A172019 -id:A172019 - cp's OEIS frontend

A152680 a(n) = 4*A005098(n) = A002144(n) - 1.

4, 12, 16, 28, 36, 40, 52, 60, 72, 88, 96, 100, 108, 112, 136, 148, 156, 172, 180, 192, 196, 228, 232, 240, 256, 268, 276, 280, 292, 312, 316, 336, 348, 352, 372, 388, 396, 400, 408, 420, 432, 448, 456, 460, 508, 520, 540, 556, 568, 576, 592, 600, 612, 616

Offset: 1

Artur Jasinski, Dec 10 2008

nonn

If we take the 4 numbers 1, A002314(n), A152676(n), A152680(n) then the multiplication table modulo A002144(n) is isomorphic with the Latin square
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
and isomorphic with the multiplication table of {1,I,-I,-1} where I is sqrt(-1), A152680(n) is isomorphic with -1, A002314(n) with I or -I and A152676(n) vice versa -I or I.
1, A002314(n), A152676(n), A152680(n) are subfields of the Galois Field [A002144(n)].
Numbers n such that A172019(n) + 1 = primes - 1. - Giovanni Teofilatto, Feb 02 2010

Cf. A002314, A152676, A152680.

aa = {}; Do[If[Mod[Prime[n], 4] == 1, AppendTo[aa, Prime[n] - 1]], {n, 1, 200}]; aa

A353768 a(n) = phi(n) mod 4; Euler totient function reduced modulo 4.

Original entry on oeis.org

1, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2

Offset: 1

Antti Karttunen, May 15 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A010873, A066499 (positions of 2's), A172019 (of 0's).

Cf. also A074942, A261872, A084300, and also A105824.

PARI
```
A353768(n) = (eulerphi(n)%4);
```

a(n) = A010873(A000010(n)).

A097987 Numbers k such that 4 does not divide phi(k), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 19, 22, 23, 27, 31, 38, 43, 46, 47, 49, 54, 59, 62, 67, 71, 79, 81, 83, 86, 94, 98, 103, 107, 118, 121, 127, 131, 134, 139, 142, 151, 158, 162, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 242, 243, 251, 254, 262, 263, 271

Offset: 1

Lekraj Beedassy, Sep 07 2004

nonn

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Essentially the same as A066499.
Cf. A000010.
Complement of A172019.

Select[Range@275, ! Divisible[EulerPhi[#], 4] &] (* Ivan Neretin, Aug 24 2016 *)

is(n)=my(o=valuation(n,2),p); (o<2 && isprimepower(n>>o,&p) && p%4>1) || n<5 \\ Charles R Greathouse IV, Feb 21 2013

a(n)=1, 2, 4, p^k, 2*p^k, with prime p == 3 (mod 4).

Corrected and extended by Vladeta Jovovic, Sep 08 2004

A358043 Numbers k such that phi(k) is a multiple of 8.

Original entry on oeis.org

15, 16, 17, 20, 24, 30, 32, 34, 35, 39, 40, 41, 45, 48, 51, 52, 55, 56, 60, 64, 65, 68, 70, 72, 73, 75, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 95, 96, 97, 100, 102, 104, 105, 110, 111, 112, 113, 115, 116, 117, 119, 120, 123, 128, 130, 132, 135, 136, 137, 140, 143

Offset: 1

Darío Clavijo, Oct 26 2022

nonn

Cf. A000010 (phi), A053574 (its 2-adic valuation), A037074 (a subsequence).

Totient multiples: A066498 (3), A172019 (4), A066500 (5), A066502 (7), A332512 (12).

Select[Range[150], Divisible[EulerPhi[#], 8] &] (* Amiram Eldar, Oct 27 2022 *)

isok(k) = Mod(eulerphi(k), 8) == 0; \\ Michel Marcus, Oct 27 2022

from sympy.ntheory import totient
def isok(n): return totient(n) % 8 == 0

A000010(a(n)) == 0 (mod 8).

cp's OEIS Frontend

A152680 a(n) = 4*A005098(n) = A002144(n) - 1.

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A353768 a(n) = phi(n) mod 4; Euler totient function reduced modulo 4.

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A097987 Numbers k such that 4 does not divide phi(k), where phi is Euler's totient function (A000010).

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A358043 Numbers k such that phi(k) is a multiple of 8.

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Author

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Crossrefs

Programs

Mathematica

PARI

Python

Formula