A353271 -id:A353271 - cp's OEIS frontend

A366291 Lexicographically earliest infinite sequence such that a(i) = a(j) => A353271(i) = A353271(j) for all i, j >= 1, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 2, 7, 10, 1, 4, 5, 11, 9, 2, 1, 12, 1, 13, 10, 14, 7, 15, 1, 16, 11, 4, 1, 7, 1, 2, 12, 17, 1, 8, 7, 18, 14, 2, 1, 9, 10, 4, 16, 19, 1, 20, 1, 21, 3, 22, 11, 23, 1, 2, 17, 24, 1, 25, 1, 26, 18, 2, 10, 27, 1, 8, 28, 29, 1, 30, 14, 31, 19, 4, 1, 32, 11, 2, 21, 33, 16

Offset: 1

Antti Karttunen, Oct 06 2023

nonn

Restricted growth sequence transform of A353271.

For all i, j: A305800(i) = A305800(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A305800, A332449, A353271.

Cf. also A365431.

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; };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353271(n) = (n / gcd(n, A332449(n)));
v366291 = rgs_transform(vector(up_to,n,A353271(n)));
A366291(n) = v366291[n];

A366292 Dirichlet inverse of A353271, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 9, -5, -9, -1, 11, -4, -11, -4, 13, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 21, -1, 21, -2, -21, -1, 5, -6, -8, -15, 25, -1, 22, -9, 7, -17, -27, -1, 3, -1, -29, 14, -1, -11, 11, -1, 33, -21, -3, -1, 16, -1, -35, -8, 37, -9, 13

Offset: 1

Antti Karttunen, Oct 06 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A332449, A353271.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A209635, A342417, A354347, A354823, A359432, A359433, A359577 for other sequences that are equal modulo 2.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353271(n) = (n / gcd(n, A332449(n)));
memoA366292 = Map();
A366292(n) = if(1==n,1,my(v); if(mapisdefined(memoA366292,n,&v), v, v = -sumdiv(n,d,if(dA353271(n/d)*A366292(d),0)); mapput(memoA366292,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA353271(n/d) * a(d).

A332449 a(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 4, 9, 10, 25, 16, 49, 30, 21, 36, 121, 22, 169, 100, 81, 90, 289, 40, 361, 250, 225, 196, 529, 66, 55, 484, 105, 490, 841, 64, 961, 270, 441, 676, 625, 154, 1369, 1156, 1089, 750, 1681, 144, 1849, 1210, 39, 1444, 2209, 198, 91, 84, 1521, 1690, 2809, 120, 1225, 1470, 2601, 2116, 3481, 34, 3721, 3364, 1029, 810, 3025, 400

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A329609 (terms sorted into ascending order).
Cf. A000290, A003961, A005117 (positions of squares), A005940, A010052, A156552, A277010, A329603, A332450, A332451, A347119, A347120, A353267 [= A348717(a(n))], A353269, A353270 [= gcd(n, a(n))], A353271, A353272, A353273.
Cf. also A332223.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332449(n) = A005940(1+(3*A156552(n)));

a(n) = A005940(1+(3*A156552(n))).
a(p) = p^2 for all primes p.
a(u) = A332451(u) and A010052(a(u)) = 1 for all squarefree numbers (A005117).
a(A003961(n)) = A003961(a(n)) = A005940(1+(6*A156552(n))).
From Antti Karttunen, Apr 10 2022: (Start)
a(n) = A347119(n) * A000290(A347120(n)) = A353270(n) * A353272(n).
a(A353269(n)) = 1 for all n.
(End)

A353270 a(n) = gcd(n, A332449(n)), where A332449(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 10, 3, 2, 23, 6, 5, 2, 3, 14, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 10, 41, 6, 43, 22, 3, 2, 47, 6, 7, 2, 3, 26, 53, 6, 5, 14, 3, 2, 59, 2, 61, 2, 21, 2, 5, 2, 67, 34, 3, 2, 71, 6, 73, 2, 3, 38, 7, 2, 79, 10, 3, 2, 83, 2, 5, 2, 3, 22, 89, 2, 7, 46, 3, 2, 5, 6

Offset: 1

Antti Karttunen, Apr 09 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A332449, A353271, A353272, A353274.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353270(n) = gcd(n, A332449(n));

a(n) = gcd(n, A332449(n)) = gcd(n, A005940(1+(3*A156552(n)))).

a(n) = n / A353271(n) = A332449(n) / A353272(n).

