cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A358448 Indices of record values of A036450(n) = d(d(d(n))).

Original entry on oeis.org

1, 2, 12, 60, 5040, 3603600, 908107200, 15437822400, 293318625600, 267154604196480000, 4935147003321575040000, 1963886355464640647040000, 74963506074440798138163840000, 65039484811827775408882461490752000000, 3415186532666352810621130006203072000000
Offset: 1

Views

Author

Charles R Greathouse IV, Nov 17 2022

Keywords

Crossrefs

Subsequence of A007416. (Not a subsequence of A002182.)
Cf. A036450.

A010553 a(n) = tau(tau(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 4, 2
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan (1915) posed the problem of finding the extreme large values of a(n). Buttkewitz et al. determined the maximal order of log a(n).
Every number eventually appears. Sequence A193987 gives the least term where each number appears. - T. D. Noe, Aug 10 2011

References

  • S. Ramanujan, Highly composite numbers. Proc. London Math. Soc., series 2, 14 (1915), 347-409. Republished in Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78-128.

Links

Crossrefs

Cf. A000005, A036450, A193987 (least number k such that tau(tau(k)) = n), A335831.

Programs

Formula

a(n) = A000005(A000005(n)). a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 3 for pq = product of two distinct primes (A006881), a(pq...z) = k + 1 for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = A000005(k+1) for p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). - Jaroslav Krizek, Jul 17 2009
a(A007947(n)) = 1 + A001221(n); (n>1). - Enrique Pérez Herrero, May 30 2010
Asymptotically, Max_{i<=n} log(tau(tau(i))) = sqrt(log(n))/log_2(n) * (c + O(log_3(n)/log_2(n)) where c = 8*Sum_{j>=1} log^2 (1 + 1/j)) ~ 2.7959802335... [Buttkewitz et al.].

A036459 Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function (A000005).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 3, 4, 1, 4, 1, 4, 3, 3, 3, 3, 1, 3, 3, 4, 1, 4, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 4, 1, 4, 3, 4, 1, 5, 1, 3, 4, 4, 3, 4, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 5, 1, 4, 4
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Iterating d for n, the prestationary prime and finally the fixed value of 2 is reached in different number of steps; a(n) is the number of required iterations.
Each value n > 0 occurs an infinite number of times. For positions of first occurrences of n, see A251483. - Ivan Neretin, Mar 29 2015

Examples

			If n=8, then d(8)=4, d(d(8))=3, d(d(d(8)))=2, which means that a(n)=3. In terms of the number of steps required for convergence, the distance of n from the d-equilibrium is expressed by a(n). A similar method is used in A018194.

Links

Crossrefs

Equals A060937 - 1. Cf. A007624, A036450, A036452, A036453, A036455, A030630.

Programs

  • Mathematica
    Table[ Length[ FixedPointList[ DivisorSigma[0, # ] &, n]] - 2, {n, 105}] (* Robert G. Wilson v, Mar 11 2005 *)
  • PARI
    for(x = 1,150, for(a=0,15, if(a==0,d=x, if(d<3,print(a-1),d=numdiv(d) )) ))
  • PARI
    a(n)=my(t);while(n>2,n=numdiv(n);t++);t \\ Charles R Greathouse IV, Apr 07 2012

Formula

a(n) = a(d(n)) + 1 if n > 2.
a(n) = 1 iff n is an odd prime.

A036452 a(n) = d(d(d(d(n)))), the 4th iterate of number-of-divisors function with initial value of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

The iterated d function rapidly converges to fixed point 2. For k=4, the first n for which a(n)>2 is 60.

Examples

			E.g., n=96 and its successive iterates are 12,6,4,3 and 2. The 4th term is a(96)=3.

Links

Crossrefs

Cf. A000005, A010553, A036450, A036453.

Programs

Formula

a(n) = d(d(d(d(n)))).

A036453 a(n) = d(d(d(d(d(n))))), the 5th iterate of the number-of-divisors function d = A000005, with initial value n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

The iterated d function rapidly converges to fixed point 2. In the 5th iterated d-sequence, the first term different from the fixed point 2 appears at n = 5040. The 6th and further iterated sequences have very long initial segment of 2's. In the 6th one the first non-stationary term is a(293318625600) = 3. In such sequences any large value occurs infinite many times and constructible.
Differs from A007395 for n = 1, 5040, 7920, 8400, 9360, 10080, 10800, etc. - R. J. Mathar, Oct 20 2008

Examples

			E.g., n = 96 and its successive iterates are 12, 6, 4, 3 and 2. The 5th term is a(96) = 2 is stationary (fixed).

Links

Crossrefs

Cf. A000005, A010553, A036450, A036452.

Programs

Extensions

Previous Mathematica program replaced by Harvey P. Dale, Jun 18 2021

A036454 Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers.

