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 1-3 of 3 results.

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206
Offset: 1

Views

Author

Antti Karttunen, Jul 14 1999

Keywords

Comments

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

Examples

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Crossrefs

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

Programs

  • Maple
    A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;
  • Mathematica
    Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Extensions

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014
Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A285706 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from prime(n), a(1) = a(2) = 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 26 2017

Keywords

Comments

Length (or size for the closed cycles: [2] and [3]) of the complete "slipping Cunningham chain of the second kind" starting with prime(n). That is, at the end of every step, the next prime q = 2p-1 "slips" by one step towards smaller primes as A064989(q).
After n = 1, 2 (primes 2 & 3) differs from A181715 for the first time at n=22, where a(22) = 2, while A181715(22) = 3, prime(22) = 79.

Examples

			See examples in A285701.

Links

Crossrefs

Cf. A000040, A064216, A064989, A181715, A246373, A285701.
Cf. A137288 (gives the positions of terms > 1 after its two initial terms).

Programs

  • Mathematica
    Table[If[n <= 2, 1, -1 + Length@ NestWhileList[Apply[Times, FactorInteger[2 # - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e] &, Prime@ n, PrimeQ@ # &]], {n, 120}] (* Michael De Vlieger, Apr 26 2017 *)
  • PARI
    A285706(n) = A285701(prime(n)); \\ The rest of code in A285701.
  • Scheme
    (define (A285706 n) (A285701 (A000040 n)))

Formula

a(n) = A285701(A000040(n)).

A285700 a(n) = Number of iterations x -> 2x-1 needed to get a nonprime number, when starting with x = n.

Original entry on oeis.org

0, 3, 2, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Apr 26 2017

Keywords

Links

Crossrefs

Cf. A000040, A005382 (positions of terms > 1), A010051, A181715, A285701.

Programs

  • Mathematica
    Array[Length@ NestWhileList[2 # - 1 &, #, PrimeQ@ # &] - 1 &, 120] (* Michael De Vlieger, Apr 26 2017 *)
  • Python
    from sympy import isprime
def a(n): return 0 if isprime(n) == 0 else 1 + a(2*n - 1) # Indranil Ghosh, Apr 26 2017
  • Scheme
    (define (A285700 n) (if (zero? (A010051 n)) 0 (+ 1 (A285700 (+ n n -1)))))

Formula

If A010051(n) = 0 [when n is a nonprime], a(n) = 0, otherwise a(n) = 1 + a((2*n)-1).
a(A000040(n)) = A181715(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.