A046022 -id:A046022 - cp's OEIS frontend

A084820 Numbers n such that n, sigma(n) and phi(n) form an integer triangle, where sigma=A000203 is the divisor sum and phi=A000010 the totient.

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137

Offset: 1

Reinhard Zumkeller, Jun 04 2003

nonn

a(n)<=A000203(a(n))+A000010(a(n)), A000203(a(n))<=a(n)+A000010(a(n)), A000010(a(n))<=a(n)+A000203(a(n)); values are odd, see A084821 for odd numbers which are not in the sequence.

			n=5, a(5)=9: phi(9)=6, sigma(9)=13: (6,9,13)=(A070080(176), A070081(176), A070082(176)).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Totient Function

Cf. A046022.

Select[Range[1, 140, 2], DivisorSigma[1, #] < EulerPhi[#] + # &] (* Amiram Eldar, Sep 12 2019 *)

is(n)=eulerphi(n)+n>sigma(n) \\ Charles R Greathouse IV, Feb 19 2013

A240471 Integer part of (n * A000005(n) / A000203(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 3, 3, 3, 2, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 3, 3, 1, 4, 3, 2, 1, 4, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 06 2014

nonn

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

Cf. A000005, A000203, A038040, A046022, A106315.

a240471 n = n * a000005 n `div` a000203 n

a(n) = n*numdiv(n)\sigma(n); \\ Michel Marcus, Dec 02 2020

a(n) = (A038040(n) - A106315(n)) / A000203(n);

a(A046022(n)) = 1.

A244329 a(n) = floor(antisigma(n) / sigma(n)) = floor(A024816(n) / A000203(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 1, 5, 3, 4, 3, 7, 3, 8, 4, 6, 6, 10, 4, 9, 7, 8, 6, 13, 5, 14, 7, 10, 10, 12, 6, 17, 11, 12, 8, 19, 8, 20, 10, 12, 14, 22, 8, 20, 12, 17, 13, 25, 11, 20, 12, 19, 18, 28, 9, 29, 19, 18, 15, 24, 14, 32, 17, 24, 16, 34, 12, 35, 23

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

RECORD transform of a(n) is A140475 (union of number 1 and primes >= 5).
Sequence of numbers n such that a(n) = floor(antisigma(n) / n) = A046022.
Sequence of numbers n such that a(n) = a(n+1) = A244666.

			For n = 10; a(10) = floor(A024816(10) / A000203(10)) = floor(37 / 18) = 2.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000203, A000217, A024816, A046022, A140475, A244327, A244328, A244666.

[Floor(((n*(n+1)div 2)-SumOfDivisors(n)) div (SumOfDivisors(n))) : n in [1..1000]];

A244329[n_] := Floor[(n*(n + 1)/2 - #)/#] & [DivisorSigma[1, n]];
Array[A244329, 100] (* Paolo Xausa, Sep 01 2024 *)

a(n) = A244327(n) - A244328(n) for n >= 7.

A067830 Primes p such that sigma(p-4) < p.

Original entry on oeis.org

5, 7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433, 1451

Offset: 1

Benoit Cloitre, Feb 08 2002

Except for the first term, terms are primes of the form p+4 with p prime, i.e., the sequence is essentially A031505, A046132. In other words, the solutions to sigma(x) < x + 4 are 1,2,4 and the odd primes. - Ralf Stephan, Feb 09 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A025584, A046022, A067833.

Select[Prime[Range[3, 230]], DivisorSigma[1, #-4] < # &] (* Amiram Eldar, Apr 25 2025 *)

isok(p) = isprime(p) && (p>4) && (sigma(p-4) < p); \\ Michel Marcus, Feb 15 2021

Edited by Charles R Greathouse IV, Mar 19 2010

A105602 Divide each Fibonacci number by its primitive part.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 24, 1, 13, 10, 21, 1, 136, 1, 165, 26, 89, 1, 1008, 5, 233, 34, 1131, 1, 26840, 1, 987, 178, 1597, 65, 46512, 1, 4181, 466, 47355, 1, 1269736, 1, 53133, 10370, 28657, 1, 2179296, 13, 825275, 3194, 364179, 1, 14927768, 445

Offset: 1

Paul Barry, Apr 15 2005

Sylvester dividends for Fibonacci numbers.

a(n)=1 for n=1, 4 and all primes, which is sequence A046022.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000045, A046022, A061446, A126015.

a[n_] := Fibonacci[n]/Product[Fibonacci[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n],{n,55}] (* James C. McMahon, Jan 25 2024 *)

a(n) = Fibonacci(n)/A061446(n).

