A270140 - OEIS

%I #21 Jul 24 2024 18:12:21

%S 1,2,4,6,8,16,18,20,24,32,42,45,54,64,72,96,100,128,162,216,256,272,

%T 288,294,320,342,352,384,486,500,512,600,648,720,832,850,864,1024,

%U 1120,1125,1152,1320,1350,1458,1512,1536,1600,1620,1806,1944,2048,2058,2500,2592,2688,3321,3456,3645,3872,4096,4176,4225,4374,4608,4624,5120,5256

%N Numbers k such that k/p_i^r_i == -1 (mod p_i) for all i = 1,...,m, where k = p_1^r_1 .... p_m^r_m.

%H Amiram Eldar, <a href="/A270140/b270140.txt">Table of n, a(n) for n = 1..1688</a> (terms up to 10^10)

%H Jose María Grau and Antonio M. Oller-Marcen, <a href="http://arxiv.org/abs/1603.05787">Power sums over commutative and unitary rings</a>, arXiv:1603.05787 [math.NT], 2016.

%e 8000 is a term since 8000 = 2^6 * 5^3 and 8000 == -2^6 (mod 2^7) and 8000 == -5^3 (mod 5^4).

%t fa = FactorInteger; mas[1]=True; mas[n_] := Union@Table[Mod[n + fa[n][[i, 1]]^ fa[n][[i,2]], fa[n][[i, 1]]^(fa[n][[i, 2]] + 1)], {i, Length[fa[n]]}] == {0}; Select[Range[10000], mas ]

%o (PARI) is(k) = {my(f = factor(k)); for(i = 1, #f~, if((k / f[i, 1]^f[i, 2] + 1) % f[i, 1], return(0))); 1;} \\ _Amiram Eldar_, Jul 24 2024

%Y Cf. A274222.

%K nonn

%O 1,2

%A _José María Grau Ribas_, Mar 12 2016