A053575 -id:A053575 - cp's OEIS frontend

A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

85, 561, 1105, 1261, 1285, 2465, 4369, 6601, 8245, 8481, 9061, 9605, 10585, 16405, 16705, 17733, 18721, 19669, 21845, 23001, 28645, 30889, 38165, 42121, 43165, 46657, 54741, 56797, 57205, 62745, 65365, 74593, 78013, 83665, 88561, 91001, 106141, 117181, 124645, 126701, 134521, 136981, 141661, 162401, 171205, 176437

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

From Antti Karttunen, Dec 26 2020: (Start)
Equally, squarefree composite numbers k of the form 4u+1 for which A336466(k) divides k-1. This follows because on squarefree n, A336466(n) = A053575(n).
No common terms with A016105, because 4xy + 2(x+y) + 1 does not divide 4xy + 3(x+y) + 2 for any distinct x, y >= 0 (where 4x+3 and 4y+3 are the two prime factors of Blum integers).
This can also seen by another way: If this sequence contained any Blum integers, then, because A016105 is a subsequence of A339817, we would have found a composite number n satisfying Lehmer's totient problem y * phi(n) = n-1, for some integer y > 1. But Lehmer proved that such solutions should have at least 7 distinct prime factors, while Blum integers have only two.
Moreover, it seems that none of the terms of A167181 may occur here, and a few of A137409 (i.e., of A125667). See A339875 for those terms.
(End)

			85 = 4*21 + 1 = 5*17, thus phi(85) = 4*16 = 64, the odd part of which is A000265(64) = 1, which certainly divides 85-1, therefore 85 is included as a term.
561 = 4*140 + 1 = 3*11*17, thus phi(561) = 2*10*16 = 320, the odd part of which is A000265(320) = 5, which divides 560, therefore 561 is included.

Antti Karttunen, Table of n, a(n) for n = 1..1316
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem

Cf. A000010, A000265, A016105, A053575, A137409, A167181, A336466.
Subsequence of A005117.
Intersection of A091113 and A339880.
Cf. A339875 (a subsequence).
Cf. also comments in A339817.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[4*Range[45000] + 1, CompositeQ[#] && Divisible[# - 1, odd[EulerPhi[#]]] &] (* Amiram Eldar, Feb 17 2021 *)

A000265(n) = (n>>valuation(n, 2));
isA339870(n) = ((n>1)&&!isprime(n)&&(1==(n%4))&&!((n-1)%A000265(eulerphi(n))));

A279186 Maximal entry in n-th row of A279185.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 2, 6, 1, 2, 4, 10, 1, 4, 2, 6, 2, 3, 1, 4, 1, 4, 1, 2, 2, 6, 6, 2, 1, 4, 2, 6, 4, 2, 10, 11, 1, 6, 4, 1, 2, 12, 6, 4, 2, 6, 3, 28, 1, 4, 4, 2, 1, 2, 4, 10, 1, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 1, 18, 4, 20, 2, 1, 6, 3, 4

Offset: 1

N. J. A. Sloane, Dec 14 2016

nonn

See A256608 for LCM of entries in row n.
From Robert Israel, Dec 15 2016: (Start)
If m and k are coprime then a(m*k) = lcm(a(m), a(k)).
If n is in A061345 and r = A053575(n) is in A167791, then a(n) = A000010(r). (End)

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000010, A053575, A063145, A167791, A279185.

Start is same as A256607 and A256608. However, all three are different.

A279186 := proc(n)
    local a,k ;
    a := 1 ;
    for k from 0 to n-1 do
        a := max(a,A279185(k,n)) ;
    end do:
    a ;
end proc : # R. J. Mathar, Dec 15 2016

T[n_, k_] := Module[{g, y, r}, If[k == 0, Return[1]]; y = n; g = GCD[k, y]; While[g > 1, y = y/g; g = GCD[k, y]]; If[y == 1, Return[1]]; r = MultiplicativeOrder[k, y]; r = r/2^IntegerExponent[r, 2]; If[r == 1, Return[1]]; MultiplicativeOrder[2, r]];
a[n_] := Table[T[n, k], {k, 0, n - 1}] // Max;
Array[a, 90] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

{ A279186(n) = my(r=lcm(znstar(n)[2])); znorder(Mod(2,r>>valuation(r,2))); } \\ Max Alekseyev, Feb 02 2024

a(n) = A007733(A002322(n)). - Max Alekseyev, Feb 02 2024

A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 15, 3, 5, 15, 1, 3, 3, 3, 1, 9, 9, 9, 15, 15, 5, 15, 9, 45, 5, 45, 1, 1, 1, 1, 3, 3, 1, 3, 5, 1, 5, 5, 5, 15, 15, 15, 3, 3, 1, 3, 9, 9, 3, 9, 1, 15, 15, 15, 15, 9, 45, 45, 1, 9, 9, 9, 9, 27, 27, 27, 45, 45, 5, 45, 135, 135, 45, 135, 9, 27, 27, 27, 3, 81, 81, 81, 135, 27, 45, 135, 405

Offset: 0

Antti Karttunen, Dec 28 2020

Compare also to the scatter plot of A339898.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000265, A053575, A019565, A160595, A339809, A339821, A339971, A339898, A339899, A340075, A340085.

