A340083 -id:A340083 - cp's OEIS frontend

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

1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 8, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 12, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 16, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340082, A340083.

Cf. also A049559, A340071.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340081(n) = gcd(n-1, A003958(n));

a(n) = gcd(n-1, A003958(n)).
a(n) = A003958(n) / A340082(n).
For n > 1, a(n) = (n-1) / A340083(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).

A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 7, 57, 1, 59, 1, 61, 31, 63, 1, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

From the second term onward, the odd part of A340083.

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

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340083, A340084, A340085.

Cf. also A160596.

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A340086(n) = if(1==n,0,my(u=A336466(n)); A000265(n-1)/gcd(n-1, u));

a(1) = 0; for n > 1, a(n) = A000265(n-1) / A340084(n) = A000265(A340083(n)).

cp's OEIS Frontend

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

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

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Author

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Crossrefs

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PARI

Formula

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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PARI

Formula

A340086 a(1) = 0, for n > 1, a(n) = A000265(n-1) / gcd(n-1, A336466(n)).

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PARI

Formula