A057635 -id:A057635 - cp's OEIS frontend

A336385 Number of k = xy such that phi(k) = n(phi(x) + phi(y)).

2, 5, 15, 14, 7, 25, 7, 36, 30, 26, 15, 68, 2, 21, 125, 78, 9, 94, 7, 90, 52, 33, 38, 208, 15, 11, 74, 69, 3, 227, 2, 166, 66, 33, 129, 276, 2, 25, 101, 228, 7, 115, 8, 76, 329, 47, 28, 482, 19, 65, 40, 50, 7, 248, 99, 234, 46, 34, 20, 572, 2, 5, 426, 356, 16

Offset: 1

Jinyuan Wang, Aug 01 2020

nonn

If phi(x*y) = n*(phi(x) + phi(y)) and phi(x) <= phi(y), then phi(x) <= 2*n and phi(y) <= n*phi(x).
a(n) >= 1 because (k, x, y) = (4*n^2, 2*n, 2*n) is a solution.
If gcd(n, 6) = 1, then a(n) >= 2 because (k, x, y) = (12*n^2, 3*n, 4*n) is also a solution. Note that a(n) = 2 when n = 1, 13, 31, 37, 61, 73, 97, 103, 149, 151, 157, 181, ...
Conjecture: a(n) > 2 if n is composite.

			a(2) = 5 because k = 16, 24, 36, 40 and 60 satisfy the equation.

Zhang Sibao and Xi Xiaozhong, Positive integer solutions on phi(ab) = k*(phi(a) + phi(b))

Cf. A000010, A057635, A336384, A336710.

f(n) = floor(n*exp(Euler)*log(log(n^2))+2.5*n/log(log(n^2)));
a(n) = {if(n==1, return(2)); my(t, v=List([])); for(x=1, f(2*n), if((t=eulerphi(x)) <= 2*n, for(y=1, f(t=n*t), if(eulerphi(x*y) == t+n*eulerphi(y), listput(v, x*y))))); #Set(v); }

A303712 a(n) is the smallest number such that there are exactly n numbers k (including a(n) itself) such that U(k) is isomorphic to U(a(n)) (or 0 if no such number exists). Here U(k) is the multiplicative group of integers modulo k.

Original entry on oeis.org

24, 1, 3, 7, 55, 129, 35, 104, 407, 707, 143, 371, 899, 665, 1144, 1771, 385, 3003, 3451, 5005, 7049, 8041, 7579, 12243, 4081, 5291, 3857, 9361, 2717, 2233

Offset: 1

Jianing Song, Apr 29 2018

Conjecture: a(n) > 0 for all n.
Other known terms: a(35) = 8855, a(39) = 6149. [Corrected by Jianing Song, Oct 04 2018]
From Jianing Song, Oct 04 2018: (Start)
a(32) = 9269, a(33) = 7315, a(37) = 15953, a(52) = 16555, a(59) = 17081.
a(31), a(34), a(36), a(38) etc. > 2*10^4 (if not equal to 0). (End)

			U(24) is isomorphic to C_2 x C_2 x C_2 and there is no other number k such that U(k) is isomorphic to U(24), so a(1) = 24.
U(1) and U(2) are both isomorphic to the trivial group.
U(3), U(4) and U(6) are isomorphic to C_2.
U(7), U(9), U(14) and U(18) are isomorphic to C_6.
U(55), U(75), U(100), U(110) and U(150) are isomorphic to C_2 x C_20.
U(129), U(147), U(172), U(196), U(258) and U(294) are isomorphic to C_2 x C_42.
U(35), U(39), U(45), U(52), U(70), U(78) and U(90) are isomorphic to C_2 x C_12.
U(104), U(105), U(112), U(140), U(144), U(156), U(180) and U(210) are isomorphic to C_2 x C_2 x C_12.

Wikipedia, Multiplicative group of integers modulo n

Cf. A057635, A289626.

b(n) = my(i=0, search_max = A057635(eulerphi(n))); for(j=eulerphi(n)+1, search_max, if(znstar(j)[2]==znstar(n)[2], i++)); i \\ search_max is the largest k such that phi(k) = phi(n). See A057635 for its program
a(n) = if(n==2, 1, my(t=3); while(b(t)!=n, t++); t) \\ Jianing Song, Oct 04 2018

a(21)-a(24) from Jianing Song, Oct 04 2018

A336710 Square array read by antidiagonals: A(n,k) is the number of ordered solutions (x_1, x_2, ..., x_n) to equation phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i), or -1 if there are infinitely many solutions, n >= 1, k >= 1.

