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Oblong numbers k for which phi(k) is also an oblong number.
2

%I #33 Feb 17 2023 02:06:08

%S 6,42,182,650,930,4830,7482,9506,12882,13572,16770,79242,167690,

%T 181902,228006,289982,380072,3480090,5209806,6872262,10102862,

%U 16068072,56002772,56648202,59174556,70299840,74831150,123287712,261517412,342601590,356322252,455459622,536223492,1057452842

%N Oblong numbers k for which phi(k) is also an oblong number.

%C Since k and k+1 are relatively prime, the calculation of phi(k)*phi(k+1) is faster than that of phi(k*(k+1)). - _Robert G. Wilson v_, Feb 14 2023

%H Robert G. Wilson v, <a href="/A359847/b359847.txt">Table of n, a(n) for n = 1..269</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.

%e 9506 is a term because 9506 = 97*98 and phi(9506) = 4032 = 63*64.

%p lastv:= 1: R:= NULL: count:= 0:

%p for n from 3 while count < 50 do

%p v:= numtheory:-phi(n);

%p if issqr(4*v*lastv+1) then

%p R:= R, n*(n-1); count:= count+1;

%p fi;

%p lastv:= v;

%p od:

%p R; # _Robert Israel_, Feb 15 2023

%t Select[Table[n*(n + 1), {n, 1, 100000}], IntegerQ @ Sqrt[4*EulerPhi[#] + 1] &] (* _Amiram Eldar_, Jan 15 2023 *)

%t k = pk0 = pk1 = 1; lst = {}; While[k < 10000, If[ IntegerQ@ Sqrt[4*pk0*pk1 + 1], AppendTo[lst, k (k + 1)]]; k++; pk0 = pk1; pk1 = EulerPhi[k + 1]]; lst (* _Robert G. Wilson v_, Feb 14 2023 *)

%o (PARI) for(k=1, 10^5, my(n=k*(k+1), p=eulerphi(n)); if(issquare(4*p+1), print1(n,", ")))

%Y Cf. A000010, A002378, A287472.

%Y Intersection of A002378 and A236386.

%K nonn

%O 1,1

%A _Alexandru Petrescu_, Jan 15 2023