A007778 -id:A007778 - cp's OEIS frontend

A143857 a(n) = n + (n+1)*(n+2)^(n+3).

8, 163, 3074, 62503, 1399684, 34588811, 939524102, 27894275215, 900000000008, 31384283767219, 1176925259169802, 47248516628391479, 2022385242251558924, 91957716979980468763, 4427218577690292387854, 225009351233083599856159

Offset: 0

Reinhard Zumkeller, Sep 03 2008

Suggested by Karl Vago's contributions to Dario Alpern's list of records.

			a(1) = 1+(1+1)*(1+2)^(1+3) = 1+2*3^4 = 163.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Dario A. Alpern, ECM Factorization applet records
Dario A. Alpern, Factorization using the Elliptic Curve Method

Cf. A007778, A061250.

List([0..20], n-> n + (n+1)*(n+2)^(n+3)); # G. C. Greubel, Nov 09 2019

[n+(n+1)*(n+2)^(n+3): n in [0..20]]; // Vincenzo Librandi, Dec 27 2010

A143857:=n->n+(n+1)*(n+2)^(n+3); seq(A143857(n), n=0..20); # Wesley Ivan Hurt, Mar 20 2014

Table[n + (n+1)*(n+2)^(n+3), {n, 0, 20}] (* Vincenzo Librandi, Mar 20 2014 *)

vector(21, n, (n-1) + n*(n+1)^(n+2)) \\ G. C. Greubel, Nov 09 2019

[n + (n+1)*(n+2)^(n+3) for n in (0..20)] # G. C. Greubel, Nov 09 2019

a(n) = n + A061250(n+3). - R. J. Mathar, Sep 04 2008

A280255 Numbers k such that tau(k^(k+1)) is a prime.

Original entry on oeis.org

3, 4, 5, 11, 17, 25, 29, 41, 49, 59, 71, 101, 107, 125, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 343, 347, 419, 431, 461, 521, 529, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319

Offset: 1

Jaroslav Krizek, Mar 07 2017

nonn

tau(k) is the number of positive divisors of k (A000005).
Numbers k such that A000005(A007778(k)) is a prime.
Lesser of twin primes (A001359) are terms. If p is lesser of twin primes then tau(p^(p+1)) = p + 2 (see A006512).
Sequence of composite terms c: 4, 25, 49, 125, 343, 529, 1369, ...; (tau(c^(c+1)): 11, 53, 101, 379, 1033, 1061, 2741, ...).
Numbers of the form p^k where p is prime and 1 + k * (1 + p^k) is prime. - Robert Israel, Sep 02 2024

			tau(4^5) = tau(1024) = 11 (prime).

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

Cf. A000005, A001359, A006512, A007778, A280256, A280257.

[n: n in [1..500] | IsPrime(NumberOfDivisors(n^(n+1)))];

N:= 10000: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
R:= {}:
for p in P do
  Qs:= select(q -> isprime(1 + q + q*p^q), {$1..ilog[p](N)});
  R:= R union map(q -> p^q, Qs)
od:
sort(convert(R,list)); # Robert Israel, Sep 02 2024

Select[Range[1319], PrimeQ@DivisorSigma[0, #^(# + 1)] &] (* Giovanni Resta, Mar 07 2017 *)

isok(n) = isprime(numdiv(n^(n+1))); \\ Michel Marcus, Mar 07 2017

A356238 a(n) = Sum_{k=1..n} (k * floor(n/k))^n.

Original entry on oeis.org

1, 8, 62, 849, 8541, 206345, 2581403, 76623522, 1617299079, 49463993875, 952905453423, 59000021366675, 1198427462876421, 54128102218676115, 2321105129608323165, 117387839988330848902, 3205342976298888473968, 268263812478494295219717

Offset: 1

Seiichi Manyama, Jul 30 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A007778, A350109, A350125, A356239, A356240.

a[n_] := Sum[(k * Floor[n/k])^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)

PARI
```
a(n) = sum(k=1, n, (k*(n\k))^n);
```

a(n) = sum(k=1, n, k^n*sumdiv(k, d, 1-(1-1/d)^n));

a(n) = Sum_{k=1..n} k^n * Sum_{d|k} (1 - (1 - 1/d)^n).

A004217 a(n) = (n^n)^(n^n).

