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 1-4 of 4 results.

A046022 Primes together with 1 and 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269
Offset: 1

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Author

Eric W. Weisstein

Keywords

Comments

Also the numbers which are incrementally largest values of A002034. - validated by Franklin T. Adams-Watters, Jul 13 2012
Solutions to A000005(x) + A000010(x) - x - 1 = 0. - Labos Elemer, Aug 23 2001
Also numbers m such that m, phi(m) and tau(m) form an integer triangle, where phi=A000010 is the totient and tau=A000005 the number of divisors (see also A084820). - Reinhard Zumkeller, Jun 04 2003
Terms > 1 are n such that n does not divide (n-1)!. - Benoit Cloitre, Nov 12 2003
Terms > 1 are the sum of their prime factors; 4 (= 2+2) is the only such composite number. - Stuart Orford (sjorford(AT)yahoo.co.uk), Aug 04 2005
From Jonathan Vos Post, Aug 23 2010, Robert G. Wilson v, Aug 25 2010, proof by D. S. McNeil, Aug 29 2010: (Start)
Also the numbers n which divide A001414(n), or equivalently divide A075254(n). Proof:
Theorem: for a multiset of m >= 2 integers a_i, each a_i >= 2, Product_{i=1..m} a_i >= Sum_{i=1..m} a_i, with equality only at (a_1,a_2) = (2,2).
Lemma: For integers x,y >= 2, if x > 2 or y > 2, x*y > x + y. This follows from distributing (x-1)*(y-1) > 1.
[Proof of the theorem by induction on m:
first consider m=2. We have equality at (2,2) and for any product(a_i) > 4 there is some a_i > 2, so the lemma gives a_1*a_2 > a_1+a_2.
Then the induction m->m+1: Product_{i=1..m+1} a_i = a_(m+1)*Product_{i=1..m} a_i >= a_(m+1) * Sum_{i=1..m} a_i.
Since a_(m+1) >= 2 and the sum >= 4, the lemma applies, and we find a_(m+1) * Sum+{i=1..m} a_i > a_(m+1) + Sum_{i=1..m} a_i = Sum_{i=1..m+1} a_i and thus Product_{i=1..m+1} a_i > Sum_{i=1..m+1} a_i, QED.]
For composite n > 4, applying the theorem to the multiset of prime factors with multiplicity yields n > sopfr(n), so there are no composite numbers greater than 4 such that they divide sopfr(n).
(End)
Numbers k such that the k-th Fibonacci number is relatively prime to all smaller Fibonacci numbers. - Charles R Greathouse IV, Jul 13 2012
Numbers k such that (-1)^k*floor(d(k)*(-1)^k/2) = 1, where d(k) is the number of divisors of k. - Wesley Ivan Hurt, Oct 11 2013
Also, union of odd primes (A065091) and the divisors of 4. Also, union of A008578 and 4. - Omar E. Pol, Nov 04 2013
Numbers k such that sigma(k!) is divisible by sigma((k-1)!). - Altug Alkan, Jul 18 2016

Links

Crossrefs

Cf. A002034, A046021, A001751, A178156, A174460, A000040.

Programs

  • Haskell
    a046022 n = a046022_list !! (n-1)
a046022_list = [1..4] ++ drop 2 a000040_list
-- Reinhard Zumkeller, Apr 06 2014
  • Maple
    A046022:=n-> `if`((-1)^n*floor(numtheory[tau](n)*(-1)^n/2) = 1, n, NULL); seq(A046022(j), j=1..260); # Wesley Ivan Hurt, Oct 11 2013
  • Mathematica
    max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]*m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, w]; max = w], {n, 1, 1000}]; a (* Artur Jasinski, Apr 06 2008 *)
  • PARI
    a(n)=if(n<6,n,prime(n-2)) \\ Charles R Greathouse IV, Apr 28 2015
  • Python
    from sympy import prime
def A046022(n): return prime(n-2) if n>4 else n # Chai Wah Wu, Oct 17 2024

Formula

A141295(a(n)) = a(n). - Reinhard Zumkeller, Jun 23 2008
A018194(a(n)) = 1. - Reinhard Zumkeller, Mar 09 2012
A240471(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2014

Extensions

Better description from Frank Ellermann, Jun 15 2001

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.

A025494 Squares which are the sum of factorials of distinct integers (probably finite).

Original entry on oeis.org

1, 4, 9, 25, 121, 144, 729, 841, 5041, 5184, 45369, 46225, 363609, 403225, 3674889, 1401602635449
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

No other elements < 31!. - Paul.Jobling(AT)WhiteCross.com, Aug 10 2000
If there are any terms in either A014597 or A025494 beyond the last one given (i.e., n = 1183893 in A014597; equivalently n^2 = 1401602635449 in A025494), then n^2 must be greater than 48! (about 1.24139*10^61). - Jon E. Schoenfield, Aug 04 2006

Links

  • Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Crossrefs

Cf. A002034, A018194, A051760, A051761, A014597, A059589.
Subsequence of A059589.

A025493 Smallest k that converges after exactly n iterations of the Kempner function A002034.

Original entry on oeis.org

1, 6, 9, 27, 177147
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

The next term, a(6) = 3^88563, is too large to include.

Crossrefs

Cf. A002034, A018194, A025492.
Showing 1-4 of 4 results.

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