A110567 -id:A110567 - cp's OEIS frontend

A123570 Duplicate of A110567.

1, 2, 9, 82, 1025, 15626, 279937, 5764802, 134217729, 3486784402, 100000000001, 3138428376722, 106993205379073, 3937376385699290, 155568095557812225, 6568408355712890626, 295147905179352825857, 14063084452067724991010

Offset: 0

dead

A342471 a(n) = Sum_{d|n} phi(d)^n.

Original entry on oeis.org

1, 2, 9, 18, 1025, 130, 279937, 65794, 10078209, 2097154, 100000000001, 16789506, 106993205379073, 156728328194, 35185445863425, 281479271743490, 295147905179352825857, 203119913861122, 708235345355337676357633, 1152923703631151106

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A029939, A110567, A309369, A338997, A342470.

a[n_] := DivisorSum[n, EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(d)^n);
```

a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-1));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-(eulerphi(k)*x)^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n-1).
G.f.: Sum_{k>=1} (phi(k)*x)^k/(1 - (phi(k)*x)^k).
If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^n/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A342473 a(n) = Sum_{d|n} phi(d)^d.

Original entry on oeis.org

1, 2, 9, 18, 1025, 74, 279937, 65554, 10077705, 1049602, 100000000001, 16777306, 106993205379073, 78364444034, 35184372089865, 281474976776210, 295147905179352825857, 101559966746186, 708235345355337676357633, 1152921504607896594, 46005119909369701746057, 10000000000100000000002

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A029939, A110567, A309369, A338997, A342470, A342471, A342485.

a[n_] := DivisorSum[n, EulerPhi[#]^# &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(d)^d);
```

a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n/gcd(k, n)-1));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-x^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n/gcd(k, n) - 1).
G.f.: Sum_{k>=1} (phi(k) * x)^k/(1 - x^k).
If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^gcd(n,k)/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A081215 a(n) = (n^(n+1)+(-1)^n)/(n+1)^2.

Original entry on oeis.org

1, 0, 1, 5, 41, 434, 5713, 90075, 1657009, 34867844, 826446281, 21794641505, 633095889817, 20088655029078, 691413758034721, 25657845139503479, 1021273028302258913, 43404581642184336392, 1961870762757168078553

Offset: 0

Vladeta Jovovic, Apr 17 2003

From Mathew Englander, Oct 19 2020: (Start)
The sum of two adjacent terms of the sequence cannot be prime.
In base n, a(n) has n-1 digits, which are (beginning from the left): n-2, 2, n-4, 4, and so on, except that if n is even the rightmost digit is 1 instead of 0. In that case, the other digits form a palindrome with every even digit from 2 to n-2 appearing twice. For example, a(14) in base 14 is c2a486684a2c1. If n is odd, then all digits from 1 to n-1 occur exactly once. For example, a(15) in base 15 is d2b496785a3c1e.
For any positive integer k, any prime p, and any positive integer h such that h*p > 2, a(h*p^k - 2) == (-1)^h * (1 - 2^(h-1)) (mod p). For example, a(7*p^k - 2) == 63 (mod p); a(10*p^k - 2) == -511 (mod p).
Suppose k and m are positive integers. If k is even, then a(k*m) == 1, a(k*m+1) == 0, and a(k*m-1) == -1 (all mod m). If k is odd, then a(k*m) == (-1)^m and a(k*m+1) == ceiling(m/2) (both mod m), while a(k*m-1) == m/2 - 1 for m even, and a(k*m-1) == 1 for m odd (mod m).
For proofs of the above, see the Englander link. (End)

Robert Israel, Table of n, a(n) for n = 0..387
Mathew Englander, Comments on OEIS A081215

Cf. A081209, A110567, A076951, A273319, A193746, and A060073.

seq((j^(j+1)+(-1)^j)/(j+1)^2, j=0..50); # Robert Israel, May 19 2016

Array[(#^(# + 1) + (-1)^#)/(# + 1)^2 &, 19, 0] (* Michael De Vlieger, Nov 13 2020 *)

a(n) = (n^(n+1)+(-1)^n)/(n+1)^2; \\ Michel Marcus, Oct 20 2020

a(n) = (-1)^n + Sum_{k=1..n} (-1)^(k+1)*(n+1)^(n-k)*C(n+1,n+2-k). - Gionata Neri, May 19 2016
E.g.f.: (Ei(1,x) - Ei(1,-LambertW(-x)))/x. - Robert Israel, May 19 2016
For n > 1, a(n) = Sum_{k=1..floor(n/2)} (n^(n-2*k) * (2*k/n + n - 2*k)). - Mathew Englander, Oct 19 2020

