A296085 -id:A296085 - cp's OEIS frontend

A296078 Least number with the same prime signature as 1+phi(n), where phi = A000010, Euler totient function.

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 4, 6, 2, 2, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 6, 6, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 6, 4, 6, 2, 6, 12, 4, 2, 4, 2, 2, 2, 2, 2, 4, 2, 6, 6, 2, 2, 4, 6, 2, 6, 2, 2, 4, 2, 12, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 4, 4

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A000010, A046523, A039649, A296079, A296080.
Cf. A039698 (positions of 2's).
Cf. also A277906, A296076, A296085, A296092.

f[n_] := Block[{ps = Last@# & /@ FactorInteger[1 + EulerPhi@n]}, Times @@ ((Prime@ Range@ Length@ ps)^ps)]; Array[f, 105] (* Robert G. Wilson v, Dec 11 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A296078(n) = A046523(1+eulerphi(n));

a(n) = A046523(A039649(n)) = A046523(1+A000010(n)).

A296092 Least number with the same prime signature as sigma(n)+1.

Original entry on oeis.org

2, 4, 2, 8, 2, 2, 4, 16, 6, 2, 2, 2, 6, 4, 4, 32, 2, 24, 6, 2, 6, 2, 4, 2, 32, 2, 2, 6, 2, 2, 6, 64, 4, 6, 4, 12, 6, 2, 6, 6, 2, 2, 12, 6, 2, 2, 4, 8, 6, 6, 2, 12, 6, 4, 2, 4, 16, 6, 2, 4, 12, 2, 30, 128, 6, 6, 6, 2, 2, 6, 2, 36, 12, 6, 8, 6, 2, 4, 16, 6, 6, 2, 6, 36, 2, 6, 4, 2, 6, 6, 2, 4, 6, 6, 4, 6, 12, 12, 2, 6, 2, 6, 30, 2, 2

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

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

Cf. A000203, A046523, A088580, A296212.

Cf. also A276528, A296076, A296078, A296085, A296091.

(define (A296092 n) (A046523 (+ 1 (A000203 n))))

a(n) = A046523(A088580(n)) = A046523(1+A000203(n)).

A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 7, 8, 2, 6, 2, 9, 4, 4, 2, 10, 11, 4, 5, 6, 2, 12, 2, 13, 14, 4, 7, 15, 2, 4, 7, 16, 2, 17, 2, 18, 9, 4, 2, 19, 3, 18, 14, 9, 2, 16, 4, 10, 4, 4, 2, 20, 2, 4, 6, 21, 7, 22, 2, 18, 23, 12, 2, 24, 2, 4, 6, 6, 4, 12, 2, 25, 26, 4, 2, 27, 14, 4, 14, 16, 2, 27, 4, 28, 4, 4, 4, 29, 2, 6, 6, 15, 2, 22, 2, 10, 12

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A046523(n), A296078(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

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

Cf. A000010, A046523, A296078.

Cf. also A286160, A296085, A300223, A300225.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A296078(n) = A046523(1+eulerphi(n));
Aux300224(n) = (1/2)*(2 + ((A296078(n)+A046523(n))^2) - A296078(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300224(n))),"b300224.txt");

A300225 Filter sequence combining A296078(n) and A296091(n), the prime signatures of phi(n)+1 and sigma(n)-1, with a(1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 4, 2, 2, 5, 2, 2, 6, 7, 2, 3, 2, 6, 2, 3, 2, 6, 8, 2, 3, 3, 2, 6, 2, 3, 9, 2, 6, 10, 2, 2, 11, 2, 2, 3, 2, 9, 11, 2, 2, 3, 12, 13, 9, 6, 2, 3, 2, 11, 2, 2, 2, 2, 2, 3, 2, 14, 6, 15, 2, 16, 17, 11, 2, 11, 2, 2, 3, 2, 3, 6, 2, 15, 18, 5, 2, 6, 9, 2, 15, 2, 2, 6, 3, 19, 2, 3, 3, 9, 2, 20, 3, 21, 2, 15, 2, 11, 6

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A296078(n), A296091(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

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

Cf. A000010, A000203, A296078, A296091.

Cf. also A296085, A300223, A300224.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A296078(n) = A046523(1+eulerphi(n));
A296091(n) = if(1==n,n,A046523(sigma(n)-1);)
Aux300225(n) = (1/2)*(2 + ((A296078(n)+A296091(n))^2) - A296078(n) - 3*A296091(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300225(n))),"b300225.txt");

cp's OEIS Frontend

A296078 Least number with the same prime signature as 1+phi(n), where phi = A000010, Euler totient function.

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A296092 Least number with the same prime signature as sigma(n)+1.

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A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

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A300225 Filter sequence combining A296078(n) and A296091(n), the prime signatures of phi(n)+1 and sigma(n)-1, with a(1) = 1.

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