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%I #139 Nov 18 2022 07:11:49
%S 1,3,5,9,7,15,11,27,25,21,13,45,17,33,35,81,19,75,23,63,55,39,29,135,
%T 49,51,125,99,31,105,37,243,65,57,77,225,41,69,85,189,43,165,47,117,
%U 175,87,53,405,121,147,95,153,59,375,91,297,115,93,61,315,67,111,275,729,119
%N Completely multiplicative with a(prime(k)) = prime(k+1).
%C 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
%C 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
%C 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
%C a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - _Michel Marcus_, Jun 13 2014
%C From _Antti Karttunen_, Nov 01 2019: (Start)
%C 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).
%C 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.
%C (End)
%D Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).
%H Indranil Ghosh, <a href="/A003961/b003961.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>.
%F If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
%F Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - _David W. Wilson_, Aug 01 2001
%F 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]
%F A064989(a(n)) = n and a(A064989(n)) = A000265(n). - _Antti Karttunen_, May 20 2014 & Nov 01 2019
%F A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - _Michel Marcus_, Jun 13 2014
%F From _Peter Munn_, Oct 31 2019: (Start)
%F a(n) = A225546((A225546(n))^2).
%F a(A225546(n)) = A225546(n^2).
%F (End)
%F 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
%e a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
%e a(A002110(n)) = A002110(n + 1) / 2.
%p a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Sep 13 2017
%t 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 *)
%t 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 *)
%o (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]))
%o (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
%o (Haskell)
%o a003961 1 = 1
%o a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
%o -- _Reinhard Zumkeller_, Apr 09 2012, Oct 09 2011
%o (MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
%o (require 'factor)
%o (define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
%o ;; _Antti Karttunen_, May 20 2014
%o (Perl) use ntheory ":all"; sub a003961 { vecprod(map { next_prime($_) } factor(shift)); } # _Dana Jacobsen_, Mar 06 2016
%o (Python)
%o from sympy import factorint, prime, primepi, prod
%o def a(n):
%o f=factorint(n)
%o return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
%o [a(n) for n in range(1, 11)] # _Indranil Ghosh_, May 13 2017
%Y See A045965 for another version.
%Y 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.
%Y 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.
%Y Cf. A191555, A252738.
%Y Cf. A249734, A249735 (bisections).
%Y 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.
%Y Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
%Y Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
%Y Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
%Y 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.
%Y A version for partition numbers is A003964, strict A357853.
%Y A permutation of A005408.
%Y Applying the same transformation again gives A045966.
%Y Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
%Y A056239 adds up prime indices, row-sums of A112798.
%Y Cf. A000720, A076610, A296150.
%K nonn,mult,nice
%O 1,2
%A _Marc LeBrun_
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