Cf. A339973 (positions of ones).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339901(n) = { my(x=A019565(2*n), y=A000265(eulerphi(x))); y/gcd((x-1),y); };

a(n) = A339971(n) / A339899(n).
a(n) = A000265(A160595(A019565(2*n))).
a(n) = A340075(A019565(n)) = A340085(A019565(2*n)).

A339971 Odd part of A339821(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 15, 15, 15, 15, 3, 3, 3, 3, 9, 9, 9, 9, 15, 15, 15, 15, 45, 45, 45, 45, 1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 15, 15, 15, 15, 3, 3, 3, 3, 9, 9, 9, 9, 15, 15, 15, 15, 45, 45, 45, 45, 9, 9, 9, 9, 27, 27, 27, 27, 45, 45, 45, 45, 135, 135, 135, 135, 27, 27, 27, 27, 81, 81, 81, 81, 135

Offset: 0

Antti Karttunen, Dec 26 2020

nonn

Cf. A000010, A000265, A003961, A019565, A053575, A057023, A336466, A339821, A339822, A339970.

Cf. A339972 (quadrisection).

A000265(n) = (n>>valuation(n,2));
A339971(n) = { my(m=1, p=2); while(n>0, p = nextprime(1+p); if(n%2, m *= A000265(p-1)); n >>= 1); (m); };

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A057023(e1) * A057023(e2) * ... * A057023(ek).
a(n) = A053575(A019565(2n)) = A336466(A003961(A019565(n))) = A000265(A339821(n)).
a(n) = A339821(n) / A000079(A339822(n)).

A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 4, 4, 1, 5, 9, 14, 0, 2, 1, 2, 0, 2, 4, 5, 7, 8, 9, 14, 10, 32, 9, 29, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 4, 4, 3, 11, 4, 14, 1, 2, 0, 2, 7, 5, 3, 2, 0, 2, 4, 14, 6, 20, 34, 14, 0, 2, 4, 5, 24, 20, 16, 23, 28, 41, 9, 29, 112, 68, 24, 74, 3, 11, 19, 5, 27, 2, 58, 14, 16, 50, 84, 119, 388, 356

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A053575, A019565, A339809, A339821, A339971, A339899, A339901.

Cf. A339973 (positions of zeros).

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339898(n) = { my(x=A019565(2*n)); ((x-1)%A000265(eulerphi(x))); };

a(n) = A339809(2*n) modulo A339971(n), where A339971(n) = A053575(A019565(2n)).

A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 9, 5, 3, 3, 3, 1, 5, 3, 27, 9, 5, 11, 9, 5, 3, 7, 9, 21, 1, 25, 15, 15, 3, 9, 81, 3, 9, 15, 15, 5, 11, 1, 27, 21, 5, 23, 9, 15, 7, 13, 27, 55, 21, 9, 3, 29, 25, 9, 45, 11, 15, 15, 9, 33, 9, 25, 243, 3, 3, 35, 27, 7, 15, 9, 45, 39, 5, 21, 33, 15, 1, 41, 81, 125, 21, 11, 15, 27, 23, 15, 27, 3, 15

Offset: 1

Antti Karttunen, Dec 29 2020

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, A000265, A003961, A003972, A005117, A053575, A151800, A339903, A340075.

A000265(n) = (n>>valuation(n,2));
A339904(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,my(q=nextprime(1+f[i,1])); A000265(q-1)*(q^(f[i,2]-1))));

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339903(n) = A000265(eulerphi(A003961(n)));

Multiplicative with a(p^e) = A000265(q-1) * q^(e-1), where q = A151800(p), the next prime larger than p.
a(n) = A000265(A003972(n)) = A053575(A003961(n)) = A000265(A000010(A003961(n))).
For all squarefree numbers k, a(k) = A339903(k).

A066670 Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).

