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A003961 Completely multiplicative with a(prime(k)) = prime(k+1). 304
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

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

From Antti Karttunen, Nov 01 2019: (Start)

More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).

Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.

(End)

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Index entries for sequences related to Heinz numbers

Index entries for sequences computed from indices in prime factorization

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_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]

A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019

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

From Peter Munn, Oct 31 2019: (Start)

a(n) = A225546((A225546(n))^2).

a(A225546(n)) = A225546(n^2).

(End)

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.

MAPLE

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):

seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

MATHEMATICA

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)

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

(Python)

from sympy import factorint, prime, primepi, prod

def a(n):

    f=factorint(n)

    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)

[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

CROSSREFS

See A045965 for another version.

Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.

Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.

Cf. A191555, A252738.

Cf. A249734, A249735 (bisections).

Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246282 (A252742).

Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).

Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.

Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.

Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.

Sequence in context: A280702 A269379 A250469 * A332818 A100463 A166722

Adjacent sequences:  A003958 A003959 A003960 * A003962 A003963 A003964

KEYWORD

nonn,mult,nice

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)