A348941 -id:A348941 - cp's OEIS frontend

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.

1, 1, 2, 11, 1, 2, 2, 3, 29, 1, 5, 22, 4, 2, 2, 49, 3, 29, 2, 11, 4, 5, 6, 6, 34, 4, 22, 22, 1, 2, 17, 55, 10, 3, 2, 319, 10, 2, 8, 3, 7, 4, 2, 55, 29, 6, 8, 98, 85, 34, 6, 44, 6, 22, 5, 6, 4, 1, 29, 22, 13, 17, 58, 1091, 4, 10, 4, 33, 12, 2, 31, 87, 3, 10, 68, 22, 10, 8, 10, 49, 469, 7, 12, 44, 3, 2, 2, 15, 25, 29, 8, 66, 34, 8

Offset: 1

Antti Karttunen, Jun 16 2019

For any other number n than those in A326182 we have a(n) < A003961(n).
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 are likely to be just a single instance, its initial 1. (Clarified Nov 07 2021).
Conjecture: There are no other fixed points than a(1) = 1. If true, then there are no odd perfect numbers. This condition is equivalent to the condition that if A161942 has no fixed points larger than one, then there are no odd perfect numbers. This follows as whenever k is a fixed point, that is, a(k) = k, then we should also have A003961(a(k)) = A003961(A064989(sigma(A003961(k)))) = A161942(A003961(k)) = A003961(k). Note that A003961 is an injective and surjective mapping from natural numbers to odd numbers, A064989 is its (left) inverse, and composition A003961(A064989(n)) is equivalent to A000265(n).
From Antti Karttunen, Aug 05 2020: (Start)
For any hypothetical odd perfect number x, we would have A003973(k) = 2 * A003961(k), with k = A064989(x) and x = A003961(k). Thus we would have a(k) = A064989(sigma(A003961(k))) = A064989(sigma(x)) = A064989(2*x) = A064989(x) = k. On the other hand, A003973(k) = sigma(A003961(k)) < A003961(A003961(k)) [see A286385 for the reason why], so a necessary condition for this is that x should be one of the terms of A246282. (Clarified Dec 01 2020).
(End)

Cf. A000037, A000203, A000265, A000593, A003961, A003973, A064989, A161942, A162284, A246282, A286385, A326041, A326182, A336702 (numbers whose abundancy index is a power of 2).
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.
Cf. also A332223 for another conjugation of sigma.

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 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));

a(n) = A064989(A003973(n)) = A064989(sigma(A003961(n))).
For k in A000037, a(k) = A064989(A003973(k)/2) = A064989((1/2)*sigma(A003961(k))).
Multiplicative with a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p). - Antti Karttunen, Nov 05 2021
a(n) = A353790(n) / A353767(n) = A353794(n) / A351456(n). - Antti Karttunen, May 13 2022

Keyword:mult added by Antti Karttunen, Nov 05 2021

A336702 Numbers whose abundancy index is a power of 2.

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Antti Karttunen, Aug 05 2020

nonn

Apart from missing 2, this sequence gives all numbers k such that the binary expansion of A156552(k) is a prefix of that of A156552(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in A005940-tree. This follows because of the two transitions x -> A005843(x) (doubling) and x -> A003961(x) (prime shift) used to generate descendants in A005940-tree, using A003961 at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.
Proof: Any left child in A005940 (i.e., A003961(k) for k) is larger than sigma(k), for any k > 2 [see A286385 for a proof], and A003961(n) > n for all n > 1. Thus, apart from A003961(2) = 3 = sigma(2), A003961^t(k) > sigma(k), where A003961^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.
The first terms of A325637 not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.
From Antti Karttunen, Nov 29 2021: (Start)
Odd terms of this sequence are given by the intersection of A349169 and A349174.
A064989 applied to the odd terms of this sequence gives the fixed points of A326042, i.e., the positions of zeros in A348736, and a subset of the positions of ones in A348941.
Odd terms of this sequence form a subsequence of A348943, but should occur neither in A348748 nor in A348749.
(End)

			For 30240, sigma(30240) = 120960 = 4*30240, therefore, as sigma(k)/k = 2^2, a power of two, 30240 is present.

Antti Karttunen, Table of n, a(n) for n = 1..769 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005843, A005940, A156552, A163511, A209229, A286385.
Cf. A000396, A027687 (subsequences).
Subsequence of A007691, and after 1, also subsequence of A325637.
Union with {2} gives the positions of zeros in A347381.
Cf. also A326042, A347391, A347392, A347393, A347394, A348736, A348748, A348749, A348941, A348943, A349169, A349174.

isA336702(n) = { my(r=sigma(n)/n); (1==denominator(r)&&!bitand(r, r-1)); }; \\ (Corrected) - Antti Karttunen, Aug 31 2021

A348942 a(n) = A326042(n) / gcd(n, A326042(n)).

Original entry on oeis.org

1, 1, 2, 11, 1, 1, 2, 3, 29, 1, 5, 11, 4, 1, 2, 49, 3, 29, 2, 11, 4, 5, 6, 1, 34, 2, 22, 11, 1, 1, 17, 55, 10, 3, 2, 319, 10, 1, 8, 3, 7, 2, 2, 5, 29, 3, 8, 49, 85, 17, 2, 11, 6, 11, 1, 3, 4, 1, 29, 11, 13, 17, 58, 1091, 4, 5, 4, 33, 4, 1, 31, 29, 3, 5, 68, 11, 10, 4, 10, 49, 469, 7, 12, 11, 3, 1, 2, 15, 25, 29, 8

Offset: 1

Antti Karttunen, Nov 04 2021

Numerator of ratio A326042(n)/n. Ratio A326042(n)/n is multiplicative because both A326042 and A000027 are.

Cf. A000203, A003961, A005820, A007691, A046060, A064989, A326042, A348739, A348940, A348941 (denominators), A348943.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (sn = s2[n])/GCD[n, sn]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };

a(n) = A326042(n) / A348940(n) = A326042(n) / gcd(n, A326042(n)).

For all n >= 1, A348943(A064989(n)) = 1.

A348940 a(n) = gcd(n, A326042(n)), where A326042 is multiplicative function A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 11, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 6, 1, 4, 1

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Cf. A326042, A348736, A348941, A348942.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := GCD[n, s2[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348940(n) = gcd(n, A326042(n));

a(n) = gcd(n, A326042(n)).
a(n) = gcd(n, A348736(n)) = gcd(A326042(n), A348736(n));
a(n) = n / A348941(n) = A326042(n) / A348942(n).

cp's OEIS Frontend

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.

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A336702 Numbers whose abundancy index is a power of 2.

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A348942 a(n) = A326042(n) / gcd(n, A326042(n)).

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A348940 a(n) = gcd(n, A326042(n)), where A326042 is multiplicative function A064989(sigma(A003961(n))).

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