A349162 -id:A349162 - cp's OEIS frontend

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

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))));

A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), 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, 7, 13, 31, 31, 91, 57, 127, 121, 31, 133, 403, 183, 133, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 427, 1093, 589, 871, 403, 993, 2047, 133, 2149, 1767, 3751, 1407, 889, 2379, 3937, 1723, 1729, 1893, 4123, 3751, 3871, 2257, 6643, 2801, 781, 3991, 1891, 2863, 7651, 589, 2413, 4953, 6097, 3541, 12493

Offset: 1

Antti Karttunen, Dec 12 2021

nonn

Conjecture: There are no 1's after the initial term. Remark: If there were some k = x^2 > 1, for which a(x) = 1, then sigma(k) would be a divisor of A003961(k). In other words, d = A350073(k) = A064989(sigma(k)) would be a divisor of k. Then, if that divisor were also a unitary divisor [with gcd(d,k/d) = 1], it would need to satisfy the equation sigma(k) = sigma(d) * sigma(k/d) = sigma(A064989(sigma(k))) * sigma(k/A064989(sigma(k))), because sigma is a multiplicative function. (Minor correction by Antti Karttunen, Jul 11 2023)

Note that if d = A064989(sigma(k)) were a unitary divisor of a square k, then sigma(k) would also be a square, the cases which are quite rare (see A008848 and A336547). Also compare to A349756. - Antti Karttunen, Jul 24 2022

Cf. A000203, A003961, A008848, A064989, A065764, A349162, A350071, A350073.

See also A069070, A336547, A349756, A364131.

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, Times @@ f2 @@@ f]; Array[a, 60] (* Amiram Eldar, Dec 12 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))); };
A350072(n) = A349162(n^2);

a(n) = A349162(n^2).

a(n) = A065764(n) / A350071(n).

A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 20, 21, 24, 27, 31, 42, 54, 57, 60, 62, 93, 114, 120, 127, 140, 160, 168, 186, 189, 216, 217, 220, 237, 254, 264, 301, 378, 381, 399, 408, 420, 434, 460, 474, 480, 513, 540, 552, 602, 620, 651, 660, 744, 762, 792, 798, 837, 840, 889, 903, 940, 1026, 1080, 1120, 1128, 1140, 1302, 1320, 1380, 1392, 1512

Offset: 1

Antti Karttunen, Dec 03 2021

nonn

Numbers k for which A161942(k) = A342671(k).
From Antti Karttunen, Jul 23 2022: (Start)
Numbers k such that k is a multiple of A350073(k).
For any square s in this sequence, A349162(s) = 1, i.e. sigma(s) divides A003961(s), and also A286385(s). Question: Is 1 the only square in this sequence? (see the conjecture in A350072).
If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
After 2, the only primes present are Mersenne primes, A000668.
(End)

Positions of 1's in A348992.
Positions where the powers of 2 (A000079) occur in A349162.
Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).
Cf. A000668, A046528 (subsequences).
Cf. also A348943.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; oddpart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := oddpart[(sigma = DivisorSigma[1, n])] == GCD[sigma, s[n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };
isA349756(n) = A355946(n);

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]);

A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 2, 7, 3, 1, 2, 3, 13, 9, 6, 14, 7, 1, 1, 31, 9, 39, 5, 21, 4, 9, 4, 1, 31, 7, 10, 14, 15, 3, 16, 9, 4, 27, 1, 7, 19, 5, 14, 9, 21, 1, 11, 6, 39, 3, 8, 62, 3, 31, 3, 49, 9, 5, 9, 1, 5, 45, 30, 7, 31, 12, 26, 127, 7, 3, 17, 63, 8, 3, 36, 39, 37, 19, 62, 35, 4, 7, 20, 93, 11, 63, 14, 28, 27, 11, 5, 9, 45, 117

Offset: 1

Antti Karttunen, Jul 22 2022

Denominator of ratio A003973(n) / A000203(n). See comments in A355933.

Cf. A000203, A003961, A003973, A355932, A355933 (numerators), A355940, A355941 (positions of 1's).

Cf. also A336849, A349162.

f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355934(n) = { my(u=sigma(n)); (u/gcd(A003973(n), u)); };

a(n) = A000203(n) / A355932(n) = A000203(n) / gcd(A000203(n), A003973(n)).

A351544 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor also divides A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 9, 1, 15, 1, 3, 5, 1, 1, 9, 1, 9, 1, 27, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 9, 1, 21, 1, 3, 1, 1, 7, 9, 1, 9, 1, 9, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 5, 9, 1, 9, 1, 3, 1, 9, 1, 9, 1, 9, 13, 7, 1, 27

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Cf. A000203, A003961, A351545, A351546.

Cf. A342671, A349161, A349162.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351544(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); prod(k=1,#f~,if(!(u%f[k,1]), f[k,1]^f[k,2], 1)); };

a(n) = Product_{p^e || A000203(n)} p^(e*[p divides A003961(n)]), where [ ] is the Iverson bracket, returning 1 if p is a divisor of A003961(n), and 0 otherwise. Here p^e is the largest power of prime p dividing sigma(n).
a(n) = A000203(n) / A351546(n).
For all n >= 1, a(n) is a multiple of A351545(n).

A351547 a(n) = sigma(n) / A351545(n).

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 15, 13, 2, 12, 28, 14, 8, 24, 31, 18, 13, 20, 6, 32, 4, 24, 12, 31, 14, 40, 56, 30, 8, 32, 63, 48, 2, 48, 91, 38, 20, 56, 90, 42, 32, 44, 84, 78, 8, 48, 124, 57, 31, 72, 98, 54, 40, 72, 120, 16, 10, 60, 24, 62, 32, 104, 127, 12, 16, 68, 14, 96, 16, 72, 195, 74, 38, 124, 140, 96, 56, 80, 186, 121

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Cf. A000203, A003961, A351545, A351546, A354997.

Cf. A286385, A342671, A349161, A349162.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351547(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); s/prod(k=1,#f~,if(!(u%f[k,1]) && (f[k,2]>=valuation(u,f[k,1])), f[k,1]^f[k,2], 1)); };

a(n) = A000203(n) / A351545(n).

a(n) = A351546(n) * A354997(n). - Antti Karttunen, Jul 09 2022

A354828 Denominators of Dirichlet inverse of fraction A003961(n) / sigma(n).

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 35, 208, 6, 12, 14, 14, 8, 24, 7595, 18, 208, 20, 3, 32, 12, 24, 7, 1116, 14, 832, 28, 30, 24, 32, 7595, 48, 18, 48, 728, 38, 20, 56, 15, 42, 32, 44, 42, 416, 24, 48, 1519, 3648, 1116, 72, 49, 54, 832, 72, 35, 16, 30, 60, 12, 62, 32, 1664, 33759775, 12, 48, 68, 63, 96, 48, 72, 182, 74, 38, 4464

Offset: 1

Antti Karttunen, Jun 07 2022

Cf. A000203, A003961, A349161, A349162.
Cf. A354827 (denominators).
Cf. also A349628, A354366.

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)));
A354828(n) = denominator(vDirInv[n]);

cp's OEIS Frontend

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

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A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), 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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A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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

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A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351544 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor also divides A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351547 a(n) = sigma(n) / A351545(n).

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A354828 Denominators of Dirichlet inverse of fraction A003961(n) / sigma(n).

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