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A003961 Completely multiplicative with a(p(k)) = p(k+1) for k-th prime p(k). 145
1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001

A064989(a(n)) = n for all n. [A064989 gives an inverse function for this injection.] - Antti Karttunen, May 20 2014

a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).

Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001

a(n) = product(A000040(A049084(A027748(n,k)+1))^A124010(n,k):k=1..A001221(n)). - Reinhard Zumkeller, Oct 09 2011

A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014

EXAMPLE

a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.

a(A002110(n)) = A002110(n + 1) / 2.

MATHEMATICA

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ ( a[First[#]] ^ Last[#] & ) /@ FactorInteger[n]; Table[ a[n], {n, 1, 65}] (* Jean-Fran├žois Alcover, Dec 01 2011 *)

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

PROG

(PARI) a(n)=local(f); if(n<1, 0, f=factor(n); prod(k=1, matsize(f)[1], nextprime(1+f[k, 1])^f[k, 2]))

(PARI) a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

(Haskell)

a003961 1 = 1

a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n

-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011

(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)

(require 'factor)

(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))

;; Antti Karttunen, May 20 2014

(Perl) use ntheory ":all";  sub a003961 { vecprod(map { next_prime($_) } factor(shift)); }  # Dana Jacobsen, Mar 06 2016

CROSSREFS

See A045965 for another version.

Row 1 of table A242378 (which gives the successive iterates of this sequence).

Cf. A064989 (inverse), A064216, A000040, A002110, A000265, A027746, A049084, A001221, A001222.

Sequence in context: A280702 A269379 A250469 * A100463 A166722 A094549

Adjacent sequences:  A003958 A003959 A003960 * A003962 A003963 A003964

KEYWORD

nonn,mult,nice,changed

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified March 27 06:40 EDT 2017. Contains 284144 sequences.