A123856 Primes p that divide A123855(p-1).
2, 3, 5, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, 211, 223, 227, 229, 269, 317, 337, 349, 353, 379, 383, 389, 401, 421, 439, 449, 457, 463, 479, 521, 523, 541, 547, 563, 569, 571, 587, 599, 613, 617, 631, 643
Offset: 1
Keywords
A123857 Composite numbers m that divide A123855(m-1) = Sum_{i=1..m-1} Sum_{j=1..m-1} prime(i)^j.
4, 8, 16, 32, 38, 64, 128, 205, 256, 316, 512, 736, 1024, 2048, 3776, 4096, 4916, 5888, 7736, 8192, 11138, 16384, 22287, 23308, 23924, 32768, 39538, 62336, 65536, 71936
Offset: 1
Comments
Programs
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Mathematica
Do[f=Mod[Sum[Sum[PowerMod[Prime[i],j,n],{i,1,n-1}],{j,1,n-1}],n];If[f==0&&!PrimeQ[n],Print[n]],{n,2,512}]
Extensions
More terms from Max Alekseyev, Sep 13 2009
A124271 a(n) = Sum_{i=1..n} (prime(i)^n - 1)/(prime(i) - 1).
1, 7, 51, 611, 19839, 603331, 32981935, 1469991559, 108336139407, 17389027481287, 1334783150250945, 222909199163881075, 31099653342061054699, 2994181661163361882651, 387134597481460117602345, 92092112661138292186297999, 26679920606217066273305101055
Offset: 1
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
Crossrefs
Programs
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Magma
[&+[(NthPrime(k)^n - 1) div (NthPrime(k) - 1): k in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Oct 21 2018
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Mathematica
Table[Sum[(Prime[i]^n-1)/(Prime[i]-1),{i,1,n}],{n,1,20}] -
PARI
a(n) = sum(i=1, n, (prime(i)^n - 1)/(prime(i) - 1)) \\ Jianing Song, Oct 20 2018
A124239 a(n) = Sum_{k=1..n} Sum_{m=1..n} (2*k - 1)^m.
1, 14, 197, 3704, 90309, 2704470, 95856025, 3921108576, 181756280697, 9413656622446, 538727822713277, 33757715581666296, 2298714540642445405, 169016703698449309846, 13345320616706684277361, 1126219424250538393789824, 101160070702700567996590513, 9636001314414804672487492878
Offset: 1
Keywords
Comments
a(3) = 197 and a(11) = 538727822713277 are primes.
p divides a(p+1) for primes p > 3.
a(2*k-1) is odd. a(2*k) is even. a(2^k) is divisible by 2^(2*k - 1) for k > 0.
Numbers n such that a(n) is divisible by n are listed in A124240.
Links
- T. D. Noe, Table of n, a(n) for n = 1..100
Programs
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Mathematica
Table[Sum[(2k-1)^m,{k,1,n},{m,1,n}],{n,1,20}] -
PARI
a(n) = sum(k=1, n, sum(m=1, n, (2*k - 1)^m)); \\ Michel Marcus, Apr 24 2022
Formula
a(n) = Sum_{k=1..n} Sum_{m=1..n} (2*k - 1)^m.
a(n) = n + Sum_{k=2..n} (2*k - 1)*((2*k - 1)^n - 1)/(2*(k - 1)).
A124275 Terms of A123856 that are not terms of A124273.
2, 5, 181
Offset: 1
Comments
Primes that divide A123855(p-1) = Sum_{j=1..p-1} Sum_{i=1..p-1} prime(i)^j but not A124271(p) = Sum_{i=1..p} (prime(i)^p - 1)/(prime(i) - 1).
The next term if it exists is greater than 1000.
The next term A124275(4), if it exists, is larger than 25000. (Checked by calculating sequences A123856 and A124273 to 1200 terms.) - M. F. Hasler, Nov 10 2006
Examples
A123856(n) begins {2, 3, 5, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, ...}.
A124273(n) begins {3, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 197, ...}.
Thus a(1) = 2, a(2) = 5, a(3) = 181.
Programs
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Maple
A124275:=proc(n) option remember; local i1,i2; i1:=1:i2:=1: # find index i1 such that A123856[i1] is > A124275[n-1]; # then continue until A124273 "jumped over" the term A123856[i1] # while n>1 and A123856(i1) <= procname(n-1) or A123856(i1) = A124273(i2) do i1:=i1+1: # find index i2 such that A124273[i2] is >= A123856(i1) # while A124273(i2) < A123856(i1) do i2:=i2+1: od: od: A123856(i1) end; # M. F. Hasler, Nov 10 2006
A124238 Terms of A123857 that are not powers of 2.
38, 205, 316, 736, 3776, 4916, 5888, 7736, 11138, 22287, 23308, 23924, 39538, 62336, 71936
Offset: 1
Comments
Extensions
More terms from Max Alekseyev, Sep 13 2009
Comments
Links
Crossrefs
Programs
Maple
Mathematica
fQ[p_] := Mod[ Sum[ PowerMod[ Prime@ i, j, p], {j, p - 1}, {i, p - 1}], p] == 0; Select[ Prime@ Range@ 117, fQ] (* Robert G. Wilson v, Jun 10 2011 *)