A160595 -id:A160595 - cp's OEIS frontend

A323374 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A323373(n) for all other numbers, except f(p) = -(p mod 2) for primes p.

1, 2, 3, 4, 3, 4, 3, 5, 6, 5, 3, 5, 3, 7, 8, 9, 3, 7, 3, 9, 10, 11, 3, 9, 12, 13, 14, 15, 3, 9, 3, 16, 12, 16, 17, 13, 3, 18, 17, 16, 3, 13, 3, 19, 20, 21, 3, 16, 22, 19, 23, 24, 3, 18, 25, 26, 27, 28, 3, 16, 3, 29, 30, 31, 32, 33, 3, 31, 34, 24, 3, 26, 3, 35, 25, 36, 37, 26, 3, 31, 38, 39, 3, 26, 40, 41, 42, 39, 3, 26, 43, 44, 37, 45, 46, 31, 3, 41, 47, 39, 3, 31, 3, 48

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039651(i) = A039651(j).

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

Cf. A039651, A049559, A160595, A323373.

Cf. also A322587, A322592, A323405.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A049559(n) = gcd(eulerphi(n), n-1);
A160595(n) = if(1==n, n, numerator(eulerphi(n)/(n-1)));
Aux323374(n) = if(isprime(n),-(n%2),[A049559(n), A160595(n)]);
v323374 = rgs_transform(vector(up_to, n, Aux323374(n)));
A323374(n) = v323374[n];

A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 6, 1, 12, 8, 2, 1, 8, 1, 15, 5, 16, 12, 24, 1, 18, 12, 4, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 16, 20, 18, 9, 28, 1, 24, 1, 30, 24, 5, 3, 4, 1, 48, 11, 8, 1, 24, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 10, 1, 32, 4, 66

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Lehmer's totient problem

Cf. A050376, A084400, A091732, A302777, A340087, A340089.

Cf. also A160595.

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); };
A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

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

For all n >= 1, a(A084400(n)) = 1.

A345938 a(n) = uphi(n) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 14, 1, 12, 1, 2, 1, 8, 1, 1, 5, 16, 12, 24, 1, 18, 12, 28, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 26, 20, 42, 9, 28, 1, 24, 1, 30, 24, 1, 3, 4, 1, 48, 11, 8, 1, 56, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 70, 1, 32, 4, 66, 15

Offset: 1

Antti Karttunen, Jun 29 2021

nonn

For all squarefree n (A005117), a(n) = A160595(n), thus if there are any composite solutions to the Lehmer's totient conjecture, then they give also a such a subset of positions of 1's in this sequence that are not powers of primes. See comments in A160595.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Lehmer's totient problem

Cf. A005117, A047994, A345937, A345939.

Cf. also A160595, A340088, A345948.

uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);
a[n_]:=uphi[n]/GCD[n-1,uphi[n]];Array[a,100]  (* Giorgos Kalogeropoulos, Jun 30 2021 *)

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345938(n) = { my(u=A047994(n)); (u/gcd(n-1, u)); };

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

a(2n-1) = A345948(2n-1), for all 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))

A339965 a(n) = sigma(n) / gcd(sigma(n),n+1).

Original entry on oeis.org

1, 1, 1, 7, 1, 12, 1, 5, 13, 18, 1, 28, 1, 8, 3, 31, 1, 39, 1, 2, 16, 36, 1, 12, 31, 14, 10, 56, 1, 72, 1, 21, 24, 54, 4, 91, 1, 20, 7, 90, 1, 96, 1, 28, 39, 72, 1, 124, 57, 31, 18, 98, 1, 24, 9, 40, 40, 90, 1, 168, 1, 32, 13, 127, 14, 144, 1, 42, 48, 144, 1, 195, 1, 38, 31, 20, 16, 168, 1, 62, 121, 126, 1, 224, 54, 44

Offset: 1

Antti Karttunen, Dec 25 2020

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A339964, A339966 (denominators).
Cf. A017665.
Cf. also A160595.

Table[DivisorSigma[1, n]/GCD[DivisorSigma[1, n], n + 1], {n, 80}] (* Wesley Ivan Hurt, Oct 10 2021 *)

A339965(n) = sigma(n)/(gcd(sigma(n),n+1));

a(n) = A000203(n) / A339964(n).

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

A342917 a(n) = A001615(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

1, 1, 1, 6, 1, 12, 1, 4, 6, 18, 1, 24, 1, 8, 3, 24, 1, 36, 1, 12, 16, 36, 1, 48, 15, 14, 9, 48, 1, 72, 1, 16, 24, 54, 4, 72, 1, 20, 7, 72, 1, 96, 1, 8, 36, 72, 1, 96, 28, 30, 18, 84, 1, 108, 9, 32, 40, 90, 1, 144, 1, 32, 3, 96, 14, 144, 1, 36, 48, 144, 1, 144, 1, 38, 30, 120, 16, 168, 1, 16, 54, 126, 1, 192, 54, 44, 15, 144

Offset: 1

Antti Karttunen, Mar 29 2021

The scatter plot shows two distinct "fans" separated by a gap. Why?

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

Cf. A001615, A342915, A342916.

Cf. also A160595.