Original entry on oeis.org

1, 1, 256, 443426488243037769948249630619149892803

Offset: 0

Henry Bottomley, Jun 08 2000

The next four terms have 617, 10922, 217833 and 4871823 digits respectively. - Rick L. Shepherd, May 07 2006

Jinyuan Wang, Table of n, a(n) for n = 0..4

(#^#)^(#^#)&/@Range[4] (* Harvey P. Dale, Dec 25 2024 *)

a(n) = A000312(A000312(n)) = n^A007778(n) = A002488(n)^n = A002489(n)^A000169(n).

a(0) prepended by Jinyuan Wang, Jan 17 2025

A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

Original entry on oeis.org

0, 1, 12, 153, 2272, 39225, 776736, 17398969, 435538944, 12058401393, 366021568000, 12090393761721, 431832459644928, 16585599200808937, 681703972229640192, 29858718555221585625, 1388451967046195347456, 68316647610168842824161, 3546179063131198669848576, 193670918442059606406896473

Offset: 0

N. J. A. Sloane

This formula is given as a solution to Exercise 1.15a in the Harary and Palmer reference on page 30. However, the formula may not be correct and could be a misprint for Sum_{k=2..n} n! * n^(n-k-1) / (n-k)! which is a formula for A000435(n). - Andrew Howroyd, Feb 06 2024
It appears that a(n) * n^-(n+1) is the mean position of the first duplicate in sequences of n elements randomly drawn with replacement. - Brian P Hawkins, Jan 06 2024
Total count over all mappings from [n] to [n] of tail length plus cycle size of all nodes, where mappings are sets of cycles of trees and tail length is the distance to the cycle that eventually traps the iterates of a node of the mapping; cycle size is the size of that cycle. Alternatively, number of elements on the trajectory of iterates of a node until a repeat is seen, summed over all nodes and mappings. - Marko Riedel, Jul 20 2024

			Example: Consider the map [1,2,3,4] -> [2,3,4,4]. The trajectory of node one is [1,2,3,4]. Hence the tail length is three and the cycle size is one, a fixed point.

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Exercise 1.15a.
P. Flajolet and A. Odlyzko, Random Mapping Statistics, INRIA RR 1114.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marko Riedel, Pusieux series of the tree function and random mapping statistics.

Cf. A000435, A001373, A007778, A262970.

a := proc(n) local k; add(n!*n^(n-k+1)/(n-k)!, k=0..n); end;
# Alternative, e.g.f.:
T := -LambertW(-x): egf := (T + T^2)/(1 - T)^4: ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n = 0..19);  # Peter Luschny, Jul 20 2024

Table[Sum[n!*n^(n-k+1)/(n-k)!, {k, 1, n}], {n, 0, 19}] (* James C. McMahon, Feb 07 2024 *)
a[n_] := n E^n Gamma[n + 1, n] - n^(n + 1);
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Jul 20 2024 *)

a(n) = sum(k=1, n, n! * n^(n-k+1) / (n-k)!) \\ Andrew Howroyd, Jan 06 2024

def a(n):
    total_sum = 0
    for k in range(1, n + 1):
        term = (math.factorial(n) / math.factorial(n - k))*(k**2)*(n**(n - k))
        total_sum += term
    return total_sum
# Brian P Hawkins, Jan 06 2024

a(n) = n^2 * A001865(n). - Gerald McGarvey, Apr 17 2008
a(n) = Sum_{k=1..n} n! * k^2 * n^(n-k) / (n-k)!. - Brian P Hawkins, Jan 06 2024
a(n) = n! * [z^n] (T+T^2)/(1-T)^4 where T is Cayley's tree function T(z) = Sum_{n >= 1} n^(n-1) * z^n/n!. - Marko Riedel, Jul 20 2024
a(n) ~ n^n * ((1/2) * n * sqrt(2 * Pi * n) - (1/3) * n) - Marko Riedel, Jul 20 2024
a(n) = n * e^n * Gamma(n + 1, n) - n^(n + 1) = 2*A262970(n) - A007778(n). - Peter Luschny, Jul 20 2024

Offset corrected by Brian P Hawkins, Jan 06 2024
Name edited by Andrew Howroyd, Feb 06 2024
Offset set to 0 and a(0) = 0 prepended by Marko Riedel, Jul 20 2024

A076482 Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 3, 8, 0, 12, 32, 81, 0, 60, 160, 405, 1024, 0, 360, 960, 2430, 6144, 15625, 0, 2520, 6720, 17010, 43008, 109375, 279936, 0, 20160, 53760, 136080, 344064, 875000, 2239488, 5764801, 0, 181440, 483840, 1224720, 3096576, 7875000, 20155392, 51883209, 134217728

Offset: 1

Henry Bottomley, Oct 14 2002

			Rows start
  0;
  0,   1;
  0,   3,   8;
  0,  12,  32,   81;
  0,  60, 160,  405, 1024;
  0, 360, 960, 2430, 6144, 15625;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Row sums are A076483.

Cf. A001710, A007778, A008279.

Table[n! (k-1)^k/k!,{n,0,10},{k,n}]//Flatten (* Harvey P. Dale, Nov 28 2019 *)

T(n,k) = n*T(n, k-1) = A007778(k-1)*A008279(n,n-k) starting with T(n,n) = (n-1)^n = A007778(n-1).

A108397 Sums of rows of the triangle in A108396.

