A339875 -id:A339875 - cp's OEIS frontend

A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

85, 561, 1105, 1261, 1285, 2465, 4369, 6601, 8245, 8481, 9061, 9605, 10585, 16405, 16705, 17733, 18721, 19669, 21845, 23001, 28645, 30889, 38165, 42121, 43165, 46657, 54741, 56797, 57205, 62745, 65365, 74593, 78013, 83665, 88561, 91001, 106141, 117181, 124645, 126701, 134521, 136981, 141661, 162401, 171205, 176437

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

From Antti Karttunen, Dec 26 2020: (Start)
Equally, squarefree composite numbers k of the form 4u+1 for which A336466(k) divides k-1. This follows because on squarefree n, A336466(n) = A053575(n).
No common terms with A016105, because 4xy + 2(x+y) + 1 does not divide 4xy + 3(x+y) + 2 for any distinct x, y >= 0 (where 4x+3 and 4y+3 are the two prime factors of Blum integers).
This can also seen by another way: If this sequence contained any Blum integers, then, because A016105 is a subsequence of A339817, we would have found a composite number n satisfying Lehmer's totient problem y * phi(n) = n-1, for some integer y > 1. But Lehmer proved that such solutions should have at least 7 distinct prime factors, while Blum integers have only two.
Moreover, it seems that none of the terms of A167181 may occur here, and a few of A137409 (i.e., of A125667). See A339875 for those terms.
(End)

			85 = 4*21 + 1 = 5*17, thus phi(85) = 4*16 = 64, the odd part of which is A000265(64) = 1, which certainly divides 85-1, therefore 85 is included as a term.
561 = 4*140 + 1 = 3*11*17, thus phi(561) = 2*10*16 = 320, the odd part of which is A000265(320) = 5, which divides 560, therefore 561 is included.

Antti Karttunen, Table of n, a(n) for n = 1..1316
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem

Cf. A000010, A000265, A016105, A053575, A137409, A167181, A336466.
Subsequence of A005117.
Intersection of A091113 and A339880.
Cf. A339875 (a subsequence).
Cf. also comments in A339817.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[4*Range[45000] + 1, CompositeQ[#] && Divisible[# - 1, odd[EulerPhi[#]]] &] (* Amiram Eldar, Feb 17 2021 *)

A000265(n) = (n>>valuation(n, 2));
isA339870(n) = ((n>1)&&!isprime(n)&&(1==(n%4))&&!((n-1)%A000265(eulerphi(n))));

A137409 Numbers that cannot be the value of 'C' in a primitive Pythagorean triple (A < B; A^2 + B^2 = C^2).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 14 2008

nonn

Complement of A008846. - R. J. Mathar, Aug 15 2010
A024362(a(n)) = 0. - Reinhard Zumkeller, Dec 02 2012
Except for the 1st term 1, complement of A004613. - Federico Provvedi, Jan 26 2019
After 1, numbers k for which A065338(k) > 1, i.e., after 1, numbers all of whose prime divisors are not of the form 4u+1. - Antti Karttunen, Dec 26 2020

			3,4,5; number 5 is not in this sequence.
5,12,13; number 13 is not in this sequence.
8,15,17; number 17 is not in this sequence.
7,24,25; number 25 is not in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004613, A008846, A020882, A024362, A065338, A137407.

Subsequences: A125667 (the odd terms), A339875.

import Data.List (elemIndices)
a137409 n = a137409_list !! (n-1)
a137409_list = map (+ 1) $ elemIndices 0 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

okQ[1] = True;
okQ[n_] := AnyTrue[FactorInteger[n][[All, 1]], Mod[#, 4] != 1&];
Select[Range[100], okQ] (* Jean-François Alcover, Mar 10 2019, after Federico Provvedi's comment *)

A065338(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f); };
isA137409(n) = ((1==n)||(A065338(n)>1)); \\ Antti Karttunen, Dec 26 2020

Extended by R. J. Mathar, Aug 15 2010

cp's OEIS Frontend

A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI