A066068 -id:A066068 - cp's OEIS frontend

A372228 a(n) is the largest prime factor of n^n + n.

2, 3, 5, 13, 313, 101, 181, 5419, 21523361, 52579, 212601841, 57154490053, 815702161, 100621, 4454215139669, 4562284561, 52548582913, 1895634885375961, 211573, 2272727294381, 415710882920521, 9299179, 1853387306082786629, 22496867303759173834520497

Offset: 1

Tyler Busby, Apr 23 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A066068, A006530, A006486, A007571, A372229.

Table[f = FactorInteger[n^n + n]; f[[Length[f]]][[1]], {n, 1, 25}] (* Vaclav Kotesovec, Apr 26 2024 *)

from sympy import primefactors
def A372228(n): return max(max(primefactors(n),default=1),max(primefactors(n**(n-1)+1))) # Chai Wah Wu, Apr 27 2024

a(n) = A006530(A066068(n)).

A372546 Number of distinct prime factors of n^n+n.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 5, 4, 3, 7, 4, 4, 4, 8, 6, 5, 5, 6, 10, 6, 6, 10, 6, 5, 6, 8, 8, 11, 6, 7, 11, 7, 7, 13, 7, 9, 8, 7, 5, 10, 7, 7, 12, 7, 9, 18, 6, 7, 10, 10, 11, 11, 10, 9, 14, 12, 12, 11, 7, 9, 13, 6, 7, 16, 5, 14, 10, 7, 7, 15, 11, 7, 13, 7, 8, 16, 9, 13

Offset: 1

Tyler Busby, May 06 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A001221, A066068, A372599, A372228, A344870, A344869, A377671.

a[n_] := PrimeNu[n^n + n]; Array[a, 40] (* Amiram Eldar, Oct 29 2024 *)

PARI
```
a(n) = omega(n^n+n);
```

from sympy.ntheory.factor_ import primenu
def A372546(n): return primenu(n*(n**(n-1)+1)) # Chai Wah Wu, May 07 2024

a(n) = A001221(A066068(n)).

A377671 Number of prime factors of n^n+n (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 4, 7, 4, 5, 4, 8, 6, 8, 5, 7, 11, 7, 6, 10, 6, 8, 7, 10, 10, 12, 6, 7, 11, 11, 7, 13, 7, 11, 8, 7, 5, 12, 7, 7, 13, 9, 10, 18, 6, 11, 11, 11, 11, 12, 10, 11, 14, 14, 12, 11, 7, 10, 13, 7, 8, 21, 5, 14, 10, 8, 7, 15, 11, 10, 13, 8, 9, 17

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A001222, A066068, A085723, A372228, A372546, A377672, A377673, A377674.

seq(numtheory:-bigomega(n^n+n),n=1..76); # Robert Israel, Nov 03 2024

a[n_] := PrimeOmega[n^n + n]; Array[a, 45] (* Amiram Eldar, Nov 04 2024 *)

PARI
```
a(n) = bigomega(n^n+n);
```

a(n) = A001222(A066068(n)).

A231712 a(n) = n^n + n - 1.

Original entry on oeis.org

0, 1, 5, 29, 259, 3129, 46661, 823549, 16777223, 387420497, 10000000009, 285311670621, 8916100448267, 302875106592265, 11112006825558029, 437893890380859389, 18446744073709551631, 827240261886336764193, 39346408075296537575441, 1978419655660313589123997

Offset: 0

Jaroslav Krizek, Nov 12 2013

nonn

Supersequence of A187605 (primes of the form n^n + n - 1).
Numbers n such that a(n) = prime: 2, 3, 19, 30, 535, 1551, ..., another term > 2300 (see A058912 and A187605).
Also generalization of the problem: "What is the minimum length of a text consisting only of the first n letters of the alphabet and containing all possible n-tuples (no blanks)?" (see Puzzleup link). Example for n = 3, length of text a(3) = 29: AAABAACABBABCACBACCBBBCBCCCAA, all triples (AAA, AAB, ..., CCC) occurring exactly once. - Jörg Zurkirchen, Sep 06 2014

Jaroslav Krizek, Table of n, a(n) for n = 0..50
PuzzleUp, Problem No. 04: Four Letters, August 20, 2014.

