A051442 -id:A051442 - cp's OEIS frontend

A331546 a(n) = |{k^(k+1)+(k+1)^k (mod prime(n)): k = 0..prime(n)-1}|.

1, 3, 5, 6, 7, 9, 11, 15, 15, 17, 19, 24, 25, 28, 28, 34, 39, 38, 41, 50, 43, 48, 55, 56, 60, 62, 70, 68, 70, 76, 76, 83, 83, 78, 88, 106, 95, 98, 105, 110, 117, 106, 114, 126, 114, 129, 138, 139, 143, 148, 146, 141, 152, 159, 164, 160, 170, 171, 176, 182, 184, 191, 192, 190, 193, 194, 216, 215, 215, 217

Offset: 1

Zhi-Wei Sun, Jan 19 2020

nonn

Conjecture: |{k^(k+1)+(k+1)^k (mod p): k = 0..p-1}| = (1-1/e)*p + O(p^(1/2)), where p denotes a prime.

			a(2) = 3 since {0^1+1^0, 1^2+2^1, 2^3+3^2} = {1, 3, 17} is a complete system of residues modulo the second prime 3.

Robert Israel, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Is it true that |{k^(k+1)+(k+1)^k (mod p): k = 0..p-1}| = (1-e^(-1))*p + O(sqrt(p))? Question 350784 at MathOverflow, Jan. 20, 2020.

Cf. A000040, A001113, A051442.

f:= proc(p) local S,k;
 nops({seq(k &^ (k+1) + (k+1) &^ k mod p, k=0..p-1)})
end proc:
seq(f(ithprime(i)),i=1..100); # Robert Israel, Jan 22 2020

p[n_]:=p[n]=Prime[n];
a[n_]:=a[n]=Length[Union[Table[Mod[PowerMod[k,k+1,p[n]]+PowerMod[k+1,k,p[n]],p[n]],{k,0,p[n]-1}]]];
Table[a[n],{n,1,70}]

A097501 p^q + q^p for twin primes p and q.

Original entry on oeis.org

368, 94932, 36314872537968, 244552822542936127033092, 2177185942561672462146321298650240665136431700, 2246585380039521951243337580678537047744572047581514711375688196554564

Offset: 2

Cino Hilliard, Aug 25 2004

nonn

Except for the first term, 6 divides a(n). Let p = 3k+2 for odd k since k even implies p even, a contradiction. Then p = 6m + 5 and q = 6m+7 = 6m1 + 1. So p^q+q^p = (6m+5)^(6m1+1) + (6m1+1)^(6m+5) = 6H + 5^odd + 1^odd. Now 5 = (6-1) and (6-1)^odd + 1 = 6G -1 + 1 = 6G as stated. Are 3 and 17 the only primes in A051442(n)?

			Consider the second twin prime pair (5,7). 5^7 + 7^5 = 94932, the 2nd entry.

Cf. A051442.

lst={}; Do[p=Prime[n]; If[PrimeQ[q=p+2],a=(p^q+q^p); AppendTo[lst,a]],{n,2*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
#[[1]]^#[[2]]+#[[2]]^#[[1]]&/@Select[Partition[Prime[Range[20]],2,1],#[[2]] - #[[1]]==2&] (* Harvey P. Dale, Sep 07 2019 *)

f(n) = for(x=1,n,p=prime(x);q=prime(x+1);if(q-p==2,v=p^q+q^p;print1(v",")))

cp's OEIS Frontend

A331546 a(n) = |{k^(k+1)+(k+1)^k (mod prime(n)): k = 0..prime(n)-1}|.

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