Original entry on oeis.org

4, 9, 16, 25, 49, 64, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Composite numbers with a prime number of divisors.

Examples

			From powers of 2: 4,16,64,1024,4096,65536 are in the sequence since exponent+1 is also prime. The same powers of any prime base are also included.

Links

Crossrefs

Intersection of A002808 and A009087.
Cf. A000005, A010051, A010553, A010055, A100995, A036450, A036452, A036459, A283262.

Programs

  • Haskell
    a009087 n = a009087_list !! (n-1)
a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list
-- Reinhard Zumkeller, Jun 05 2013
  • Magma
    [n: n in [1..20000] | not IsPrime(n) and IsPrime(DivisorSigma(0, n))]; // Vincenzo Librandi, May 19 2015
  • Maple
    N:= 10^5:
P1:= select(isprime,[2,seq(2*i+1,i=1..floor((sqrt(N)-1)/2))]):
P2:= select(`<=`,P1,1+ilog2(N))[2..-1]:
S:= {seq(seq(p^(q-1), q = select(`<=`,P2,1+floor(log[p](N)))),p=P1)}:
sort(convert(S,list)); # Robert Israel, May 18 2015
  • Mathematica
    specialPrimePowerQ[n_] := With[{f = FactorInteger[n]}, Length[f] == 1 && PrimeQ[f[[1, 1]]] && f[[1, 2]] > 1 && PrimeQ[f[[1, 2]] + 1]]; Select[Range[20000], specialPrimePowerQ]  (* Jean-François Alcover, Jul 02 2013 *)
Select[Range[20000], ! PrimeQ[#] && PrimeQ[DivisorSigma[0, #]] &] (* Carlos Eduardo Olivieri, May 18 2015 *)
  • PARI
    for(n=1,34000, if(isprime(n), n++,x=numdiv(n); if(isprime(x),print(n))))
  • PARI
    list(lim)=my(v=List(),t);lim=lim\1+.5;forprime(p=3,log(lim)\log(2) +1, t=p-1; forprime(q=2,lim^(1/t),listput(v,q^t))); vecsort(Vec(v))
\\ Charles R Greathouse IV, Apr 26 2012
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A036454(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p-1)[0]) for p in primerange(3,x.bit_length()+1)))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Formula

d(d(a(n))) = 2, where d(x) = tau(x) = sigma_0(x) is the number of divisors of x.
a(n) = (n log n)^2 + 2n^2 log n log log n + O(n^2 log n). - Charles R Greathouse IV, Apr 26 2012
(1 - A010051(a(n))) * A010055(a(n)) * A010051(A100995(a(n))+1) = 1. - Reinhard Zumkeller, Jun 05 2013
A036459(a(n)) = 2. - Ivan Neretin, Jan 25 2016
a(n) = A283262(n)^2. - Amiram Eldar, Jul 04 2022
Sum_{n>=1} 1/a(n) = Sum_{k>=2} P(prime(k)-1) = 0.54756961912815344341..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Compare with sequence A007422 and A030513 -- the resemblance is rather strong. Still this sequence is different. For example, 36, 100, 120, and 168 are here.

Examples

			a(15) = 39 and d(39) = 4, d(d(39)) = d(4) = 3 and d(d(d(39))) = 2. After 3 iteration the equilibrium is reached.

Links

Crossrefs

Cf. A000005, A007422, A030513, A036450, A036452, A036454, A036456, A036457, A036458.

Programs

  • Maple
    filter:= proc(n) local r;
  r:= numtheory:-tau(numtheory:-tau(n));
  r::odd and isprime(r)
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2016
  • Mathematica
    fQ[n_] := Module[{d2 = DivisorSigma[0, DivisorSigma[0, n]]}, d2 > 2 && PrimeQ[d2]]; Select[Range[200], fQ] (* T. D. Noe, Jan 22 2013 *)
  • PARI
    is(n)=isprime(n=numdiv(numdiv(n))) && n>2 \\ Charles R Greathouse IV, Jan 22 2013

Formula

d(d(d(a(n)))) = 2 for all n.
A036459(a(n)) = 3. - Ivan Neretin, Jan 25 2016

Extensions

Definition clarified by R. J. Mathar and Charles R Greathouse IV, Jan 22 2013

A036457 Numbers k for which exactly 5 applications of A000005 are needed to reach 2.

Original entry on oeis.org

60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 200, 204, 220, 224, 228, 234, 240, 252, 260, 276, 288, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 392, 396, 414, 416, 420, 432, 444, 450, 460, 468, 476, 480
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Subsequences include A030630 (numbers with 12 divisors), A030636 (numbers with 18 divisors), A030638 (numbers with 20 divisors), A137491 (numbers with 28 divisors), etc. [edited by Jon E. Schoenfield, May 12 2018]

Examples

			a(13)=180; the successive iterates are 18, 6, 4, 3, and finally the 5th is 2;
a(3)=84; divisor numbers are 12, 6, 4, 3, and 2.

