A286530 -id:A286530 - cp's OEIS frontend

A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.

3, 5, 6, 10, 11, 12, 17, 22, 29, 34, 35, 41, 44, 51, 58, 59, 60, 65, 70, 71, 72, 82, 84, 87, 91, 92, 96, 101, 102, 107, 111, 115, 118, 119, 125, 128, 129, 130, 137, 141, 142, 147, 149, 155, 174, 179, 182, 183, 191, 197, 201, 202, 205, 209, 213, 214, 215, 217, 222

Offset: 1

Leroy Quet, Mar 24 2010

nonn

The sequence contains the smaller member of every pair of twin primes (A001359) and all squarefree semiprimes m such that m+4 is also a squarefree semiprime (A255746). Can one prove that this is an infinite sequence? - Vladimir Shevelev, Jul 11 2015
The sequence does not contain perfect squares. Indeed, let a(m)=k^2. Then d(k^2+d(k^2)) = d(k^2). Note that d(k^2) is odd. On the other hand, it is known (A046522) that d(k^2)<2*k. Hence, (k+1)^2 - k^2 > d(k^2). Thus k^2Vladimir Shevelev, Feb 10 2017
If p is prime and t+1 is odd prime, then p^t is not in the sequence. Indeed, if d(p^t+t+1)=t+1, then p^t+t+1=q^t, where q is prime > p (if p^t+t+1= say q^l*r^m, then (l+1)*(m+1)=t+1 which is impossible by the condition). But q>=p+2 and p^t+t+1>=p^t+2*t*p^(t-1) or t+1>=2*t*p^(t-1) which trivially has only solution t=1; however, by the condition t>=2. - Vladimir Shevelev, Feb 18 2017
If an odd integer k is in this sequence, so is 2k. - Charlie Neder, Jan 14 2019

			10 is in the sequence because d(10)=4 and d(10+d(10))=d(14)=4. - _Emeric Deutsch_, Apr 08 2010

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000005, A062249, A001359, A255746, A259934, A282175, A282231, A286529.

Positions of zeros in A286530.

with(numtheory): a := proc (n) if tau(n+tau(n)) = tau(n) then n else end if end proc: seq(a(n), n = 1 .. 230); # Emeric Deutsch, Apr 08 2010

Select[Range@ 224, Function[n, DivisorSigma[0, n + #] == # &@ DivisorSigma[0, n]]](* Michael De Vlieger, Sep 27 2015 *)
Position[#, 0][[All, 1]] &@ Table[DivisorSigma[0, n + DivisorSigma[0, n]] - DivisorSigma[0, n], {n, 222}] (* Michael De Vlieger, May 21 2017 *)

is(n)=numdiv(n+n=numdiv(n))==n \\ M. F. Hasler, Sep 27 2015

More terms from Emeric Deutsch, Apr 08 2010

A286529 a(n) = d(n+d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 3, 2, 2, 2, 4, 3, 6, 6, 4, 2, 6, 4, 6, 2, 4, 2, 8, 4, 4, 3, 4, 3, 6, 6, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 6, 4, 8, 2, 10, 2, 6, 6, 6, 4, 6, 3, 4, 6, 8, 4, 4, 4, 4, 2, 7, 2, 4, 2, 12, 6, 8, 4, 2, 4, 4, 4, 4, 2, 8, 2, 12, 6, 8, 5, 4, 5, 4, 5, 12, 4, 4, 4, 12, 2, 12, 4, 12, 4, 8, 4, 6, 2, 6, 6, 12, 6, 8, 8, 2, 2, 8, 8, 10, 2, 8, 2, 16, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, An asymptotic formula involving the enumerating function of finite Abelian groups, Publikacije Elektrotehničkog fakulteta, Serija Matematika 3 (1992), pp. 61-66.
Imre Kátai, On a problem of A. Ivic, Mathematica Pannonica, Vol. 18, No. 1 (2007), pp. 11-18.

Cf. A000005, A062249, A175304, A286479, A286530.

Table[DivisorSigma[0, n + DivisorSigma[0, n]], {n, 117}] (* Michael De Vlieger, May 21 2017 *)

PARI
```
A286529(n) = numdiv(n+numdiv(n));
```

from sympy import divisor_count as d
def a(n): return d(n + d(n)) # Indranil Ghosh, May 21 2017

(define (A286529 n) (A000005 (+ n (A000005 n))))

a(n) = A000005(A062249(n)) = A000005(n+A000005(n)).

Sum_{k=1..n} a(k) ~ D*n*log(n) + O(n*log(n)/log(log(n))), where D > 0 is a constant (conjectured with an error O(n) by Ivić, 1992; proven by Kátai, 2007). - Amiram Eldar, Jul 08 2020

A286480 Compound filter (prime signature of n & prime signature of n+d(n)): a(n) = P(A046523(n), A286479(n)), where P(n,k) is sequence A000027 used as a pairing function and d(n) is number of divisors of n (A000005).

Original entry on oeis.org

2, 12, 5, 14, 5, 61, 12, 179, 109, 61, 5, 265, 23, 142, 27, 226, 5, 607, 23, 148, 42, 61, 12, 1509, 109, 601, 44, 148, 5, 625, 23, 698, 27, 61, 61, 1117, 23, 601, 27, 2509, 5, 850, 80, 265, 148, 142, 12, 1426, 109, 607, 61, 148, 23, 430, 27, 3765, 27, 61, 5, 8575, 80, 601, 148, 2144, 61, 625, 23, 148, 27, 1741, 5, 8587, 80, 601, 363, 148, 216, 625, 138, 5719

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000005, A046523, A282231, A286479, A286530.

f[n_] := If[n == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {f@ n, f[n + DivisorSigma[0, n]]}, {n, 80}] (* Michael De Vlieger, May 21 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286479(n) = A046523(n+numdiv(n));
A286480(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n));

from sympy import factorint, divisor_count
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a286479(n): return a046523(n + divisor_count(n))
def a(n): return T(a046523(n), a286479(n)) # Indranil Ghosh, May 21 2017

(define (A286480 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A286479 n)) 2) (- (A046523 n)) (- (* 3 (A286479 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n)).

cp's OEIS Frontend

A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.

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A286529 a(n) = d(n+d(n)), where d(n) is the number of divisors of n (A000005).

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A286480 Compound filter (prime signature of n & prime signature of n+d(n)): a(n) = P(A046523(n), A286479(n)), where P(n,k) is sequence A000027 used as a pairing function and d(n) is number of divisors of n (A000005).

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Programs

Mathematica

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Python

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Formula