A353272 a(n) = A332449(n) / gcd(n, A332449(n)), where A332449(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 2, 3, 5, 5, 8, 7, 15, 7, 18, 11, 11, 13, 50, 27, 45, 17, 20, 19, 25, 75, 98, 23, 11, 11, 242, 35, 35, 29, 32, 31, 135, 147, 338, 125, 77, 37, 578, 363, 75, 41, 24, 43, 55, 13, 722, 47, 33, 13, 42, 507, 65, 53, 20, 245, 105, 867, 1058, 59, 17, 61, 1682, 49, 405, 605, 200, 67, 85, 1083, 162, 71, 77, 73, 1922, 63

Offset: 1

Antti Karttunen, Apr 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A332449, A353270, A353271, A353273, A353274 [= gcd(n, a(n))].

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353272(n) = { my(u=A332449(n)); (u / gcd(n, u)); };

a(n) = A332449(n) / A353270(n) = A332449(n) / gcd(n, A332449(n)).

a(p) = p for all primes p.

A353273 a(n) = (A332449(n) / gcd(n, A332449(n))) - n, where A332449(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 7, -2, 8, 0, -1, 0, 36, 12, 29, 0, 2, 0, 5, 54, 76, 0, -13, -14, 216, 8, 7, 0, 2, 0, 103, 114, 304, 90, 41, 0, 540, 324, 35, 0, -18, 0, 11, -32, 676, 0, -15, -36, -8, 456, 13, 0, -34, 190, 49, 810, 1000, 0, -43, 0, 1620, -14, 341, 540, 134, 0, 17, 1014, 92, 0, 5, 0, 1848, -12, 19, 266, 314, 0, 145

Offset: 1

Antti Karttunen, Apr 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A332449, A353270, A353271, A353272, A353274, A353275.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353273(n) = { my(u=A332449(n)); (u / gcd(n, u))-n; };

a(n) = A353272(n) - n.

A353274 a(n) = gcd(n, A353272(n)).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 1, 13, 2, 3, 1, 17, 2, 19, 5, 3, 2, 23, 1, 1, 2, 1, 7, 29, 2, 31, 1, 3, 2, 5, 1, 37, 2, 3, 5, 41, 6, 43, 11, 1, 2, 47, 3, 1, 2, 3, 13, 53, 2, 5, 7, 3, 2, 59, 1, 61, 2, 7, 1, 5, 2, 67, 17, 3, 2, 71, 1, 73, 2, 3, 19, 7, 2, 79, 5, 1, 2, 83, 1, 5, 2, 3, 11, 89, 2, 7, 23, 3, 2, 5, 3, 97

Offset: 1

Antti Karttunen, Apr 09 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A332449, A353270, A353271, A353272, A353273.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353274(n) = { my(u=A332449(n)); gcd(n, (u / gcd(n, u))); };

a(n) = gcd(n, A353272(n)) = gcd(n, A353273(n)) = gcd(A353272(n), A353273(n)).

a(p) = p for all primes p.

cp's OEIS Frontend

A366291 Lexicographically earliest infinite sequence such that a(i) = a(j) => A353271(i) = A353271(j) for all i, j >= 1, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A366292 Dirichlet inverse of A353271, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A332449 a(n) = A005940(1+(3*A156552(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A353270 a(n) = gcd(n, A332449(n)), where A332449(n) = A005940(1+(3*A156552(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A353272 a(n) = A332449(n) / gcd(n, A332449(n)), where A332449(n) = A005940(1+(3*A156552(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A353273 a(n) = (A332449(n) / gcd(n, A332449(n))) - n, where A332449(n) = A005940(1+(3*A156552(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A353274 a(n) = gcd(n, A353272(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula