A326041 -id:A326041 - 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

A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Original entry on oeis.org

1, 2, 9, 4, 20, 18, 42, 8, 63, 40, 88, 36, 156, 84, 180, 16, 238, 126, 342, 80, 378, 176, 460, 72, 325, 312, 405, 168, 696, 360, 930, 32, 792, 476, 840, 252, 1184, 684, 1404, 160, 1558, 756, 1806, 352, 1260, 920, 2068, 144, 1519, 650, 2142, 624, 2544, 810, 1760, 336, 3078, 1392, 3186, 720, 3660, 1860, 2646, 64, 3120

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A151799, A326041, A341528, A341529 [= a(A003961(n))], A342662, A342663, A342664.

f[p_, e_] := If[p == 2, 2^e, Module[{q = NextPrime[p, -1]}, p^e*(q^(e + 1) - 1)/(q - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

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) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));

Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
a(n) = n * A326041(n) = n * A000203(A064989(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/9) * Product_{p prime > 2} (p^3/((p+1)*(p^2-prevprime(p)))) = 0.1815217..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

A342664 Denominator of ratio A342661(n)/A342662(n).

Original entry on oeis.org

1, 3, 8, 7, 9, 4, 20, 15, 52, 27, 21, 14, 77, 10, 4, 31, 117, 26, 170, 63, 160, 63, 114, 5, 279, 77, 64, 5, 115, 6, 464, 63, 28, 351, 6, 13, 589, 85, 308, 27, 777, 80, 902, 147, 26, 171, 516, 31, 1425, 837, 104, 539, 423, 32, 189, 25, 1360, 345, 530, 7, 1829, 232, 1040, 127, 231, 14, 2074, 117, 304, 9, 1206, 65, 2627

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A326041, A341526 [= a(A003961(n))], A341527, A342661, A342662, A342663 (numerators), A342667 [largest prime factor of a(A003961(n))], A342670.

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) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342664(n) = { my(v=A342662(n)); (v/gcd(A342661(n),v)); };
\\ Alternatively as:
A342664(n) = denominator(A342661(n)/A342662(n));

a(n) = A342662(n) / A342670(n) = A342662(n) / gcd(A342661(n), A342662(n)).

A342663 Numerator of ratio A342661(n)/A342662(n): a(n) = A342661(n) / gcd(A342661(n), A342662(n)).

Original entry on oeis.org

1, 2, 9, 4, 10, 3, 21, 8, 63, 20, 22, 9, 78, 7, 5, 16, 119, 21, 171, 40, 189, 44, 115, 3, 325, 52, 81, 3, 116, 5, 465, 32, 33, 238, 7, 9, 592, 57, 351, 16, 779, 63, 903, 88, 35, 115, 517, 18, 1519, 650, 119, 312, 424, 27, 220, 14, 1539, 232, 531, 5, 1830, 155, 1323, 64, 260, 11, 2077, 68, 345, 7, 1207, 42, 2628, 1184

Offset: 1

Antti Karttunen, Mar 23 2021

Let r(row,col) = A341605(row,col)/A341606(row,col) and d(n) = A342661(n)/A342661(n) = A342663(n)/A342664(n). Then for row > 1, r(row-1,col) = d(A246278(row,col)) * r(row,col).

Cf. A000203, A064989, A246278, A326041, A341526, A341527 [= a(A003961(n))], A341605, A341606, A342661, A342662, A342664 (denominators), A342670.

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) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342663(n) = { my(u=A342661(n)); (u/gcd(u, A342662(n))); };
\\ Alternatively as:
A342663(n) = numerator(A342661(n)/A342662(n));

a(n) = A342661(n) / A342670(n) = A342661(n) / gcd(A342661(n), A342662(n)).

A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

Original entry on oeis.org

1, 2, 3, 13, 5, 2, 7, 2, 31, 5, 11, 13, 13, 7, 3, 11, 17, 31, 19, 13, 7, 11, 23, 2, 19, 13, 13, 13, 29, 2, 31, 13, 11, 17, 5, 31, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 11, 7, 19, 17, 13, 53, 13, 11, 7, 19, 29, 59, 13, 61, 31, 31, 1093, 13, 11, 67, 17, 23, 5, 71, 31, 73, 37, 19, 19, 7, 13, 79, 11, 71, 41, 83, 13

Offset: 1

Antti Karttunen, Mar 23 2021

nonn

Equally, largest prime in the denominator of ratio A342661(A003961(n)) / A342662(A003961(n)).

Cf. A000203, A003961, A006530, A064989, A341526, A341528, A341529, A326041, A342661, A342662, A342664, A342668.

Cf. also A341607.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };
A342667(n) = A006530(A341526(n));

a(n) = A006530(A341526(n)).

a(n) = A006530(A342664(A003961(n))).

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 2, 1, 1, 2, 4, 4, 2, 12, 36, 1, 2, 6, 2, 2, 2, 4, 4, 24, 1, 6, 5, 56, 6, 72, 2, 1, 24, 2, 120, 28, 2, 12, 4, 10, 2, 12, 2, 4, 36, 8, 4, 8, 1, 1, 18, 2, 6, 30, 8, 24, 2, 6, 6, 144, 2, 12, 2, 1, 12, 144, 2, 14, 12, 240, 4, 12, 2, 2, 9, 4, 336, 24, 2, 2, 1, 2, 4, 56, 4, 12, 24, 4, 6, 72, 56, 8, 2, 8, 360

Offset: 1

Antti Karttunen, Mar 24 2021

nonn

Cf. A000203, A064989, A326041, A341530 [ = a(A003961(n))], A342661, A342662, A342663, A342664.

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) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342670(n) = gcd(A342661(n), A342662(n));

a(n) = gcd(A342661(n), A342662(n)).

a(n) = gcd(n*A000203(A064989(n)), A000203(n)*A064989(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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A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

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A342664 Denominator of ratio A342661(n)/A342662(n).

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A342663 Numerator of ratio A342661(n)/A342662(n): a(n) = A342661(n) / gcd(A342661(n), A342662(n)).

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A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

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A342670 a(n) = gcd(n*sigma(A064989(n)), sigma(n)*A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

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A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.