A354201 -id:A354201 - cp's OEIS frontend

A354203 Fully multiplicative with a(p^e) = A354201(A000720(p))^e.

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 7, 1, 5, 3, 2, 1, 13, 1, 11, 2, 3, 7, 19, 1, 4, 5, 1, 3, 17, 2, 23, 1, 7, 13, 6, 1, 29, 11, 5, 2, 37, 3, 31, 7, 2, 19, 43, 1, 9, 4, 13, 5, 41, 1, 14, 3, 11, 17, 47, 2, 53, 23, 3, 1, 10, 7, 59, 13, 19, 6, 67, 1, 61, 29, 4, 11, 21, 5, 71, 2, 1, 37, 79, 3, 26, 31, 17, 7, 73, 2, 15, 19, 23

Offset: 1

Antti Karttunen, May 23 2022

Index entries for sequences computed from indices in prime factorization

Left inverse of A354202.

Cf. A354201, A354206.

A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };

A354200 Prime map that sends 2 to 5, and each 4k+1 and 4k+3 prime to the next larger prime of the same type.

Original entry on oeis.org

5, 7, 13, 11, 19, 17, 29, 23, 31, 37, 43, 41, 53, 47, 59, 61, 67, 73, 71, 79, 89, 83, 103, 97, 101, 109, 107, 127, 113, 137, 131, 139, 149, 151, 157, 163, 173, 167, 179, 181, 191, 193, 199, 197, 229, 211, 223, 227, 239, 233, 241, 251, 257, 263, 269, 271, 277, 283, 281, 293, 307, 313, 311, 331, 317, 337, 347, 349, 359

Offset: 1

Antti Karttunen, May 23 2022

nonn

			A000040(2) = 3, and the next larger prime of the form 4k+3 is 7, therefore a(2) = 7.
A000040(3) = 5, and the next larger prime of the form 4k+1 is 13, therefore a(3) = 13.
A000040(4) = 7, and the next larger prime of the form 4k+3 is 11, therefore a(4) = 11.

Cf. A000040, A000720, A354201, A354202.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));

For all n >= 1, A354201(A000720(a(n))) = A000040(n).

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

1, 1, 1, 23, 3, 1, 1, 5, 11, 3, 2, 23, 1, 1, 3, 469, 2, 11, 1, 69, 1, 2, 1, 5, 53, 1, 4, 23, 11, 3, 7, 69, 2, 2, 3, 253, 3, 1, 1, 15, 1, 1, 1, 46, 33, 1, 2, 469, 33, 53, 2, 23, 23, 4, 6, 5, 1, 11, 13, 69, 29, 7, 11, 19507, 3, 2, 1, 46, 1, 3, 2, 55, 2, 3, 53, 23, 2, 1, 3, 1407, 2797, 1, 5, 23, 6, 1, 11, 10, 9, 33

Offset: 1

Antti Karttunen, May 23 2022

Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
Cf. A354361 (positions of 1's).
Cf. also A326042, A348750, A354088, A354096 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354206(n) = A354203(sigma(A354202(n)));

Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
a(n) = n - A354207(n).

A354207 a(n) = n - A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

0, 1, 2, -19, 2, 5, 6, 3, -2, 7, 9, -11, 12, 13, 12, -453, 15, 7, 18, -49, 20, 20, 22, 19, -28, 25, 23, 5, 18, 27, 24, -37, 31, 32, 32, -217, 34, 37, 38, 25, 40, 41, 42, -2, 12, 45, 45, -421, 16, -3, 49, 29, 30, 50, 49, 51, 56, 47, 46, -9, 32, 55, 52, -19443, 62, 64, 66, 22, 68, 67, 69, 17, 71, 71, 22, 53, 75, 77, 76

Offset: 1

Antti Karttunen, May 23 2022

sign

Cf. A000203, A354200, A354201, A354202, A354203, A354205, A354206.

Cf. also A348736.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354207(n) = (n-A354203(sigma(A354202(n))));

a(n) = n - A354206(n).

cp's OEIS Frontend

A354203 Fully multiplicative with a(p^e) = A354201(A000720(p))^e.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A354200 Prime map that sends 2 to 5, and each 4k+1 and 4k+3 prime to the next larger prime of the same type.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A354207 a(n) = n - A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula