1,2

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021

Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021

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)

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

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

Index entries for sequences computed from indices in prime factorization.

Index entries for sequences related to Heinz numbers.

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)

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

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.

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

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

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 *)

(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

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)), A341529 (= a(n)*sigma(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, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), 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.

A version for partition numbers is A003964, strict A357853.

A permutation of A005408.

Applying the same transformation again gives A045966.

Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.

A056239 adds up prime indices, row-sums of A112798.

Cf. A000720, A076610, A296150.

Sequence in context: A280702 A269379 A250469 * A332818 A348437 A100463

Adjacent sequences: A003958 A003959 A003960 * A003962 A003963 A003964

nonn,mult,nice

Marc LeBrun

approved