The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A326042

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A326042

a(n) = A064989(sigma(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.
(history; published version)

#63 by Michael De Vlieger at Sun May 15 23:51:03 EDT 2022

STATUS

proposed

approved

#62 by Antti Karttunen at Sun May 15 14:00:36 EDT 2022

STATUS

editing

proposed

#61 by Antti Karttunen at Sun May 15 14:00:06 EDT 2022

CROSSREFS

Cf. A000037, A000203, A000265, A000593, A003961, A003973, A064989, A161942, A162284, A246282, A286385, A326041, A326182, A336702 (numbers whose abundancy index is a power of 2).

#60 by Antti Karttunen at Fri May 13 15:16:02 EDT 2022

FORMULA

a(n) = A353790(n) / A353767(n) = A353794(n) / A351456(n). - Antti Karttunen, May 13 2022

CROSSREFS

Cf. A348736 [n - a(n)], A348738 [a(n) < n], A348739 [a(n) > n], A348750 [= A064989(a(A003961(n)))], A348940 [gcd(n,a(n))], A348941, A348942, A351456, A353767, A353790, A353794.

STATUS

approved

editing

#59 by Susanna Cuyler at Mon Nov 08 08:17:53 EST 2021

STATUS

proposed

approved

#58 by Amiram Eldar at Sun Nov 07 15:14:44 EST 2021

STATUS

editing

proposed

#57 by Amiram Eldar at Sun Nov 07 15:14:25 EST 2021

MATHEMATICA

f1[p_, e_] := NextPrime[p]^e; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a2[DivisorSigma[1, a1[n]]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)

STATUS

proposed

editing

#56 by Antti Karttunen at Sun Nov 07 12:33:18 EST 2021

STATUS

editing

proposed

#55 by Antti Karttunen at Sun Nov 07 12:28:18 EST 2021

NAME

a(n) = A064989(sigma(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

#54 by Antti Karttunen at Sun Nov 07 12:26:17 EST 2021

COMMENTS

Fixed points k (for which a(k) = k) satisfy A003973(k) = 2^e * A003961(k) for some exponent e >= 0. Applying A003961 to such numbers gives the odd terms in A336702, of which there is most are likely to be just a single instance, its initial 1. (Clarified Nov 07 2021).