A340072 -id:A340072 - cp's OEIS frontend

A340075 The odd part of A340072(n).

1, 1, 1, 3, 1, 1, 1, 9, 5, 3, 1, 3, 1, 5, 3, 27, 1, 5, 1, 9, 5, 3, 1, 9, 7, 1, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 1, 27, 1, 5, 1, 9, 5, 7, 1, 27, 11, 21, 9, 3, 1, 25, 1, 45, 11, 15, 1, 9, 1, 9, 25, 243, 3, 3, 1, 27, 7, 3, 1, 45, 1, 5, 21, 33, 15, 1, 1, 81, 125, 21, 1, 15, 9, 23, 15, 27, 1, 15, 5, 21, 9, 13, 33, 81

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Each term a(n) is a multiple of A340149(n), therefore, as both sequences have only positive terms, it follows that if a(n) = 1 then A340149(n) = 1 also, but not necessarily vice versa.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A019565, A339901, A339904, A340072, A340074, A340076 (positions of ones), A340149 (differs from the first time at n=85).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };
A340075(n) = A000265(A340072(n));

a(n) = A000265(A340072(n)).
a(n) = A339904(n) / A340074(n) = A339904(n) / gcd(A003961(n)-1, A339904(n)).
For all n >= 0, a(A019565(n)) = A339901(n).

A340071 a(n) = gcd(A003961(n)-1, phi(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

1, 2, 4, 2, 6, 2, 10, 2, 4, 4, 12, 4, 16, 4, 2, 2, 18, 2, 22, 2, 2, 2, 28, 2, 6, 2, 4, 2, 30, 8, 36, 2, 16, 4, 4, 8, 40, 4, 4, 4, 42, 4, 46, 4, 6, 2, 52, 4, 10, 2, 2, 8, 58, 2, 18, 4, 2, 4, 60, 2, 66, 2, 2, 2, 2, 2, 70, 2, 16, 10, 72, 2, 78, 2, 4, 2, 2, 2, 82, 2, 4, 4, 88, 2, 12, 4, 2, 2, 96, 4, 2, 4, 8, 2, 4, 2, 100

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A049559.

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

Cf. A000010, A003961, A003972, A049559, A253885, A340072, A340073.

Cf. also A340081.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340071(n) = { my(u=A003961(n)); gcd(u-1, eulerphi(u)); };

a(n) = A049559(A003961(n)).
a(n) = gcd(A253885(n-1), A003972(n)) = gcd(A003961(n)-1, A000010(A003961(n))).
a(n) = A003972(n) / A340072(n).
For n > 1, a(n) = (A003961(n)-1) / A340073(n).

A340082 a(n) = A003958(n) / gcd(n-1, A003958(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 4, 1, 1, 4, 1, 4, 3, 10, 1, 2, 2, 12, 4, 2, 1, 8, 1, 1, 5, 16, 12, 4, 1, 18, 12, 4, 1, 12, 1, 10, 4, 22, 1, 2, 3, 16, 16, 4, 1, 8, 20, 6, 9, 28, 1, 8, 1, 30, 12, 1, 3, 4, 1, 16, 11, 8, 1, 4, 1, 36, 16, 6, 15, 24, 1, 4, 1, 40, 1, 12, 16, 42, 28, 10, 1, 16, 4, 22, 15, 46, 36, 2, 1, 36

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340081, A340083, A340085 (gives the odd part).

Cf. also A160595, A340072.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340082(n) = { my(u=A003958(n)); u/gcd(n-1, u); };

a(n) = A003958(n) / A340081(n) = A003958(n) / gcd(n-1, A003958(n)).

A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 13, 6, 5, 1, 11, 1, 8, 17, 40, 1, 37, 1, 31, 27, 19, 1, 67, 8, 25, 31, 49, 1, 13, 1, 121, 4, 14, 19, 28, 1, 17, 21, 47, 1, 41, 1, 29, 29, 43, 1, 101, 12, 73, 47, 19, 1, 187, 5, 74, 57, 23, 1, 157, 1, 55, 137, 364, 59, 97, 1, 85, 9, 23, 1, 337, 1, 61, 61, 103, 71, 127, 1, 283, 156, 32, 1, 247, 11

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160596.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160596, A253885, A340071, A340072.

Cf. also A340083.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340073(n) = { my(x=A003961(n)); (x-1)/gcd(x-1, eulerphi(x)); };

a(n) = A160596(A003961(n)).

a(n) = A253885(n-1) / A340071(n) = (A003961(n)-1) / A340071(n).

A340147 a(n) = A247074(A003961(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 9, 5, 3, 1, 3, 1, 5, 6, 27, 1, 10, 1, 9, 10, 6, 1, 18, 7, 8, 25, 15, 1, 3, 1, 81, 3, 9, 15, 15, 1, 11, 4, 27, 1, 5, 1, 9, 10, 14, 1, 27, 11, 21, 18, 6, 1, 50, 2, 45, 22, 15, 1, 18, 1, 18, 50, 243, 24, 12, 1, 27, 7, 3, 1, 90, 1, 20, 21, 33, 30, 16, 1, 81, 125, 21, 1, 30, 3, 23, 30, 54, 1, 15, 40, 21

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Prime shifted analog of A247074.

Each term a(n) is a divisor of A340072(n).

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A003972, A247074, A340072, A340148, A340149 (odd part).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340147(n) = A247074(A003961(n));

a(n) = A247074(A003961(n)).

a(n) = A003972(n) / A340148(n).

cp's OEIS Frontend

A340075 The odd part of A340072(n).

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A340071 a(n) = gcd(A003961(n)-1, phi(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes.

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A340082 a(n) = A003958(n) / gcd(n-1, A003958(n)).

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A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

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A340147 a(n) = A247074(A003961(n)).

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