Original entry on oeis.org

3, 3, 5, 3, 3, 3, 3, 5, 11, 5, 3, 3, 7, 5, 3, 3, 3, 5, 3, 5, 3, 11, 23, 5, 3, 13, 5, 3, 7, 29, 3, 5, 11, 3, 3, 5, 3, 5, 41, 3, 7, 5, 11, 3, 11, 23, 3, 5, 3, 3, 13, 53, 5, 3, 7, 11, 7, 29, 3, 5, 3, 5, 17, 11, 3, 23, 3, 7, 37, 5, 3, 3, 13, 5, 5, 41, 83, 3, 43, 7, 5, 29, 11, 89, 3, 11, 5, 23, 3, 3

Offset: 1

Labos Elemer, Dec 18 2001

nonn

			A066669(9) = 23, phi(23) = 2*11, so a(9)=11.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000265, A053575, A006530, A065966, A066669, A066671, A066672, A066673.

Select[Array[#/2^IntegerExponent[#, 2] &@ EulerPhi@ # &, 200], PrimeQ]  (* Michael De Vlieger, Dec 08 2018 *)

lista(nn) = {for (n=1, nn, en=eulerphi(n); if (isprime(p=en>>valuation(en, 2)), print1(p, ", ")); ); } \\ Michel Marcus, Dec 08 2018

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A053575(A066669(n)).
a(n) = A000265(A000010(A066669(n))) = A006530(A000010(A066669(n))). (End)

A227946 Smallest m such that the number of iterations of "take odd part of phi" to reach 1 from m (A227944) is n.

Original entry on oeis.org

1, 2, 7, 19, 47, 163, 487, 1307, 2879, 19683, 39367, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 258280327, 688747547, 3486784401, 10460353203

Offset: 0

Max Alekseyev, Oct 03 2013

The odd part of a number is its largest odd divisor (A000265), phi is Euler's totient function (A000010). - Alonso del Arte, Oct 13 2013

			a(1) = 2 because just one step is needed to reach 1 from 2, since phi(2) = 1. The numbers 3, 4, 5 and 6 also take one step.
a(2) = 7 because two steps are needed to reach 1 from 7: phi(7) = 6, the odd part of which is 3, and phi(3) = 2, the odd part of which is 1. The numbers from 8 to 18 take one or two steps to reach 1.
a(3) = 19 because three steps are needed to reach 1 from 19: phi(19) = 18, the odd part of which is 9, and phi(9) = 6, the odd part of which is 3, and phi(3) = 2, the odd part of which is 1.

A variant of A049117. - R. J. Mathar, Oct 06 2013

Cf. A007755, A053575, A000010.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a227946 = (+ 1) . fromJust . (`elemIndex` a227944_list)
-- Reinhard Zumkeller, Nov 10 2013

a(n) = smallest m such that A227944(m)=n.

a(15) through a(21) copied over from A049117 by Max Alekseyev, Oct 13 2013

A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 15, 1, 1, 1, 3, 5, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 27, 1, 1, 1, 1, 5, 3, 1, 1, 1, 81, 1, 1, 1, 3, 1, 1, 1, 9, 1, 3, 1

Offset: 0

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000010, A000265, A049559, A053575, A019565, A339809, A339821, A339971, A339898, A339901.

A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339899(n) = { my(x=A019565(2*n)); gcd((x-1),A000265(eulerphi(x))); };

a(n) = gcd(A339809(2*n), A339971(n)), where A339971(n) = A053575(A019565(2n)).
a(n) = gcd(A339971(n), A339898(n)).
a(n) = A339971(n) / A339901(n).
a(n) = A000265(A049559(A019565(2*n))).

A193453 Number of odd divisors of phi(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 4, 1, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 4, 2, 2, 2, 2, 1, 4, 2, 1, 2, 2, 3, 2, 2, 3, 2, 2, 1, 4, 4, 3, 1, 2, 2, 4, 1, 2, 2, 4, 2, 3, 3, 2, 3, 4, 2, 4, 1, 4, 2, 2, 2, 1, 4, 2, 2, 2, 2, 3, 2, 4, 2, 3, 1, 2, 4, 4, 2, 3, 1, 4, 2, 2

Offset: 1

Michel Lagneau, Jul 26 2011

nonn

phi(n) : A000010 is the Euler totient function. This sequence equals A193169 (n) for n < 63.

			a(63) = 3 because phi(63) = 36 with 3 odd divisors {1, 3, 9}.

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

Cf. A000010, A001227, A053575.

f[n_] := Block[{d = Divisors[EulerPhi[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 80}]

A193453(n) = sumdiv(eulerphi(n), d, d%2); \\ Antti Karttunen, Dec 04 2017

a(n) = A001227(A000010(n)) = A000005(A053575(n)). - Antti Karttunen, Dec 04 2017

More terms from Antti Karttunen, Dec 04 2017

cp's OEIS Frontend

A339870 Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1.

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A279186 Maximal entry in n-th row of A279185.

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A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

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A339971 Odd part of A339821(n).

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A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

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A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

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A066670 Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).

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A227946 Smallest m such that the number of iterations of "take odd part of phi" to reach 1 from m (A227944) is n.

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