A236386 -id:A236386 - cp's OEIS frontend

A359847 Oblong numbers k for which phi(k) is also an oblong number.

6, 42, 182, 650, 930, 4830, 7482, 9506, 12882, 13572, 16770, 79242, 167690, 181902, 228006, 289982, 380072, 3480090, 5209806, 6872262, 10102862, 16068072, 56002772, 56648202, 59174556, 70299840, 74831150, 123287712, 261517412, 342601590, 356322252, 455459622, 536223492, 1057452842

Offset: 1

Alexandru Petrescu, Jan 15 2023

nonn

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

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

Robert G. Wilson v, Table of n, a(n) for n = 1..269
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A002378, A287472.

Intersection of A002378 and A236386.

lastv:= 1: R:= NULL: count:= 0:
for n from 3 while count < 50 do
  v:= numtheory:-phi(n);
  if issqr(4*v*lastv+1) then
    R:= R, n*(n-1); count:= count+1;
    fi;
  lastv:= v;
od:
R; # Robert Israel, Feb 15 2023

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

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

A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682

Offset: 1

Bernard Schott, Feb 26 2023

nonn

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.

Subsequence of triangular numbers is A287472.

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000217, A055469, A287472.

Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);

Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)

isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023

from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023

cp's OEIS Frontend

A359847 Oblong numbers k for which phi(k) is also an oblong number.

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