A166684 -id:A166684 - cp's OEIS frontend

A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

-1, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 2, 1, 0, 3, 0, 3, 2, 2, 0, 5, 0, 2, 1, 3, 0, 6, 0, 2, 2, 2, 2, 4, 0, 2, 2, 5, 0, 6, 0, 3, 3, 2, 0, 6, 0, 3, 2, 3, 0, 5, 2, 5, 2, 2, 0, 9, 0, 2, 3, 2, 2, 6, 0, 3, 2, 6, 0, 7, 0, 2, 3, 3, 2, 6, 0, 6, 1, 2, 0, 9, 2, 2, 2, 5, 0, 9, 2, 3, 2, 2, 2, 8, 0, 3, 3, 4, 0, 6, 0, 5, 6

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2017

sign

The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s.

First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - Robert G. Wilson v, Nov 28 2017

			a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One less than A056595.
Cf. A000430, A032741, A046951, A166684, A080257.
Cf. A001620, A013661.

f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* Robert G. Wilson v, Nov 28 2017 *)

a(n) = A032741(n) - A046951(n).
a(n) = A056595(n) - 1. - Antti Karttunen, Oct 30 2017
a(n) = 0 iff n is a prime or a square of a prime, A000430. - Robert G. Wilson v, Nov 28 2017
Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A280619 Integers m such that sigma(m) - eulerphi(m) <= 4*sqrt(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Michel Marcus, Jan 06 2017

Equals {1, 8} union A000430 (primes and square of primes).

Equals 8 union A166684.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Mathematical Reflections, Solution to Problem O261, Issue 3, 2013, p. 24.

A000430 is the main entry for this sequence.

Cf. A000010, A000203, A166684.

Select[Range[250],DivisorSigma[1,#]-EulerPhi[#]<=4Sqrt[#]&] (* Harvey P. Dale, Jul 10 2018 *)

isok(n) = sigma(n) - eulerphi(n) <= 4*sqrt(n); \\ Michel Marcus, Jan 06 2017

A283970 Integers m such that m divides sigma_2(m) - k where k is some divisor of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 17, 19, 23, 25, 29, 30, 31, 35, 36, 37, 40, 41, 43, 47, 48, 49, 50, 53, 59, 60, 61, 65, 67, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 121, 127, 130, 131, 132, 136, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 175, 179, 180

Offset: 1

Juri-Stepan Gerasimov, Mar 18 2017

nonn

			2 is in this sequence because 2 divides A001157(2) - 1 = 5 - 1 = 4.

Supersequence of A166684.

Cf. A001157 (sigma_2(n): sum of squares of divisors of n), A205523 (integers n such that n divides sigma_1(n) - i where i is some divisor of n), A284082.

[[n: k in [1..n] | Denominator(n/k) eq 1 and
Denominator(((DivisorSigma(2, n))-k)/n) eq 1]: n in [1..100]];

Select[Range@ 180, Function[n, Total@ Boole@ Map[Divisible[ DivisorSigma[2, n] - #, n] &, Divisors@ n] > 0]] (* Michael De Vlieger, Mar 19 2017 *)

isok(n) = fordiv(n, d, if (!((sigma(n, 2) - d) % n), return (1))); \\ Michel Marcus, Mar 18 2017

A307360 A sequence in which every divisor other than 1 is used at most three times.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Joshua R. Tint, Apr 04 2019

In other words, for every k > 1, there are at most 3 multiples of k in the sequence. - Rémy Sigrist,  Apr 08 2019
The sequence begins at 1. The smallest integer greater than the last term which is not divisible by a divisor already used three times (excluding one) is added to the sequence.
Contains all prime numbers (A000040), given that the prime numbers only have the divisors of themselves and one, by definition, therefore the only divisor which could exist in the sequence already to disqualify the number from inclusion in the sequence would be the prime number itself, but a number cannot have a divisor higher than itself (the prime numbers), so given that the sequence increases, the divisor could not exist in the sequence, and any prime number would be included.
Terms are {1} or primes or squares of primes (A000430) or numbers of the form prime(2k + 1) * prime(2k + 2) (A089581) where k >= 0. - David A. Corneth, Apr 09 2019

			For instance, 8 is not in the sequence because 2, 4, and 6 are all divisible by 2 and appear previously in the sequence. The sequence, then, skips to nine. After 9, no more numbers divisible by three appear in the sequence, given that after 3 and 6, it is the third number divisible by three to appear in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Aaron Greicius, The Prime Numbers, Lecture Fall 2012.

See A166684 for the variant in which every divisor other than one is used at most twice.

Union of {1}, A000430 and A089581.

N:= 1000: # for terms <= N
M:= Vector(N):
Candidates:= {$2..N}:
A[1]:= 1:
for n from 2 while Candidates <> {} do
  A[n]:= min(Candidates):
  Candidates:= Candidates minus {A[n]};
  for d in numtheory:-divisors(A[n]) minus {1} do
     M[d]:= M[d]+1;
     if M[d] = 3 then Candidates:= Candidates minus {seq(i,i=2*d..N, d)} fi;
  od;
od:
seq(A[i],i=1..n-1); # Robert Israel, Apr 09 2019

Select[Range@ 229, Or[# == 1, PrimeQ@ #, PrimeQ@ Sqrt@ #, And[SquareFreeQ@ #, If[PrimeNu@ # == 2, And[OddQ@ First@ #, Apply[SameQ, (# - {1, 2})/2]] &@ PrimePi[FactorInteger[#][[All, 1]]], False]]] &] (* Michael De Vlieger, Apr 11 2019 *)

is(n) = if(n==1, return(1)); my(f=factor(n)); if(f[, 2] == [1]~ || f[, 2] ==[2]~, return(1)); if(f[,2] == [1,1]~ && nextprime(f[1,1]+1) == f[2,1] && primepi(f[1,1]) % 2 == 1, return(1)); 0 \\ David A. Corneth, Apr 09 2019

More terms from Jinyuan Wang, Apr 07 2019

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A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

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A280619 Integers m such that sigma(m) - eulerphi(m) <= 4*sqrt(m).

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A283970 Integers m such that m divides sigma_2(m) - k where k is some divisor of m.

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Magma

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A307360 A sequence in which every divisor other than 1 is used at most three times.

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