
d = 1/(1 - exp(-1)) = A185393
f = (exp(1) + 1) / (2*(exp(1) - 1)^2) = 0.629685240773129106752912520161993823...
c = 1.038111196610478473178942324022485064169644880240145128332184584611...
a(n) ~ A * c * d^n * n^(n*(n+1)/2 + 1/12 - f) / exp(n^2/4), where A is the Glaisher-Kinkelin constant A074962.

A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).

Original entry on oeis.org

3, 5, 7, 13, 19, 31, 41, 43, 47, 61, 71, 73, 83, 101, 103, 107, 109, 113, 131, 139, 151, 167, 181, 193, 197, 199, 227, 229, 233, 241, 271, 281, 283, 311, 313, 317, 337, 349, 353, 359, 373, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 547, 563, 569, 571, 577, 593, 599

Offset: 1

Alain Rocchelli, Mar 13 2025

nonn

All terms of this sequence are contained in A060770.

a(n) ~ prime(round(n*e/(e-1))) as n tends to infinity, cf. A185393.

			3 is a term because 2*log(2)/3 > 1*log(1)/2 and 3 is the 2nd prime following 2.
5 is a term because 3*log(3)/5 > 2*log(2)/3 and 5 is the 3rd prime following 3.

Cf. A382051, A060770, A068996, A185393.

Select[Prime[Range[2, 109]],PrimePi[#]*Log[PrimePi[#]]/#>(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 14 2025 *)

my(N=1); forprime(P=3, 600, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1-1/e (A068996).

A306811 Decimal expansion of Pi/(Pi - 1) = 1 + 1/Pi + 1/Pi^2 + ... .

Original entry on oeis.org

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

Offset: 1

Giovanni Zedda, Mar 11 2019

Apart from the first digit the same as A201775. - R. J. Mathar, Apr 09 2019

			1.4669422069242598599833948...

Index entries for transcendental numbers.

Cf. A000796 (Pi), A185393 (e/(e-1)), A201775.

RealDigits[Pi/(Pi - 1), 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

PARI
```
Pi/(Pi-1) \\ Michel Marcus, Mar 12 2019
```

Equals Sum_{j>=0} 1/(Pi^j).

cp's OEIS Frontend

A349117 a(n) = Sum_{m=1..n} (Sum_{k=1..m} (Sum_{j=1..k} j^k)).

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A349964 a(n) = Sum_{k=0..n} (k*n)^n.

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A052824 A simple grammar: cycles of pairs of cycles.

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A366342 a(n) = Product_{k=1..n} Sum_{j=1..k} j^k.

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A382052 Primes prime(k) such that k*log(k)/prime(k) > (k-1)*log(k-1)/prime(k-1).

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A306811 Decimal expansion of Pi/(Pi - 1) = 1 + 1/Pi + 1/Pi^2 + ... .

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Mathematica

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A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).