A182938 -id:A182938 - cp's OEIS frontend

A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 162, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 324, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400

Offset: 1

José María Grau Ribas, Jan 21 2012

nonn

k is a term if and only if k = Product_{i=1..t} p_i^e_i with e_i > p_i for some i.
A182938(a(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - 1/p^(p+1) = 0.13585792767780221591... - Amiram Eldar, Nov 24 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. Hampel, The length of the shortest period of rests of numbers n^n, Ann. Polon. Math. 1 (1955), 360-366.

Cf. A027748, A124010, A008590 (subsequence), A185358, A207481 (complement).

a185359 n = a185359_list !! (n-1)
a185359_list = [x | x <- [1..], or $ zipWith (<)
                    (a027748_row x) (map toInteger $ a124010_row x)]
-- Reinhard Zumkeller, Feb 18 2012

j[p_,e_]:=e>p;j[n_]:={False}==Union@Module[{fa=FactorInteger[n]},Table[j[fa[[i,1]],fa[[i,2]]],{i,1,Length[fa]}]];Select[Range[1000],!j[#]&]

A104126 a(n) = prime(n)^(prime(n)+1).

Original entry on oeis.org

8, 81, 15625, 5764801, 3138428376721, 3937376385699289, 14063084452067724991009, 37589973457545958193355601, 480250763996501976790165756943041

Offset: 1

Cino Hilliard, Mar 06 2005

Sum of reciprocals rapidly converges to 0.1374098524791901212366977116..

A182938(a(n)) = 0. [Reinhard Zumkeller, Feb 18 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..75

Cf. A051674, A000040.

a104126 n = p ^ (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Feb 18 2012

#^(#+1)&/@Prime[Range[10]] (* Harvey P. Dale, Dec 12 2021 *)

ptopp1(n) = { local(x,y,z,sr=0); forprime(x=1,n, y=x^(x+1); z=(x+1)^x; sr+=1./y; print1(y","); ); print(); print(sr) }

Offset corrected by Reinhard Zumkeller, Feb 18 2012

A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 6, 10, 11, 9, 13, 14, 15, 5, 17, 12, 19, 15, 21, 22, 23, 12, 15, 26, 10, 21, 29, 30, 31, 6, 33, 34, 35, 18, 37, 38, 39, 20, 41, 42, 43, 33, 30, 46, 47, 15, 28, 30, 51, 39, 53, 20, 55, 28, 57, 58, 59, 45, 61, 62, 42, 7, 65, 66, 67, 51, 69, 70, 71, 24, 73, 74, 45

Offset: 1

Ilya Gutkovskiy, Oct 26 2019

Number of ways to factor n into 2 kinds of 2, 3 kinds of 3, 5 kinds of 5, ... , p kinds of p.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A008480, A050367, A059481, A135323, A182938.

a:= n-> mul(binomial(i[1]+i[2]-1, i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 26 2019

a[n_] := Times @@ (Binomial[#[[1]] + #[[2]] - 1, #[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^p)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

If n = Product (p_j^k_j) then a(n) = Product (binomial(p_j + k_j - 1, k_j)).

Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.40373... - Vaclav Kotesovec, Mar 28 2025

A329445 Dirichlet inverse of A328745.

Original entry on oeis.org

1, -2, -3, 1, -5, 6, -7, 0, 3, 10, -11, -3, -13, 14, 15, 0, -17, -6, -19, -5, 21, 22, -23, 0, 10, 26, -1, -7, -29, -30, -31, 0, 33, 34, 35, 3, -37, 38, 39, 0, -41, -42, -43, -11, -15, 46, -47, 0, 21, -20, 51, -13, -53, 2, 55, 0, 57, 58, -59, 15, -61, 62, -21, 0, 65, -66, -67, -17

Offset: 1

Werner Schulte, Nov 13 2019

Signed version of A182938.

Cf. A008836, A182938, A328745.

from math import prod, comb
from sympy import factorint
def A329445(n): return prod(-comb(p,e) if e&1 else comb(p,e) for p,e in factorint(n).items()) # Chai Wah Wu, Dec 23 2022

Multiplicative with a(p^e) = (-1)^e*binomial(p,e) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n)/n^s = Product_{p prime} (1-p^(-s))^p.
a(n) = A182938(n) * A008836(n) for n > 0.

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A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.

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A104126 a(n) = prime(n)^(prime(n)+1).

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A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.

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A329445 Dirichlet inverse of A328745.

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