Original entry on oeis.org

-1, 0, 3, 0, 9, 15, 0, 35, 39, 118, 0, 33, 31, 463, 90, 0, 17, 138, 558, 200, 435, 0, 63, 57, 1080, 580, 1580, 644, 0, 15, 198, 750, 1375, 2400, 1820, 294, 0, 91, 87, 1200, 570, 4695, 3535, 3024, 792, 0, 79, 411, 528, 2490, 1680, 8386, 12292, 5256, 3285, 0, 67, 183, 2584, 685, 7710, 2555, 15568, 14364, 16605, 1595, 0, 39, 294, 1346, 6565, 2790, 21070, 6160, 42030, 28305, 21780, 15708, 0

Offset: 1

Jinyuan Wang, Aug 10 2020

For n = 1, we have phi(x_1) = k * phi(x_1), thus A(1, k) = 0 iff k >= 2.

For n >= 2, if phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i) and phi(x_1) <= phi(x_2) <= ... <= phi(x_n), then phi(x_(n-1)) <= n*k and phi(x_n) <= k*(n-1)*phi(x_(n-1)). It implies that the equation has finite solutions iff n >= 2 or k >= 2.

			The square array A(n,k) begins:
  -1,   0,   0,    0,   0,  ...
   3,   9,  35,   33,  17,  ...
  15,  39,  31,  138,  57,  ...
 118, 463, 558, 1080, 732,  ...
 ...

Shi Baohuai and Pan Xiaowei, On the arithmetic functional equation phi(x_1*...*x_(n-1)*x_n) = m*(phi(x_1) + ... + phi(x_(n-1)) + phi(x_n)), Mathematics Practice and Understanding, 2014, Issue 24, Pages 307-310.

Cf. A000010, A057635, A336385.

Terms a(16) onward from Max Alekseyev, Feb 01 2025

A380578 Number of nonisomorphic groups appearing as the group of units of the ring Z/kZ for every k such that phi(k) = n.

Original entry on oeis.org

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

Offset: 1

Miles Englezou, Mar 26 2025

nonn

Every group of units is abelian.

			a(4) = 2 because of the 4 distinct k such that phi(k) = 4 there are 2 nonisomorphic group of units Z/kZ*: C_4, and C_2 x C_2.
a(40) = 3 because of the 9 distinct k such that phi(k) = 40 there are 3 nonisomorphic group of units Z/kZ*: C_40, C_20 x C_2, and C_10 x C_2 x C_2.
a(41) = 0 because there are no k such that phi(k) = 41.

Cf. A000010, A000688, A005277, A007617, A014197, A049283, A057635.

groupcount(n) = b=[]; if(n==1, b=concat(b,2), forstep(k=floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, b=concat(b,k)); if(k==n, b=concat(b,0)))); Z=[]; if(istotient(n)==0, return(0), for(m=2, b[1], if(eulerphi(m)<>n, next, W=[]; U=[]; D=divisors(eulerphi(m)); lambda=lcm(znstar(m)[2]); for(k=1, m-1, if(gcd(k,m)==1, U=concat(U, k))); for(j=1, length(D), if(D[j]>lambda, break); S=[]; for(r=1, eulerphi(m), if(znorder(Mod(U[r], m))==D[j], S=concat(S, U[r]))); W=concat(W, length(S)))); Z=concat(Z,[W]); Z=Set(Z)); return(length(Z)))

a(n) <= A014197(n).

a(n) = 0 for every n belonging to A007617.

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A336385 Number of k = x*y such that phi(k) = n*(phi(x) + phi(y)).

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A303712 a(n) is the smallest number such that there are exactly n numbers k (including a(n) itself) such that U(k) is isomorphic to U(a(n)) (or 0 if no such number exists). Here U(k) is the multiplicative group of integers modulo k.

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A336710 Square array read by antidiagonals: A(n,k) is the number of ordered solutions (x_1, x_2, ..., x_n) to equation phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i), or -1 if there are infinitely many solutions, n >= 1, k >= 1.

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A380578 Number of nonisomorphic groups appearing as the group of units of the ring Z/kZ for every k such that phi(k) = n.

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A336385 Number of k = xy such that phi(k) = n(phi(x) + phi(y)).