Cf. A187605, A058912, A066279, A066068.

[n^n+n-1 : n in [0..20]]; // Wesley Ivan Hurt, Sep 23 2014

A231712:=n->n^n+n-1: seq(A231712(n), n=0..20); # Wesley Ivan Hurt, Sep 23 2014

Mathematica
```
Join[{0}, Table[n^n + n - 1, {n, 18}]]
```

a(n)=n^n+n-1 \\ Edward Jiang, Sep 06 2014

a(n) = A066279(n) - 2 = A066068(n) - 1.

E.g.f.: 1/(1 + LambertW(-x)) + (x-1)*exp(x). - Alois P. Heinz, Jun 15 2018

A377672 a(n) is the number of divisors of n^n + n.

Original entry on oeis.org

2, 4, 8, 12, 8, 32, 48, 48, 12, 128, 16, 24, 16, 256, 64, 80, 32, 96, 1536, 96, 64, 1024, 64, 96, 96, 512, 512, 3072, 64, 128, 2048, 384, 128, 8192, 128, 1152, 256, 128, 32, 2048, 128, 128, 6144, 288, 768, 262144, 64, 480, 1536, 1536, 2048, 3072, 1024, 1024

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A000005, A066068, A344859, A372228, A372546, A377671, A377673, A377674.

a[n_] := DivisorSigma[0, n^n + n]; Array[a, 45] (* Amiram Eldar, Nov 04 2024 *)

PARI
```
a(n) = numdiv(n^n+n);
```

a(n) = A000005(A066068(n)).

A377673 a(n) is the sum of the divisors of n^n + n.

Original entry on oeis.org

3, 12, 72, 588, 5652, 117504, 1895712, 46503600, 839411118, 25440307200, 474527311344, 22404560101168, 489294047662728, 30902868417576960, 1096805935992340800, 38000593697802058224, 1318965178069293272496, 90596485743469636057920, 3578317312662511683264000

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A000203, A066068, A366820, A372228, A372546, A377671, A377672, A377674.

a[n_] := DivisorSigma[1, n^n + n]; Array[a, 20] (* Amiram Eldar, Nov 04 2024 *)

PARI
```
a(n) = sigma(n^n+n);
```

a(n) = A000203(A066068(n)).

A377674 a(n) = phi(n^n + n) where phi is the Euler totient function.

Original entry on oeis.org

1, 2, 8, 96, 1248, 12000, 259200, 5461344, 129140160, 2725643520, 127561104000, 2743415522496, 139778722137600, 2504616361228800, 111747349423990784, 8644660582219776000, 387774574486565683200, 12306643656809728412160, 816897235219321957908480

Offset: 1

Sean A. Irvine, Nov 03 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..116

Cf. A000010, A066068, A366822, A372228, A372546, A377671, A377672, A377673.

a[n_] := EulerPhi[n^n + n]; Array[a, 20] (* Amiram Eldar, Nov 04 2024 *)

PARI
```
a(n) = eulerphi(n^n+n);
```

a(n) = A000010(A066068(n)).

A117694 a(n) = (n^n + n)/2.

Original entry on oeis.org

1, 3, 15, 130, 1565, 23331, 411775, 8388612, 193710249, 5000000005, 142655835311, 4458050224134, 151437553296133, 5556003412779015, 218946945190429695, 9223372036854775816, 413620130943168382097, 19673204037648268787721

Offset: 1

Alexander Adamchuk, Apr 12 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..380

Cf. A066068.

