A349627 -id:A349627 - cp's OEIS frontend

A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

1, 1, 5, 9, 7, 5, 11, 9, 25, 7, 13, 45, 17, 11, 35, 81, 19, 25, 23, 3, 55, 13, 29, 9, 49, 17, 25, 99, 31, 35, 37, 27, 65, 19, 77, 225, 41, 23, 85, 21, 43, 55, 47, 39, 175, 29, 53, 405, 121, 49, 95, 153, 59, 25, 91, 99, 23, 31, 61, 15, 67, 37, 275, 729, 17, 65, 71, 19, 145, 77, 73, 45, 79, 41, 245, 207, 143, 85, 83

Offset: 1

Antti Karttunen, Nov 09 2021

Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
Both ratios are multiplicative because the constituent sequences are.
No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).

Cf. A000203, A003961, A161942, A286385, A319626, A319627, A326042, A336702, A342671, A349163, A349164, A349627, A349628.

Cf. A348992, A349162 (denominators).

Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349161(n) = { my(u=A003961(n)); (u/gcd(u,sigma(n))); };

from math import prod, gcd
from sympy import nextprime, factorint
def A349161(n):
    f = factorint(n).items()
    a = prod(nextprime(p)**e for p, e in f)
    b = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return a//gcd(a,b) # Chai Wah Wu, Mar 17 2023

a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).

a(n) = A003961(A349164(n)).

A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 5, 13, 6, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 12, 24, 4, 31, 14, 8, 56, 30, 24, 32, 7, 48, 18, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 24, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 30, 60, 8, 62, 32, 104, 127, 12, 48, 68, 14, 96, 48, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 42

Offset: 1

Antti Karttunen, Nov 09 2021

Denominator of ratio A003961(n) / A000203(n).
Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.
Question: Would it be possible to prove that a(n) > 1 for all n > 2?
Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - Antti Karttunen, Feb 16 2022

Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].

Cf. also A349745, A351551, A351554.

Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

a(n) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

A349628 Denominators of the Möbius transform of ratio A003961(n)/sigma(n).

Original entry on oeis.org

1, 1, 4, 7, 6, 1, 8, 35, 52, 1, 12, 14, 14, 1, 24, 155, 18, 1, 20, 21, 32, 1, 24, 70, 186, 1, 104, 28, 30, 1, 32, 217, 48, 1, 16, 26, 38, 1, 56, 35, 42, 1, 44, 42, 312, 1, 48, 310, 456, 1, 72, 49, 54, 1, 72, 140, 80, 1, 60, 84, 62, 1, 416, 889, 28, 1, 68, 63, 96, 1, 72, 26, 74, 1, 744, 70, 32, 1, 80, 155, 968, 1, 84

Offset: 1

Antti Karttunen, Nov 26 2021

Because the ratio A003961(n)/A000203(n) is multiplicative, so is also its Möbius transform. This sequence gives the denominator of that ratio when presented in its lowest terms.

Cf. A000203, A003961.
Cf. A349627 (numerators).
Cf. also A342671, A349161, A349162, A349634.

f[p_, e_] := NextPrime[p]^e*(p - 1)/(p^(e + 1) - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Denominator @ DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349628(n) = denominator(sumdiv(n,d,moebius(n/d)*(A003961(d)/sigma(d))));

A354827 Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).

Original entry on oeis.org

1, -1, -5, -2, -7, 5, -11, -8, -75, 7, -13, 5, -17, 11, 35, -1648, -19, 75, -23, 1, 55, 13, -29, 2, -245, 17, -225, 11, -31, -35, -37, -1664, 65, 19, 77, 75, -41, 23, 85, 4, -43, -55, -47, 13, 175, 29, -53, 412, -847, 245, 95, 17, -59, 225, 91, 11, 23, 31, -61, -5, -67, 37, 825, -7662464, 17, -65, -71, 19, 145, -77

Offset: 1

Antti Karttunen, Jun 07 2022

Because the ratio A003961(n) / A000203(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354828 gives the denominators. See the examples.

			The ratio a(n)/A354828(n) for n = 1..21: 1, -1, -5/4, -2/7, -7/6, 5/4, -11/8, -8/35, -75/208, 7/6, -13/12, 5/14, -17/14, 11/8, 35/24, -1648/7595, -19/18, 75/208, -23/20, 1/3, 55/32.

Cf. A000203, A003961, A349161, A349162.
Cf. A354828 (denominators).
Cf. also A349627, A354365.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA354827(n) = (A003961(n)/sigma(n));
vDirInv = DirInverseCorrect(vector(up_to,n,AuxA354827(n)));
A354827(n) = numerator(vDirInv[n]);
A354828(n) = denominator(vDirInv[n]);

A349633 Numerators of the Möbius transform of ratio A003961(n)/n.

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 4, 9, 10, 1, 2, 1, 4, 2, 4, 27, 2, 5, 4, 3, 8, 1, 6, 3, 14, 2, 50, 3, 2, 2, 6, 81, 4, 1, 8, 5, 4, 2, 8, 9, 2, 4, 4, 3, 4, 3, 6, 9, 44, 7, 4, 3, 6, 25, 4, 9, 8, 1, 2, 1, 6, 3, 40, 243, 8, 2, 4, 3, 4, 4, 2, 5, 6, 2, 28, 3, 8, 4, 4, 27, 250, 1, 6, 2, 4, 2, 4, 9, 8, 2, 16, 9, 4, 3, 8, 27, 4, 22, 20

Offset: 1

Antti Karttunen, Nov 28 2021

Because the ratio A003961(n)/n is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349634 gives the denominators. See the examples.

			The ratio a(n)/A349634(n) for n = 1..15: 1/1, 1/2, 2/3, 3/4, 2/5, 1/3, 4/7, 9/8, 10/9, 1/5, 2/11, 1/2, 4/13, 2/7, 4/15.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A008683, A349634 (denominators).

Cf. also A349627.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, MoebiusMu[n/#] * s[#]/# &]; Array[a, 100] (* Amiram Eldar, Nov 28 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349633(n) = numerator(sumdiv(n,d,moebius(n/d)*(A003961(d)/d)));

A378659 Positions of records for ratio A003961(n)/sigma(n), where A003961 is fully multiplicative with a(p) = nextprime(p) and sigma is the sum of the divisors function.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 16, 27, 32, 64, 128, 243, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

			  Term k       A003961(k)/A000203(k)    ratio
       1                1/1          =  1
       3                5/4          =  1.25
       4                9/7          =  1.2857143
       7               11/8          =  1.375
       8               27/15         =  1.8
       9               25/13         =  1.9230769
      16               81/31         =  2.6129032
      27              125/40         =  3.125
      32              243/63         =  3.8571429
      64              729/127        =  5.7401575
     128             2187/255        =  8.5764706
     243             3125/364        =  8.5851648
     256             6561/511        =  12.839530
     ...
33554432     847288609443/67108863   =  12625.584
67108864    2541865828329/134217727  =  18938.376

Cf. A000203, A003961.
Cf. A001359 (apparently gives the positions of successive minima of the ratio, for n > 2).
Cf. also A341529, A349627, A349628, A354827, A354828.

cp's OEIS Frontend

A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A349628 Denominators of the Möbius transform of ratio A003961(n)/sigma(n).

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A354827 Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).

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A349633 Numerators of the Möbius transform of ratio A003961(n)/n.

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A378659 Positions of records for ratio A003961(n)/sigma(n), where A003961 is fully multiplicative with a(p) = nextprime(p) and sigma is the sum of the divisors function.

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