A178156 Numbers m such that (m-1)! is not divisible by m^2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163

Offset: 1

Reinhard Zumkeller, Dec 17 2010

nonn

Union of {8, 9} and A001751.

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1972), Part Eight, Chap. 3, Sect. 1, Problem 133b.

Cf. A046022, A000142, A174460.

Cf. A001751.

import Data.List (insert)
a178156 n = a178156_list !! (n-1)
a178156_list = insert 9 $ insert 8 a001751_list
-- Reinhard Zumkeller, Oct 14 2014

Select[Range[200],!Divisible[(#-1)!,#^2]&] (* Harvey P. Dale, Mar 06 2016 *)

for(m=1,3e2,if((m-1)!%m^2,print1(m", "))) \\ Charles R Greathouse IV, Aug 21 2011

from sympy import primepi
def A178156(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: 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+x-primepi(x)-primepi(x>>1)-(x>=8)-(x>=9))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024

Entries corrected by Charles R Greathouse IV, Aug 21 2011

A300858 a(n) = A243823(n) - A243822(n).

Original entry on oeis.org

0, 0, 0, 0, 0, -1, 0, 1, 1, -1, 0, -1, 0, 1, 2, 4, 0, -1, 0, 3, 4, 3, 0, 3, 3, 5, 6, 7, 0, -5, 0, 11, 6, 7, 6, 6, 0, 9, 8, 11, 0, 1, 0, 13, 12, 13, 0, 13, 5, 13, 12, 17, 0, 13, 10, 19, 14, 19, 0, 5, 0, 21, 18, 26, 12, 11, 0, 23, 18, 15, 0, 25, 0, 25, 24, 27

Offset: 1

Michael De Vlieger, Mar 14 2018

sign

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.
Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.
For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.
For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.
Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

			a(6) = -1 since the only nondivisor in the cototient of 6 is 4, which divides 6^e with e > 1; therefore 0 - 1 = -1.
a(8) = 1 since the only nondivisor in the cototient of 8 is 6, and 6 does not divide 8^e with e > 1, therefore 1 - 0 = 1.
Some values of a(n) and related sequences:
   n  a(n) A243823(n) A243822(n)    A272619(n)       A272618(n)
  -------------------------------------------------------------
   1   0          0          0      -                -
   2   0          0          0      -                -
   3   0          0          0      -                -
   4   0          0          0      -                -
   5   0          0          0      -                -
   6  -1          0          1      -                {4}
   7   0          0          0      -                -
   8   1          1          0      {6}              -
   9   1          1          0      {6}              -
  10  -1          1          2      {6}              {4,8}
  11   0          0          0      -                -
  12  -1          1          2      {10}             {8,9}
  13   0          0          0      -                -
  14   1          3          2      {6,10,12}        {4,8}
  15   2          3          1      {6,10,12}        {9}
  16   4          4          0      {6,10,12,14}     -
  17   0          0          0      -                -
  18  -1          3          4      {10,14,15}       {4,8,12,16}
  19   0          0          0      -                -
  20   3          5          2      {6,12,14,15,18}  {8,16}
  ...

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

Cf. A000005, A000010, A000961, A010846, A046022, A121998, A133995, A173540, A243822, A243823, A272618, A272619, A295523, A300859, A300861.

f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 76]

a(n) = 1 + n + numdiv(n) - eulerphi(n) - 2*sum(k=1, n, if(gcd(n,k)-1, 0, moebius(k)*(n\k))); \\ Michel Marcus, Mar 17 2018

a(n) = 1 + n - A000010(n) - 2*A010846(n) + A000005(n).

A244666 Numbers n such that floor(antisigma(n) / sigma(n)) = floor(antisigma(n+1) / sigma(n+1)).

Original entry on oeis.org

1, 2, 3, 9, 21, 33, 81, 261, 897, 1334, 1364, 2974, 4364, 14282, 26937, 46593, 64665, 74918, 79833, 92685, 145215, 147454, 161001, 162602, 166934, 289454, 347738, 383594, 422073, 430137, 440013, 443402, 445874, 621027, 649154, 655005, 1174305, 1187361, 1670955

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

Also numbers n such that floor((n*(n+1)/2) / sigma(n)) = floor(((n+1)*(n+2)/2) / sigma(n+1)).
Numbers n such that A244327(n) = A244327(n+1).
Numbers n such that A244329(n) = A244329(n+1).

