A125667 -id:A125667 - cp's OEIS frontend

A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Len Smiley, Nov 01 2005

This sequence might also be called "The Non-Pythagorean integers" since no primitive Pythagorean triangle (PPT) exists containing them. Numbers of the form 4n-2 cannot be a leg or hypotenuse of PPT [a,b,c]. This excludes all even members of the present sequence. Integers 1 and zero are excluded because they form a 'degenerate triangle' with angles = 0. Compare A125667. - H. Lee Price, Feb 02 2007

Besides the first term this sequence is the denominator of Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 14 2011

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A125667. Complement of the union of {1}, A020882, A020883 and A020884.

Cf. A016825, A161718.

Table[If[n == 1, 1, 4n - 6], {n, 60}] (* Robert G. Wilson v, Nov 04 2005 *)

a(n) = 4*n-6, n>=2.
a(n) = A016825(n-2), n>1. - R. J. Mathar, Aug 18 2008
G.f.: x(1+3x^2)/(1-x)^2. - R. J. Mathar, Nov 10 2008
a(n^2 - 2n + 3)/2 = Sum_{i=1..n} a(i). - Ivan N. Ianakiev, Apr 24 2013
a(n) = 2*a(n-1) - a(n-2), n>3. - Rick L. Shepherd, Jul 06 2017
a(n) = |A161718(n-1)| = (-1)^(n-1)*A161718(n-1), n>0. - Rick L. Shepherd, Jul 06 2017
E.g.f.: 3*(x + 2) + exp(x)*(4*x - 6). - Stefano Spezia, Feb 02 2023

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

Original entry on oeis.org

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

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A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

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

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

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