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.

A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2014

Keywords

Links

Crossrefs

Inverse: A250245.
Other similar permutations: A250243, A250248, A250250, A163511, A252756.
Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.
Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

Programs

  • PARI
    up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

Formula

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015 - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018
As a composition of related permutations:
a(n) = A163511(A252756(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].
A001221(a(n)) = A302041(n).
A001222(a(n)) = A253557(n).
A008683(a(n)) = A302050(n).
A000005(a(n)) = A302051(n)
A010052(a(n)) = A302052(n), for n >= 1.
A056239(a(n)) = A302039(n).

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

Original entry on oeis.org

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

Views

Author

Henry Bottomley, Dec 31 2002

Keywords

Comments

Fernandez calls this the order of primeness of n.
a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013
When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014
Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

Examples

			a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.

Links

Crossrefs

A left inverse of A007097.
One less than A049076.
a(A000040(n)) = A049076(n).
Cf. A373338 (mod 2), A018252 (positions of zeros).
Cf. A000720, A049084, A061773.
Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.

Programs

  • Haskell
    a078442 n = fst $ until ((== 0) . snd)
                        (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013
  • Maple
    A078442 := proc(n)
    if not isprime(n) then
        0 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc: # R. J. Mathar, Jul 07 2012
  • Mathematica
    a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)
  • PARI
    A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010

Formula

a(n) = A049076(n)-1.
a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013
For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014

A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83, 84, 175, 86, 105
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

a(n) tells which number in square array A246278 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.
Equally: a(n) tells which number in array A246279 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

Links

Crossrefs

Inverse: A249817.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A078898(n), namely: A250246, A250248 and A250250.
Other similar or related permutations: A249816.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249822.
Differs from its inverse A249817 for the first time at n=33, where a(33) = 45, while A249817(33) = 39.

Programs

  • Mathematica
    lim = 87; a003961[p_?PrimeQ] := a003961[p] = Prime[PrimePi@ p + 1]; a003961[1] = 1; a003961[n_] :=  a003961[n] = Times @@ (a003961[First@ #]^Last@ # &) /@ FactorInteger@ n; a055396[n_] := PrimePi[FactorInteger[n][[1, 1]]]; a078898 = Block[{nn = 90, spfs}, spfs = Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[Count[Take[spfs, i], spfs[[i]]], {i, nn}]]; a246278 = NestList[Map[a003961, #] &, Table[2 k, {k, lim}], lim]; Table[a246278[[a055396@ n, a078898[[n]]]], {n, 2, lim}]
(* Michael De Vlieger, Jan 04 2016, after Harvey P. Dale at A055396 and A078898 *)

Formula

a(1) = 1, a(n) = A246278(A055396(n), A078898(n)).
a(1) = 1, a(n) = A246278(A055396(n), A249822(A055396(n), A246277(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249816(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 78, 79, 80, 45, 82
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2014

Keywords

Comments

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored at the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

Examples

			As a(21) = 27, and A000040(21) = 73 and A000040(27) = 103, a(73) = 103.

Links

Crossrefs

Inverse: A250248.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A083221, A246277.
Differs from its inverse A250248 for the first time at n = 33, where a(33) = 39, while A250248(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=73, where a(73) = 103, while A249817(73) = 73.
Differs from "doubly recursed" version A250249 for the first time at n=42, where a(42) = 42, while A250249(42) = 54, thus the first prime where they get different values is p_42 = 181, where a(181) = 181, while A250249(181) = 251 = p_54.

Formula

a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
Showing 1-4 of 4 results.

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