Jaroslav Krizek, Table of n, a(n) for n = 1..39

Cf. A000203, A000217, A024816, A046022, A244327, A244328, A244329.

[n: n in [1..10^6] | Floor((n*(n+1)div 2) div (SumOfDivisors(n))) eq Floor(((n+1)*(n+2)div 2) div (SumOfDivisors(n+1)))]

A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.

Original entry on oeis.org

-1, -2, -2, -2, -2, -4, -2, 0, 0, -4, -2, 0, -2, -4, 2, 8, -2, 0, -2, 8, 6, -4, -2, 16, 10, -4, 18, 16, -2, 4, -2, 32, 14, -4, 26, 36, -2, -4, 18, 48, -2, 12, -2, 32, 54, -4, -2, 64, 28, 20, 26, 40, -2, 36, 50, 80, 30, -4, -2, 72, -2, -4, 90, 96, 62, 28, -2, 56, 38, 52, -2, 144, -2, -4, 90, 64, 86, 36, -2, 160, 108, -4, -2, 120, 86, -4

Offset: 1

Labos Elemer, Jul 20 2001

sign

It can be shown that phi(n)*tau(n) >= n, which means that quotient = n/tau(n) <= phi(n); note: a(n)+5 is positive.

The value is always positive except when a(n) = 0 for {8,9,12}; or a(n) = -2 for primes together with 4 (i.e., for A046022 but without 1); or a(n) = -4 for A001747 (without 2 and 4); or a(n) = -1 for n = 1.

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A000010, A000005, A062355, A062516, A046022, A001747, A036762, A036763, A051728, A051729, A051730.

Table[EulerPhi[n]DivisorSigma[0,n]-2n,{n,90}] (* Harvey P. Dale, Feb 03 2021 *)

a(n)={eulerphi(n)*numdiv(n) - 2*n} \\ Harry J. Smith, Aug 11 2009

a(n) = A062355(n) - 2*n. - Amiram Eldar, Jul 10 2024

Offset changed from 0 to 1 by Harry J. Smith, Aug 11 2009

A109853 a(n) = A109852(2^n).

Original entry on oeis.org

1, 2, 5, 9, 13, 19, 29, 37, 43, 53, 61, 71, 79, 89, 101, 107, 113, 131, 139, 151, 163, 173, 181, 193, 199, 223, 229, 239, 251, 263, 271, 281, 293, 311, 317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491, 503, 521, 541, 557, 569, 577, 593

Offset: 0

Amarnath Murthy, Jul 07 2005

nonn

Conjecture: a(n) is prime if n is not 0 nor 2.
Conjecture: a(n) is the (2n-2)nd prime for n>1. A109852(2^n-1): 1,3,5,11,17,23,31,41,47,59,67,73. - Robert G. Wilson v, Jun 14 2006
Conjecture: the Union of A109852(2^n-1) & A109852(2^n) is A046022: {1,2,3,4,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79, ...,} and except for 4, equals A008578: The noncomposite numbers. - Robert G. Wilson v, Jun 14 2006

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Rémy Sigrist, PARI program for A109853

Cf. A008578, A031215, A046022, A109852.

f[s_] := Block[{k = 2, len = Length@s}, exp = Ceiling[Log[2, len]]; m = s[[2^exp - len + 1]]; While[MemberQ[s, k*m], k++ ]; Append[s, k*m]]; Rest@Nest[f, {1, 1}, 70]; t = Rest@Nest[f, {1, 1}, 2^14 + 3]; Table[t[[2^n]], {n, 0, 14}] (* Robert G. Wilson v, Jun 14 2006 *)

PARI
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See Links section.
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More terms from Robert G. Wilson v, Jun 14 2006

More terms from Rémy Sigrist, May 19 2019

cp's OEIS Frontend

A084820 Numbers n such that n, sigma(n) and phi(n) form an integer triangle, where sigma=A000203 is the divisor sum and phi=A000010 the totient.

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A240471 Integer part of (n * A000005(n) / A000203(n)).

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A244329 a(n) = floor(antisigma(n) / sigma(n)) = floor(A024816(n) / A000203(n)).

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Magma

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A067830 Primes p such that sigma(p-4) < p.

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Extensions

A105602 Divide each Fibonacci number by its primitive part.

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Mathematica

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A178156 Numbers m such that (m-1)! is not divisible by m^2.

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A300858 a(n) = A243823(n) - A243822(n).

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Formula

A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.