A332826 -id:A332826 - cp's OEIS frontend

A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

2, 4, 6, 12, 10, 36, 30, 60, 90, 100, 42, 540, 70, 900, 210, 420, 22, 8100, 66, 2100, 3150, 1764, 330, 18900, 550, 4900, 2970, 94500, 770, 44100, 2310, 4620, 6930, 484, 11550, 4252500, 130, 4356, 16170, 115500, 182, 9922500, 546, 291060, 242550, 108900, 2730, 1455300, 8190, 302500, 858, 1131900, 1430, 8820900, 19110

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000010, A019565, A097248, A318834, A332825, A332826, A332827.

Cf. A048675 (a left inverse).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };

a(n) = Product_{d|n} A332825(d).
a(n) = A318834(n) * A332825(n).
A048675(a(n)) = n.
A097248(a(n)) = A019565(n).

A332827 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A003557(A332824(n)) for all other numbers, except f(1) = 0.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 5, 6, 3, 7, 3, 8, 3, 2, 3, 9, 3, 6, 10, 11, 3, 12, 13, 14, 15, 16, 3, 17, 3, 2, 5, 18, 13, 19, 3, 20, 21, 22, 3, 23, 3, 24, 25, 26, 3, 27, 5, 28, 3, 29, 3, 30, 21, 31, 5, 32, 3, 33, 3, 34, 35, 2, 13, 36, 3, 18, 10, 37, 3, 38, 3, 39, 40, 41, 25, 42, 3, 43, 44, 45, 3, 46, 3, 47, 48, 49, 3, 50, 40, 51, 52, 53, 3, 54, 3, 55, 56, 57, 3, 58, 3, 59, 60

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j).

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

Cf. A000010, A003557, A019565, A305801, A332824, A332826.

Cf. also A324538.

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
Aux332827(n) = if(1==n,0,A003557(A332824(n)));
v332827 = rgs_transform(vector(up_to,n,Aux332827(n)));
A332827(n) = v332827[n];

cp's OEIS Frontend

A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

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