A023037
a(n) = n^0 + n^1 + ... + n^(n-1), or a(n) = (n^n-1)/(n-1) with a(0)=0; a(1)=1.
Original entry on oeis.org
0, 1, 3, 13, 85, 781, 9331, 137257, 2396745, 48427561, 1111111111, 28531167061, 810554586205, 25239592216021, 854769755812155, 31278135027204241, 1229782938247303441, 51702516367896047761, 2314494592664502210319, 109912203092239643840221
Offset: 0
a(3) = 3^0 + 3^1 + 3^2 = 1+3+9 = 13.
- Alois P. Heinz, Table of n, a(n) for n = 0..387 (first 101 terms from T. D. Noe)
- Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
- W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
-
A023037:=n->add(n^i, i=0..n-1): seq(A023037(n), n=0..25); # Wesley Ivan Hurt, May 28 2016
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Join[{0,1},Table[(n^n-1)/(n-1),{n,2,20}]] (* Harvey P. Dale, Aug 01 2014 *)
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a(n) = if(n==1, 1, (n^n-1)/(n-1)); \\ Altug Alkan, Oct 04 2017
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def A023037(n): return (n**n-1)//(n-1) if n>1 else n # Chai Wah Wu, Sep 28 2023
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[lucas_number1(n,n+1,n) for n in range(0, 19)] # Zerinvary Lajos, May 16 2009
Entry improved by Tobias Nipkow (nipkow(AT)in.tum.de).
A214812
Largest prime factor of (p^p-1)/(p-1) where p = prime(n).
Original entry on oeis.org
3, 13, 71, 4733, 1806113, 1803647, 2699538733, 109912203092239643840221, 1920647391913, 549334763, 568972471024107865287021434301977158534824481, 41903425553544839998158239, 5926187589691497537793497756719, 19825223972382274003506149120708429799166030881820329892377241, 194707033016099228267068299180244011637
Offset: 1
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FactorInteger[#][[-1,1]]&/@Table[(p^p-1)/(p-1),{p,Prime[Range[15]]}] (* Harvey P. Dale, Aug 27 2016 *)
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a(n) = my(p=prime(n)); vecmax(factor((p^p-1)/(p-1))[,1]); \\ Daniel Suteu, May 26 2022
A056852
a(n) = (p^p + 1)/(p + 1), where p = prime(n).
Original entry on oeis.org
7, 521, 102943, 23775972551, 21633936185161, 45957792327018709121, 98920982783015679456199, 870019499993663001431459704607, 85589538438707037818727607157700537549449, 533411691585101123706582594658103586126397951, 277766709362573247738903423315679814371773581141321037961
Offset: 2
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a := n -> (ithprime(n)^ithprime(n)+1)/(ithprime(n)+1): # Lorenzo Sauras Altuzarra, Nov 27 2022
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Table[ (Prime[ n ]^Prime[ n ] + 1)/(Prime[ n ] + 1), {n, 2, 11} ]
(#^#+1)/(#+1)&/@Prime[Range[2,20]] (* Harvey P. Dale, Apr 23 2015 *)
A214811
Triangle read by rows: row n lists prime factors of (p^p-1)/(p-1) where p = prime(n).
Original entry on oeis.org
3, 13, 11, 71, 29, 4733, 15797, 1806113, 53, 264031, 1803647, 10949, 1749233, 2699538733, 109912203092239643840221, 461, 1289, 831603031789, 1920647391913, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763, 568972471024107865287021434301977158534824481, 149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239
Offset: 1
Triangle begins:
[3]
[13]
[11, 71]
[29, 4733]
[15797, 1806113]
[53, 264031, 1803647]
[10949, 1749233, 2699538733]
[109912203092239643840221]
[461, 1289, 831603031789, 1920647391913]
[59, 16763, 84449, 2428577, 14111459, 58320973, 549334763]
[568972471024107865287021434301977158534824481]
[149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239]
...
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f:=proc(n) local i,t1,p,B,F;
p:=ithprime(n);
B:=(p^p-1)/(p-1);
F:=ifactors(B)[2];
lprint(n,p,B,F);
t1:=[seq(F[i][1],i=1..nops(F))];
sort(t1);
end;
A351657
Period of the Fibonacci n-step sequence mod n.
Original entry on oeis.org
1, 3, 13, 10, 781, 728, 137257, 36, 273, 212784, 28531167061, 42640
Offset: 1
For n = 4, take the tetranacci sequence (A000078), 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (mod 4), which gives 0, 0, 0, 1, 1, 2, 0, 0, 3, 1, 0, 0, 0, 1, 1, 2, ... This repeats a pattern of length 10, so a(4) = 10.
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from math import lcm
from itertools import count
from sympy import factorint
def f(n,pe): # period of the Fibonacci n-step sequence mod pe
a = b = (0,)*(n-1)+(1%pe,)
s = 1 % pe
for m in count(1):
b, s = b[1:] + (s,), (s+s-b[0]) % pe
if a == b:
return m
def A351657(n): return 1 if n == 1 else lcm(*(f(n,p**e) for p, e in factorint(n).items())) # Chai Wah Wu, Feb 23-27 2022
Original entry on oeis.org
85, 781, 137257, 28531167061
Offset: 1
-
f1[n_]:=Module[{s=0},Do[s+=n^a,{a,0,n-1}];s]; f2[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; Select[Table[f1[n],{n,50}],f2[ # ]&]
A248843
Table read by rows in which row n lists divisors of (p^p-1)/(p-1) where p = prime(n).
Original entry on oeis.org
1, 3, 1, 13, 1, 11, 71, 781, 1, 29, 4733, 137257, 1, 15797, 1806113, 28531167061, 1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021, 1, 10949, 1749233, 2699538733, 19152352117, 29557249587617, 4722122236541789
Offset: 1
Table begins:
[1, 3],
[1, 13],
[1, 11, 71, 781],
[1, 29, 4733, 137257],
[1, 15797, 1806113, 28531167061],
[1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021],
...
A354226
a(n) is the number of distinct prime factors of (p^p - 1)/(p - 1) where p = prime(n).
Original entry on oeis.org
1, 1, 2, 2, 2, 3, 3, 1, 4, 7, 1, 7, 5, 3, 3, 5, 3, 4, 6, 4, 10, 5, 4, 6, 6, 9, 5, 4, 5, 8, 6, 4, 11
Offset: 1
a(3)=2, since (5^5 - 1)/(5 - 1) = 11*71.
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a(n) = my(p=prime(n)); omega((p^p-1)/(p-1)); \\ Michel Marcus, May 22 2022
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from sympy import factorint, prime
def a(n): p = prime(n); return len(factorint((p**p-1)//(p-1)))
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, May 23 2022
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