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
A342917(n) = { my(u=A001615(n)); (u/gcd(1+n,u)); };

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

A339871 Number of primes p for which the p-adic valuation of phi(n) exceeds the p-adic valuation of n-1, with a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 2, 0, 1, 1, 2, 1, 1, 0, 2, 2, 2, 1, 2, 0, 1, 0, 3, 2, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 1, 1, 2, 0, 2, 1, 3, 2, 2, 0, 2, 1, 2, 2, 2, 2, 1, 0, 3, 3, 2, 0, 1, 0, 2, 2

Offset: 1

Antti Karttunen, Dec 20 2020

nonn

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

Cf. A001221, A055734, A160595, A339817, A339872.

A339871(n) = if(1==n,0,my(s=0); for(k=1,n,my(p=prime(k)); if(valuation(eulerphi(n),p)>valuation(n-1,p), s++)); (s));

A339871(n) = if(1==n,0,my(f=factor(eulerphi(n))); sum(i=1,#f~,f[i,2]>valuation(n-1,f[i,1])));

A339871(n) = omega(eulerphi(n)/gcd(n-1,eulerphi(n)));

a(n) = A001221(A160595(n)).

a(n) <= A055734(n).

A339872 Index k of the least prime(k) such that prime(k)-adic valuation of phi(n) exceeds the prime(k)-adic valuation of n-1, or 0 if no such k exists (for example, when n = 1 or a prime).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 3, 1, 2, 1, 0, 1, 0, 1, 3, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 5, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Antti Karttunen, Dec 20 2020

nonn

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

Cf. A055396, A160595, A339817, A339871.

A339872(n) = if(1==n,0,for(k=1,n,my(p=prime(k)); if(valuation(eulerphi(n),p)>valuation(n-1,p), return(k))); (0));

A339872(n) = if(1==n,0,my(f=factor(eulerphi(n))); for(i=1,#f~,if(f[i,2]>valuation(n-1,f[i,1]), return(primepi(f[i,1])))); (0));

A339872(n) = { my(t=eulerphi(n), x=t/gcd(n-1,t)); if(1==x,0,primepi(factor(x)[1, 1])); };

a(n) = A055396(A160595(n)).

A195418 a(n) = phi(C(n)) / gcd(C(n)-1, phi(C(n))), where C(n) is the n-th Cullen number.

Original entry on oeis.org

1, 1, 3, 5, 3, 33, 5, 33, 341, 1045, 189, 1299, 891, 4437, 9477, 581, 3855, 105525, 27825, 23751, 173043, 10531345, 56511, 2386125, 380955, 256861, 24926139, 5108467, 32397379, 930343095, 930291, 36512775

Offset: 0

Alonso del Arte, Sep 20 2011

When C(n) is prime (or 1), then a(n) = 1; that is, n is in A005849.

On the penultimate page of their paper, Grau and Luca ask for "a good (large) lower bound on this quantity which is valid for all n and which tends to infinity with n."

			a(2) = 3 because the second Cullen number is 9; phi(9) = 6, therefore 6/gcd(8, 6) = 6/2 = 3.

Amiram Eldar, Table of n, a(n) for n = 0..848
José María Grau Ribas and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134, preprint, arXiv:1103.3578 [math.NT], Mar 18, 2011.

Cf. A000010, A002064, A005849, A160595.

cullen[n_] := n(2^n) + 1; Table[EulerPhi[cullen[n]]/GCD[cullen[n] - 1, EulerPhi[cullen[n]]], {n, 0, 39}]

a(n)=my(C=n<Charles R Greathouse IV, Feb 05 2013

A340093 Composite numbers k such that A003958(k) divides k-1.

Original entry on oeis.org

4, 8, 9, 16, 32, 64, 81, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 180225, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 1

Antti Karttunen, Dec 31 2020

Composite numbers k for which A340082(k) = 1.
Are there any other non-powers of 2 apart from 9, 81, 180225 (= 3^4 * 5^2 * 89) present?
If there are no squarefree numbers in this sequence, then Lehmer's Totient problem has no composite solutions.

Wikipedia, Lehmer's totient problem

Cf. A003958, A005117, A340082.
Cf. A000079 (subsequence from its term a(2)=4 onward).
Cf. also A160595.

f[n_] := Times @@ (((fct = FactorInteger[n])[[;; , 1]] - 1)^fct[[;; , 2]]); Select[Range[10^7], CompositeQ[#] && Divisible[# - 1, f[#]] &] (* Amiram Eldar, Dec 31 2020 *)

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
isA340093(n) = ((n>1)&&!isprime(n)&&!((n-1)%A003958(n)));

More terms from Amiram Eldar, Dec 31 2020

cp's OEIS Frontend

A323374 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A323373(n) for all other numbers, except f(p) = -(p mod 2) for primes p.

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A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A345938 a(n) = uphi(n) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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

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A339965 a(n) = sigma(n) / gcd(sigma(n),n+1).

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A342917 a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A339871 Number of primes p for which the p-adic valuation of phi(n) exceeds the p-adic valuation of n-1, with a(1) = 0 by convention.

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A339872 Index k of the least prime(k) such that prime(k)-adic valuation of phi(n) exceeds the prime(k)-adic valuation of n-1, or 0 if no such k exists (for example, when n = 1 or a prime).

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