[(n+n^n)/2: n in [1..30]]; // G. C. Greubel, Jul 23 2023

Mathematica
```
Table[n^n+n, {n,30}]/2
```

[(n+n^n)/2 for n in range(1,31)] # G. C. Greubel, Jul 23 2023

a(n) = (n^n + n)/2.
a(n) = A066068(n)/2, for n > 0.
E.g.f.: (1/2)*(x*exp(x) - LambertW(-x)/(1 + LambertW(-x))). - G. C. Greubel, Jul 23 2023

A342707 T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 1, 3, 3, 1, 0, 1, 2, 4, 4, 4, 1, 0, 2, 2, 5, 27, 5, 5, 1, 0, 0, 3, 6, 28, 256, 6, 6, 1, 0, 1, 1, 7, 30, 257, 3125, 7, 7, 1, 0, 0, 2, 8, 31, 260, 3126, 46656, 8, 8, 1, 0, 1, 2, 9, 81, 261, 3130, 46657, 823543, 9, 9, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

			Array T(n, k) begins:
  n\k|  0  1   2   3     4      5       6        7          8           9
  ---+-------------------------------------------------------------------
    0|  0  0   0   0     0      0       0        0          0           0
    1|  1  1   1   1     1      1       1        1          1           1
    2|  0  1   2   3     4      5       6        7          8           9
    3|  1  2   3   4     5      6       7        8          9          10
    4|  1  1   4  27   256   3125   46656   823543   16777216   387420489
    5|  2  2   5  28   257   3126   46657   823544   16777217   387420490
    6|  1  2   6  30   260   3130   46662   823550   16777224   387420498
    7|  2  3   7  31   261   3131   46663   823551   16777225   387420499
    8|  0  1   8  81  1024  15625  279936  5764801  134217728  3486784401
    9|  1  2   9  82  1025  15626  279937  5764802  134217729  3486784402
   10|  0  2  10  84  1028  15630  279942  5764808  134217736  3486784410

Wikipedia, Hereditary base-n notation

See A341907 for a similar sequence.

Cf. A000120, A000312, A002488, A014566, A066068, A066279, A222112, A343255, A345021.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^T(e,k)); v }

T(n, n) = A343255(n).
T(n, 0) = A345021(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A222112(n-1).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^k = A000312(k).
T(5, k) = k^k + 1 = A014566(k).
T(6, k) = k^k + k = A066068(k).
T(7, k) = k^k + k + 1 = A066279(k).
T(16, k) = k^k^k = A002488(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A302989 a(n) = n^n + n*n + n.

Original entry on oeis.org

1, 3, 10, 39, 276, 3155, 46698, 823599, 16777288, 387420579, 10000000110, 285311670743, 8916100448412, 302875106592435, 11112006825558226, 437893890380859615, 18446744073709551888, 827240261886336764483, 39346408075296537575766, 1978419655660313589124359

Offset: 0

Alex Ratushnyak, Apr 17 2018

			a(3) = 3^3 + 3*3 + 3 = 27 + 9 + 3 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n=1..100 from Muniru A Asiru)

Cf. A000007, A000290, A000312, A001477, A066068, A066141, A214647.

List([1..30],n->n^n+n*n+n); # Muniru A Asiru, May 27 2018

seq(n^n+n*n+n,n=0..30); # Muniru A Asiru, May 27 2018

Join[{1},Table[n^n+n^2+n,{n,20}]] (* Harvey P. Dale, Jun 13 2020 *)

a(n) = A066141(n) * n + A000007(n).
From Alois P. Heinz, May 28 2018: (Start)
E.g.f.: 1/(1+LambertW(-x)) + exp(x)*(2*x+x^2).
a(n) = A001477(n) + A000290(n) + A000312(n). (End)

a(0)=1 prepended by Alois P. Heinz, May 28 2018

cp's OEIS Frontend

A372228 a(n) is the largest prime factor of n^n + n.

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A372546 Number of distinct prime factors of n^n+n.

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A377671 Number of prime factors of n^n+n (counted with multiplicity).

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A231712 a(n) = n^n + n - 1.

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Magma

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A377672 a(n) is the number of divisors of n^n + n.

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A377673 a(n) is the sum of the divisors of n^n + n.

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Mathematica

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A377674 a(n) = phi(n^n + n) where phi is the Euler totient function.

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A117694 a(n) = (n^n + n)/2.

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