A110676 Number of prime factors with multiplicity of 1 + n^(n+1).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 6, 3, 5, 4, 5, 4, 9, 3, 4, 9, 3, 6, 10, 6, 7, 6, 11, 5, 11, 10, 5, 10, 8, 3, 12, 6, 10, 8, 5, 6, 13, 8, 6, 11, 6, 10, 16, 4, 4, 6, 9, 6, 11, 8, 4, 10, 10, 5, 13, 10, 7, 11, 6, 6, 21, 4, 23, 8, 6, 8, 16, 15, 7, 12, 7, 8, 19, 8, 13, 14, 5, 6, 20, 6, 10, 13, 12, 7, 9, 9, 6, 21

Offset: 1

Jonathan Vos Post, Sep 14 2005

As also noticed by T. D. Noe, for odd n: 2 | a(n), for even n: (n+1)^2 | a(n). Coincidentally, a(74) includes 13 multidigit prime factors all of which end with the digit 1. There is no upper limit to this sequence, which rapidly becomes slow to compute. The derived sequences of n such that a(n) = k for any constant k > 2 do not yet appear in the OEIS. For instance, a(n) = 3 for n = 4, 5, 7, 9, 15, 18, 31, ... Is each such derived sequence finite?

			a(1) = 1 because 1+1^2 = 2 is prime (and the only such prime).
a(2) = 2 because 1 + 2^3 = 9 = 3^2 which has (with multiplicity) two prime factors.
a(3) = 2 because 1 + 3^4 = 82 = 2 * 41 (the last such semiprime?).
a(4) = 3 because 1 + 4^5 = 1025 = 5^2 * 41 which has (with multiplicity) 3 prime factors.
a(8) = 6 because 1 + 8^9 = 134217729 = 3^4 * 19 * 87211.
a(14) = 9 because 1 + 14^15 = 155568095557812225 = 3^2 * 5^2 * 61 * 71 * 101 * 811 * 1948981.
a(1000) > 52.

Cf. A001222, A007778, A110567.

a(n) = bigomega(1+(n^(n+1))) \\ Georg Fischer, Jun 21 2024

a(n) = A001222(A110567(n)) = A001222(1 + A007778(n)) = A001222(1 + (n^(n+1))).

Trivially a(n) << n log n. At most n^(n+1) + 1 is of the form 2*3^k. - Charles R Greathouse IV, Jun 24 2024

a(13) and 3 other terms corrected by Georg Fischer, Jun 21 2024

A110874 Number of prime factors of 2 + n^(n+1) counted with multiplicity.

Original entry on oeis.org

1, 2, 1, 5, 2, 2, 4, 5, 2, 5, 4, 4, 5, 3, 1, 4, 5, 3, 4, 6, 3, 8, 4, 5, 4, 4, 2, 6, 3, 6, 5, 5, 5, 6, 6, 8, 6, 6, 4, 5, 4, 6, 4, 5, 3, 8, 4, 3, 5, 5, 5, 7, 7, 11, 4, 5, 4, 13, 4, 6, 2, 5, 2, 6, 6, 5, 8, 9, 5, 9, 4, 7, 4, 4, 5, 7, 6, 7, 6, 9, 4, 9, 5, 8, 5, 8

Offset: 1

Jonathan Vos Post, Sep 18 2005

nonn

Compared with A110676, number of prime factors with multiplicity of 2 + n^(n+1), this seems to have an unlimited number of primes (n = 1, 3, 15, ...) and semiprimes (n = 2, 5, 6, 9, 27, ...). Of course, n even gives n | a(n).

			a(1) = 1 because 2 + 1^2 = 3 is prime (one prime factor).
a(2) = 2 because 2 + 2^3 = 10 = 2 * 5 is semiprime (two prime factors).
a(3) = 1 because 2 + 3^4 = 83 is prime.
a(4) = 5 because 2 + 4^5 = 1026 = 2 * 3^3 * 19 has five prime factors (3 has multiplicity of 3).
a(5) = 2 because 2 + 5^6 = 15627 = 3 * 5209 is semiprime (two prime factors).
a(6) = 2 because 2 + 6^7 = 279938 = 2 * 139969 is semiprime (two prime factors).
a(15) = 1 because 2 + 15^16 = 6568408355712890627 is prime. What is the next prime?

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. A001222, A007778, A110567, A110676.

Table[PrimeOmega[2+n^(n+1)],{n,41}] (* Harvey P. Dale, Nov 08 2020 *)

a(n) = A001222(1 + A110567(n)) = A001222(2 + A007778(n)) = A001222(2 + n^(n+1)).

More terms from Sean A. Irvine, Sep 17 2023

cp's OEIS Frontend

A123570 Duplicate of A110567.

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A342471 a(n) = Sum_{d|n} phi(d)^n.

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A342473 a(n) = Sum_{d|n} phi(d)^d.

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A081215 a(n) = (n^(n+1)+(-1)^n)/(n+1)^2.

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A110676 Number of prime factors with multiplicity of 1 + n^(n+1).

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A110874 Number of prime factors of 2 + n^(n+1) counted with multiplicity.

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