A118306 - OEIS

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A118306

If n = product{k>=1} p(k)^b(n,k), where p(k) is the k-th prime and where each b(n,k) is a nonnegative integer, then: If n occurs earlier in the sequence, then a(n) = product{k>=2} p(k-1)^b(n,k); If n does not occur earlier in the sequence, then a(n) = product{k>=1} p(k+1)^b(n,k).

1, 3, 2, 9, 7, 15, 5, 27, 4, 21, 13, 45, 11, 33, 6, 81, 19, 75, 17, 63, 10, 39, 29, 135, 49, 51, 8, 99, 23, 105, 37, 243, 14, 57, 77, 225, 31, 69, 22, 189, 43, 165, 41, 117, 12, 87, 53, 405, 25, 147, 26, 153, 47, 375, 91, 297, 34, 93, 61, 315, 59, 111, 20, 729, 119, 195, 71

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Sequence is a permutation of the positive integers and it is its own inverse permutation.

From Antti Karttunen, Nov 05 2016: (Start)

A016945 gives the positions of even terms.

A007310 is closed with respect to this permutation. See A277911 for the permutation induced.

A029744 (without 3) seems to give the positions of records in this sequence (note that it gives the record positions in related A003961 and A048673) which implies that A083658 (without its term 5) would then give the record values.

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences computed from prime indices

FORMULA

From Antti Karttunen, Nov 05 2016: (Start)

a(1) = 1; and for n > 1, if n = a(k) for some k = 1 .. n-1, then a(n) = A064989(n), otherwise a(n) = A003961(n). [After the original definition and R. J. Mathar's Maple-code]

a(1) = 1, and for n > 1, if A055396(n) is odd, a(n) = A003961(n), otherwise a(n) = A064989(n). [The above reduces to this.]

a(n) = product{k>=1} prime(k-((-1)^A055396(n)))^e(k) when n = product{k>=1} prime(k)^e(k).

a(2n) = A249734(n) and a(A249734(n)) = 2n.

A126760(a(A007310(n))) = A277911(n).

For n > 1, A055396(a(n)) = A055396(n) - (-1)^A055396(n). [Permutation sends the terms on any odd row of A246278 to the next even row just below, and vice versa.]

A246277(a(n)) = A246277(n). [While keeping them in the same column.]

a(n) = A064989(A064989(a(A003961(A003961(n))))).

(End)

MAPLE

A064989 := proc(n) local a, ifs, p ; a := 1 ; ifs := ifactors(n)[2] ; for p in ifs do if op(1, p) > 2 then a := a* prevprime(op(1, p))^op(2, p) ; fi ; od; RETURN(a) ; end: A003961 := proc(n) local a, ifs, p ; a := 1 ; ifs := ifactors(n)[2] ; for p in ifs do a := a* nextprime(op(1, p))^op(2, p) ; od; RETURN(a) ; end: A118306 := proc(nmin) local a, anxt, i, n ; a := [1] ; while nops(a) < nmin do n := nops(a)+1 ; if n in a then anxt := A064989(n) ; else anxt := A003961(n) ; fi ; a := [op(a), anxt] ; od; a ; end: A118306(100) ; # R. J. Mathar, Sep 06 2007

PROG

(PARI)

A118306(n) = { if(1==n, 1, my(f = factor(n)); my(d = (-1)^primepi(f[1, 1])); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-d)); factorback(f)); }; \\ Antti Karttunen, Nov 06 2016

for(n=1, 10001, write("b118306.txt", n, " ", A118306(n)));

(Scheme) (define (A118306 n) (cond ((= 1 n) n) ((odd? (A055396 n)) (A003961 n)) (else (A064989 n)))) ;; Antti Karttunen, Nov 05 2016

CROSSREFS

Cf. A003961, A064989, A007310, A016945, A029744, A048673, A055396, A083221, A083658, A246277, A246278, A249734, A277911.

Sequence in context: A021756 A274925 A084398 * A237651 A124003 A159588

Adjacent sequences: A118303 A118304 A118305 * A118307 A118308 A118309

KEYWORD

nonn,look

AUTHOR

Leroy Quet, May 14 2006

EXTENSIONS

More terms from R. J. Mathar, Sep 06 2007

A small omission in the definition corrected by Antti Karttunen, Nov 05 2016

STATUS

approved