A177712 -id:A177712 - cp's OEIS frontend

A054741 Numbers m such that totient(m) < cototient(m).

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136

Offset: 1

Labos Elemer, Apr 26 2000

nonn

For powers of 2, the two function values are equal.
Numbers m such that m/phi(m) > 2. - Charles R Greathouse IV, Sep 13 2013
Numbers m such that A173557(m)/A007947(m) < 1/2. - Antti Karttunen, Jan 05 2019
Numbers m such that there are powers of m that are abundant. This follows from abundancy and totient being multiplicative, with the abundancy for prime p of p^k being asymptotically p/(p-1) as k -> oo; given that p/(p-1) = p^k/phi(p^k) for k >= 1. - Peter Munn, Nov 24 2020

			For m = 20, phi(20) = 8, cototient(20) = 20 - phi(20) = 12, 8 = phi(20) < 20-phi(20) = 12; for m = 21, the opposite holds: phi = 12, 21-phi = 8.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mitsuo Kobayashi, A generalization of a series for the density of abundant numbers, International Journal of Number Theory, Vol. 12, No. 3 (2016), pp. 671-677.

A177712 is a subsequence. Complement: A115405.
Positions of negative terms in A083254.
Cf. A000010, A007947, A051953, A005408, A036798, A089684, A173557.
Cf. A323170 (characteristic function).
Complement of A000079\{1} within A119432.

Select[ Range[300], 2EulerPhi[ # ] < # &] (* Robert G. Wilson v, Jan 10 2004 *)

is(n)=n>2*eulerphi(n) \\ Charles R Greathouse IV, Sep 13 2013

m such that A000010(m) < A051953(m).

a(n) seems to be asymptotic to c*n with c=1.9566...... - Benoit Cloitre, Oct 20 2002 [It is an old theorem that a(n) ~ cn for some c, for any sequence of the form "m/phi(m) > k". - Charles R Greathouse IV, May 28 2015] [c is in the interval (1.9540, 1.9562) (Kobayashi, 2016). - Amiram Eldar, Feb 14 2021]

Erroneous comment removed by Antti Karttunen, Jan 05 2019

A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

Original entry on oeis.org

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180

Offset: 1

Michael Minic (Rassilon6(AT)aol.com)

At least one exponent in the canonical prime factorization (cf. A124010) of n is odd, and at least one exponent is greater than 1. - Reinhard Zumkeller, Jan 24 2013

Compare this sequence, as a set, with A177712, numbers that have an odd factor, but are not odd. The self-inverse function defined by A225546, maps the members of either one of these sets 1:1 onto the other set. - Peter Munn, Jul 31 2020

			63 is included because 63 = 3^2 * 7.
64 is not included because it is a perfect square (8^2).
65 is not included because it is squarefree (5 * 13).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number.
Eric Weisstein's World of Mathematics, Squarefree.
Wikipedia, Square number.
Wikipedia, Squarefree integer.

Cf. A210490 (complement), intersection of A013929 and A000037.

Related to A177712 via A225546.

a051144 n = a051144_list !! (n-1)
a051144_list = filter ((== 0) . a008966) a000037_list
-- Reinhard Zumkeller, Sep 02 2013, Jan 24 2013

[k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // Marius A. Burtea, Dec 29 2019

N:= 10000;  # to get all entries up to N
A051144:= remove(numtheory:-issqrfree,{$1..N}) minus {seq(i^2,i=1..floor(sqrt(N)))}:
# Robert Israel, Mar 30 2014

searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* Alonso del Arte, Dec 20 2019 *)

is(n)=!issquare(n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A051144(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+(y:=isqrt(x))+sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 23 2025

(1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 24 2013

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(2*s) - zeta(s)/zeta(2*s), for s > 1. - Amiram Eldar, Dec 03 2022

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

Offset corrected by Reinhard Zumkeller, Jan 24 2013

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

Original entry on oeis.org

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011
Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - R. J. Mathar, Dec 04 2011
Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

			5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

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

Cf. A177712, A136724, A177713.
Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
Contains A004766, A017137 and nonzero terms of A008588.
Disjoint from A002145.
Subsequence of A138591.

f:= proc(n) local r,k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

A363101 Even numbers that are neither prime powers nor squarefree.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 48, 50, 52, 54, 56, 60, 68, 72, 76, 80, 84, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 120, 124, 126, 132, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 192, 196, 198, 200, 204, 208, 212, 216, 220, 224, 228, 232, 234, 236, 240, 242

Offset: 1

Michael De Vlieger, May 19 2023

nonn

Even numbers k such that A001222(k) > A001221(k) > 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A005418, A024619, A126706, A360769.

Intersection of A013929 and A177712.

Select[Range[2, 242, 2], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]

A177732 The sums of two or more consecutive positive numbers, the largest being even.

Original entry on oeis.org

3, 7, 9, 10, 11, 15, 18, 19, 20, 21, 23, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 45, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 63, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 87, 90, 91, 93, 95, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form (j+2l)*(2l-j+1)/2 with j>=1 and 2l>j. Subsequences are A014105 where >=3, (j=1), A014107 where >=9 (j=2). - R. J. Mathar, Jul 14 2012

			3=1+2, 7=3+4, 9=2+3+4, 10=1+2+3+4, 11=5+6,..

Cf. A177712, A177713, A177731.

z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}];Union@lst2
With[{upto=108},Select[Union[Flatten[Table[Accumulate[Range[2n-1,1,-1]]+ 2n,{n,upto/4}]]],#<=upto&]] (* Harvey P. Dale, May 19 2019 *)

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A054741 Numbers m such that totient(m) < cototient(m).

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A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

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A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

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A363101 Even numbers that are neither prime powers nor squarefree.

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A177732 The sums of two or more consecutive positive numbers, the largest being even.

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Mathematica