Original entry on oeis.org

0, 2, 10, 66, 692, 9780, 167982, 3362828, 76695880, 1961316270, 55555555610, 1726135607262, 58359930206844, 2132745542253872, 83767436069591302, 3518790190560477240, 157412216095654840592, 7471013615160978901626

Offset: 0

Reinhard Zumkeller, Jun 02 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A007778, A062970.

a108397 0 = 0
a108397 1 = 2
a108397 n = n * (n^(n+1) + n^2 - 2) `div` (2 * (n-1))
-- Reinhard Zumkeller, Mar 31 2015

a(n) = n*(n^(n+1) + n^2 - 2) / (2*(n-1)) for n>1.

A350202 Number T(n,k) of nodes in the k-th connected component of all endofunctions on [n] when components are ordered by increasing size; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 7, 1, 61, 19, 1, 709, 277, 37, 1, 9911, 4841, 811, 61, 1, 167111, 91151, 19706, 1876, 91, 1, 3237921, 1976570, 486214, 60229, 3739, 127, 1, 71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1, 1780353439, 1257567127, 380291461, 62248939, 5971291, 340729, 11197, 217, 1

Offset: 1

Alois P. Heinz, Dec 19 2021

			Triangle T(n,k) begins:
         1;
         7,        1;
        61,       19,        1;
       709,      277,       37,       1;
      9911,     4841,      811,      61,      1;
    167111,    91151,    19706,    1876,     91,    1;
   3237921,  1976570,   486214,   60229,   3739,  127,   1;
  71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Column k=1 gives A350157.
Row sums give A007778.
T(n+1,n) gives A003215 for n>=1.
Cf. A001865, A319298, A322383.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(g(i)^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..10);

multinomial[n_, k_List] := n!/Times @@ (k!);
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][g[i]^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
T[n_, k_] := b[n, 1, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

A020955 a(n) = n^(2^n - n - 1).

Original entry on oeis.org

1, 1, 2, 81, 4194304, 1490116119384765625, 226267027688376192080197927193400943822503936, 258086210989349276047917817413172383631691140276099547911280598425927853437317437263620645695945672001

Offset: 0

Yasutoshi Kohmoto

nonn

Number of finite models of natural number on the free class with n members.

			For n=2, a(2)=2, universe class={0,1}.
1 : { }ma 0, {0}ma 1, {0,1}ma 1, {1}ma 0,
2 : { }ma 0, {0}ma 1, {0,1}ma 1, {1}ma 1.

Yasutoshi Kohmoto, The Other Side of Mathematics [broken link?]
Yasutoshi Kohmoto, The Other Side of Mathematics [via Internet Archive Wayback-machine]

Cf. A097547.

a(n) = A097547(n)/A007778(n), n > 0. - R. J. Mathar, Jan 12 2017

a(0)=1 and a(7) corrected by Vincenzo Librandi, Apr 25 2011

A060375 a(n) = (n+2)^(n+3) - n^(n+1).

Original entry on oeis.org

8, 80, 1016, 15544, 278912, 5749176, 133937792, 3481019600, 99865782272, 3134941592320, 106893205379072, 3934237957322568, 155461102352433152, 6564470979327191336, 294992337083795013632, 14056516043712012100384

Offset: 0

Jason Earls, Apr 02 2001

nonn

a(n) is divisible by 8. 2^(3+4k) divides a(4k) and a(4k+2). 8k divides a(4k-1). 4(4k+2) divides a(4k+1). - Alexander Adamchuk, Nov 18 2006

			a(1) = |0^1 - 2^3| = 8, a(2) = |1^2 - 3^4| = 80.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A007778 (n^(n+1)).

Table[(-n^(n+1)+(n+2)^(n+3)),{n,0,18}] (* Alexander Adamchuk, Nov 18 2006 *)
#[[3]]^#[[4]]-#[[1]]^#[[2]]&/@Partition[Range[0,20],4,1] (* Harvey P. Dale, Oct 07 2023 *)

{ for (n=0, 100, write("b060375.txt", n, " ", (n + 2)^(n + 3) - n^(n + 1)); ) } \\ Harry J. Smith, Jul 04 2009

a(n) = A007778(n+2) - A007778(n). - Alexander Adamchuk, Nov 18 2006

More terms from Larry Reeves (larryr(AT)acm.org), Apr 20 2001

Better description from Alexander Adamchuk, Nov 18 2006

cp's OEIS Frontend

A143857 a(n) = n + (n+1)*(n+2)^(n+3).

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A280255 Numbers k such that tau(k^(k+1)) is a prime.

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A356238 a(n) = Sum_{k=1..n} (k * floor(n/k))^n.

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A004217 a(n) = (n^n)^(n^n).

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A036360 a(n) = Sum_{k=1..n} n! * n^(n-k+1) / (n-k)!.

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A076482 Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.

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A108397 Sums of rows of the triangle in A108396.

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