Links

Crossrefs

Cf. A000005, A007624, A036450, A036452, A036453, A036455, A030630.

Programs

  • Maple
    A036459:= proc(n) option remember;
  if n <= 2 then 0 else 1 + procname(numtheory:-tau(n)) fi
end proc:
select(A036459 = 5, [$1..1000]); # Robert Israel, Jan 25 2016
  • Mathematica
    Select[Range@ 480, Last@ # == 2 && #[[5]] != 2 &@ NestList[DivisorSigma[0, #] &, #, 5] &] (* Michael De Vlieger, Jan 26 2016 *)
  • PARI
    is(n)=for(i=1,4,n=numdiv(n); if(n<3, return(0))); numdiv(n)==2 \\ Charles R Greathouse IV, Sep 17 2015

Formula

d(d(d(d(d(a(n)))))) = 2 for all n.
A036459(a(n)) = 5. - Ivan Neretin, Jan 25 2016

Extensions

New name from Robert Israel, Jan 25 2016

A036456 Numbers k for which exactly 4 applications of A000005 are needed to reach 2.

Original entry on oeis.org

12, 18, 20, 24, 28, 30, 32, 40, 42, 44, 45, 48, 50, 52, 54, 56, 63, 66, 68, 70, 75, 76, 78, 80, 88, 92, 98, 99, 102, 104, 105, 110, 112, 114, 116, 117, 124, 128, 130, 135, 136, 138, 144, 147, 148, 152, 153, 154, 162, 164, 165, 170, 171, 172, 174, 175, 176, 182
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Similar to but different from A007624. Terms like 60, 72, 84, 90, 96, 108, 126, etc. are not present here.

Examples

			a(3)=20 and a(17)=63; for both x=20 and 63, d(x)=6 and d(d(x))=4, the 3rd iterates are 3 and the equilibrium value, i.e., 2 appears as 4th iterates.

Links

Crossrefs

Cf. A000005, A007624, A036450, A036452, A036453, A036455.

Programs

  • PARI
    isok(n) = ((nd=numdiv(n)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) == 2); \\ Michel Marcus, Dec 30 2013 & Jan 26 2015

Formula

With d(n) = number of divisors(n), d(d(d(d(a(n))))) = 2 and d(d(d(a(n)))) > 2.
A036459(a(n)) = 4. - Ivan Neretin, Jan 25 2016

Extensions

New name (using new name for A036457 from Robert Israel) from Jon E. Schoenfield, May 12 2018

A053026 Maximum power of 2 arising when A000005 is applied repeatedly to n!.

Original entry on oeis.org

1, 2, 4, 8, 16, 8, 4, 4, 4, 16, 8, 8, 8, 8, 8, 2, 8, 4, 8, 4, 4, 4, 4, 4, 4, 4, 16, 16, 4, 16, 8, 16, 4, 4, 16, 4, 4, 2, 4, 16, 8, 16, 16, 4, 8, 8, 4, 8, 8, 16, 8, 8, 32, 32, 4, 32, 4, 4, 8, 4, 2, 32, 2, 8, 4, 8, 4, 8, 8, 8, 8, 2, 8, 8, 8, 32, 32, 8, 4, 8, 8, 4, 8, 8, 8, 8, 8, 32, 8, 8, 2, 4, 2, 4, 8
Offset: 1

Views

Author

Labos Elemer, Feb 24 2000

Keywords

Comments

Unlike the iteration of Euler phi (A000010) or cototient (A051953) functions, here the emerging powers of 2 are not accumulated at the terminal phase of iteration sequence. Non-2-powers can be intercalated.

Examples

			For n = 53, the iterations are {53!, 16174080000, 840, 32, 6, 4, 3, 2}, so a(53) = 32.
For n = 130, the iterations are {130!, 287298761874053529600, 38016, 64, 7, 2}, so a(130) = 64.
For n = 563, the iterations are {563!, 2875041108020454013464609906430286933482949481627276804096000000000, 77051520, 512, 10, 4, 3, 2}, so a(563) = 512.

Crossrefs

Cf. A000005, A000142, A000010, A051953, A036450-A036459, A380802, A380803.

Programs

  • Mathematica
    Join[{1,2},Table[SelectFirst[Rest[NestWhileList[DivisorSigma[0,#]&,n!,#>2&]],IntegerQ[Log[2,#]]&],{n,3,100}]] (* Harvey P. Dale, Jul 02 2018 *)
  • PARI
    a(n) = {my(m = n!); while(1 << valuation(m, 2) != m, m = numdiv(m)); m;} \\ Amiram Eldar, Feb 04 2025

Formula

a(n) = 2^A380802(n). - Amiram Eldar, Feb 04 2025
Showing 1-10 of 14 results. Next

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