A160596 -id:A160596 - cp's OEIS frontend

A340083 a(n) = (n-1) / gcd(n-1, A003958(n)).

0, 1, 1, 3, 1, 5, 1, 7, 2, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 4, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 63, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89, 5, 91, 23

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340081, A340082.

Cf. also A160596, A340073.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340083(n) = ((n-1)/gcd(n-1, A003958(n)));

a(n) = (n-1) / A340081(n) = (n-1) / gcd(n-1, A003958(n)).

A339966 a(n) = (n+1) / gcd(sigma(n),n+1).

Original entry on oeis.org

2, 1, 1, 5, 1, 7, 1, 3, 10, 11, 1, 13, 1, 5, 2, 17, 1, 19, 1, 1, 11, 23, 1, 5, 26, 9, 7, 29, 1, 31, 1, 11, 17, 35, 3, 37, 1, 13, 5, 41, 1, 43, 1, 15, 23, 47, 1, 49, 50, 17, 13, 53, 1, 11, 7, 19, 29, 59, 1, 61, 1, 21, 8, 65, 11, 67, 1, 23, 35, 71, 1, 73, 1, 25, 19, 11, 13, 79, 1, 27, 82, 83, 1, 85, 43, 29, 11, 89, 1, 7

Offset: 1

Antti Karttunen, Dec 25 2020

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A339964, A339965 (numerators).
Cf. A017666.
Cf. also A160596.

PARI
```
A339966(n) = (n+1)/(gcd(sigma(n),n+1));
```

a(n) = (n+1) / A339964(n).

a(n) = denominator(sigma(n)/(n+1)). - Michel Marcus, Jan 07 2023

A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 13, 6, 5, 1, 11, 1, 8, 17, 40, 1, 37, 1, 31, 27, 19, 1, 67, 8, 25, 31, 49, 1, 13, 1, 121, 4, 14, 19, 28, 1, 17, 21, 47, 1, 41, 1, 29, 29, 43, 1, 101, 12, 73, 47, 19, 1, 187, 5, 74, 57, 23, 1, 157, 1, 55, 137, 364, 59, 97, 1, 85, 9, 23, 1, 337, 1, 61, 61, 103, 71, 127, 1, 283, 156, 32, 1, 247, 11

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160596.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160596, A253885, A340071, A340072.

Cf. also A340083.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340073(n) = { my(x=A003961(n)); (x-1)/gcd(x-1, eulerphi(x)); };

a(n) = A160596(A003961(n)).

a(n) = A253885(n-1) / A340071(n) = (A003961(n)-1) / A340071(n).

A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 7, 57, 1, 59, 1, 61, 31, 63, 1, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

From the second term onward, the odd part of A340083.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340083, A340084, A340085.

Cf. also A160596.

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A340086(n) = if(1==n,0,my(u=A336466(n)); A000265(n-1)/gcd(n-1, u));

a(1) = 0; for n > 1, a(n) = A000265(n-1) / A340084(n) = A000265(A340083(n)).

A340089 a(n) = (n-1) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 21, 1, 23, 1, 25, 13, 3, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 7, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 81, 1, 83, 21, 85, 43, 29, 1, 89

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A091732, A340087, A340088.

Cf. also A160596.

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340089(n) = ((n-1)/gcd(n-1, A091732(n)));

a(n) = (n-1) / A340087(n) = (n-1) / gcd(n-1, A091732(n)).

A342916 a(n) = (1+n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

2, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 1, 5, 2, 17, 1, 19, 1, 7, 11, 23, 1, 25, 13, 9, 7, 29, 1, 31, 1, 11, 17, 35, 3, 37, 1, 13, 5, 41, 1, 43, 1, 5, 23, 47, 1, 49, 25, 17, 13, 53, 1, 55, 7, 19, 29, 59, 1, 61, 1, 21, 2, 65, 11, 67, 1, 23, 35, 71, 1, 73, 1, 25, 19, 77, 13, 79, 1, 9, 41, 83, 1, 85, 43, 29, 11, 89, 1, 91, 23, 31

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

It is conjectured that a(n) = 1 only when n is a prime, A000040. See Thomas Ordowski's May 21 2017 problem in A001615.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000040, A001615, A342915, A342917.
Cf. also A160596.
After n=1 differs from A342918 for the first time at n=44, where a(44) = 5, while A342918(44) = 15.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342916(n) = ((1+n)/gcd(1+n,A001615(n)));

a(n) = (1+n) / A342915(n) = (1+n) / gcd(1+n, A001615(n)).

Incorrect A-number in the formula corrected by Antti Karttunen, May 31 2021

A345939 a(n) = (n-1) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 9, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 21, 1, 23, 1, 25, 1, 3, 1, 29, 1, 1, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 1, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 81, 1, 83, 21, 85, 43, 87, 1, 89, 5, 91

Offset: 1

Antti Karttunen, Jun 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A047994, A345937, A345938.

Cf. also A160596, A340089, A345949.

uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);
a[n_]:=(n-1)/GCD[n-1,uphi[n]];Array[a,100] (* Giorgos Kalogeropoulos, Jul 02 2021 *)

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345939(n) = ((n-1) / gcd(n-1, A047994(n)));

a(n) = (n-1) / A345937(n) = (n-1) / gcd(n-1, A047994(n)).

a(2n-1) = A345949(2n-1), for n > 1.

A160598 Numerator of coresilience C(n) = (n - phi(n))/(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 8, 1, 8, 1, 8, 1, 12, 1, 12, 9, 4, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 6, 11, 24, 1, 20, 15, 8, 1, 30, 1, 24, 21, 8, 1, 32, 7, 30, 19, 28, 1, 36, 5, 32, 3, 10, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 2, 1, 48, 1, 38, 35, 8, 17, 54, 1, 48, 27, 14, 1, 60, 1

Offset: 2

M. F. Hasler, May 23 2009

Obviously C(p) = 1/(p-1), i.e., a(p)=1, for all primes p. Sequence A160599 lists composite numbers for which this is the case.

			a(10)=2 since for n=10, we have (n - phi(n))/(n-1) = (10-4)/9 = 2/3.

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
Project Euler, Problem 245: resilient fractions, May 2009

Cf. A160595, A160596, A160597, A160599.

[Numerator((n-EulerPhi(n))/(n-1)): n in [2..80]]; // Vincenzo Librandi, Dec 27 2016

Numerator[Table[(n - EulerPhi[n])/(n - 1), {n, 2, 90}]] (* Vincenzo Librandi, Dec 27 2016 *)

A160598(n)=numerator((n-eulerphi(n))/(n-1))

cp's OEIS Frontend

A340083 a(n) = (n-1) / gcd(n-1, A003958(n)).

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A339966 a(n) = (n+1) / gcd(sigma(n),n+1).

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A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

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A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

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A340089 a(n) = (n-1) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A342916 a(n) = (1+n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A345939 a(n) = (n-1) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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Mathematica

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A160598 Numerator of coresilience C(n) = (n - phi(n))/(n-1).

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