A296078 -id:A296078 - cp's OEIS frontend

A296085 Filter sequence combining A296078(n) and A296092(n), the prime signatures of 1+phi(n) and 1+sigma(n).

1, 2, 1, 3, 1, 1, 2, 4, 5, 1, 1, 1, 5, 2, 6, 7, 1, 8, 5, 9, 5, 1, 2, 9, 10, 1, 1, 5, 1, 9, 5, 11, 12, 5, 6, 13, 5, 1, 14, 5, 1, 1, 13, 15, 9, 1, 2, 3, 5, 15, 16, 17, 5, 2, 1, 6, 4, 5, 1, 2, 13, 1, 18, 19, 14, 15, 5, 16, 20, 14, 1, 21, 13, 5, 3, 5, 1, 6, 4, 15, 15, 1, 5, 21, 16, 5, 12, 1, 5, 14, 1, 22, 5, 5, 2, 15, 13, 13, 1, 5, 1, 15, 18, 9, 9

Offset: 1

Antti Karttunen, Dec 08 2017

nonn

Restricted growth sequence transform of P(A296078(n), A296092(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
For all i, j:
  a(i) = a(j) => A296213(i) = A296213(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A000010, A000203, A046523, A296078, A296092, A296213.

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));
A296092(n) = A046523(1+sigma(n));
Anotsubmitted5(n) = (1/2)*(2 + ((A296078(n)+A296092(n))^2) - A296078(n) - 3*A296092(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted5(n))),"b296085.txt");

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");

A296079 a(n) = 1 if 1+phi(n) is prime, 0 otherwise, where phi = A000010, Euler totient function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

Out of the first 65537 values, 26197 are 1's (indicating primes), and 39340 are 0's, indicating nonprimes.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Characteristic function of A039698.
Cf. A039689 (positions of zeros).
Cf. also A296077, A296078, A296080.

Table[If[PrimeQ[EulerPhi[n]+1],1,0],{n,120}] (* Harvey P. Dale, Apr 23 2020 *)

PARI
```
A296079(n) = isprime(1+eulerphi(n));
```

a(n) = A010051(A039649(n)) = A010051(1+A000010(n)).

For all n, a(n) >= A010051(n) and a(2n) >= A010051(n).

A296076 Least number with the same prime signature as 1 + A002322(n), where A002322 is Carmichael's lambda.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A002322, A046523, A263027, A263028 (positions of 2's), A296077, A296078.

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
A296076(n) = A046523(1+lcm(znstar(n)[2]));

a(n) = A046523(A263027(n)) = A046523(1+A002322(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)).

A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 4, 4, 4, 7, 4, 8, 4, 6, 5, 9, 4, 10, 6, 8, 6, 11, 4, 12, 7, 10, 7, 13, 6, 14, 8, 13, 7, 15, 6, 16, 10, 13, 9, 17, 7, 16, 10, 18, 13, 19, 8, 15, 13, 14, 11, 20, 7, 21, 12, 14, 18, 16, 10, 22, 18, 23, 13, 24, 13, 25, 14, 15, 14, 21, 13, 26, 18, 27, 15, 28, 13, 29, 16, 30, 15, 31, 13, 25, 23, 21, 17, 25, 18, 32, 16, 21, 15

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A000010, A039649, A289625, A289626, A296078.

Cf. A039650.

allocatemem(2^30);
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A289625(1+eulerphi(n)))),"b296080.txt");

cp's OEIS Frontend

A296085 Filter sequence combining A296078(n) and A296092(n), the prime signatures of 1+phi(n) and 1+sigma(n).

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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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A296079 a(n) = 1 if 1+phi(n) is prime, 0 otherwise, where phi = A000010, Euler totient function.

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Mathematica

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A296076 Least number with the same prime signature as 1 + A002322(n), where A002322 is Carmichael's lambda.

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

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A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

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