cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A339901 a(n) = A339971(n) / gcd(A339809(2*n), A339971(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 15, 3, 5, 15, 1, 3, 3, 3, 1, 9, 9, 9, 15, 15, 5, 15, 9, 45, 5, 45, 1, 1, 1, 1, 3, 3, 1, 3, 5, 1, 5, 5, 5, 15, 15, 15, 3, 3, 1, 3, 9, 9, 3, 9, 1, 15, 15, 15, 15, 9, 45, 45, 1, 9, 9, 9, 9, 27, 27, 27, 45, 45, 5, 45, 135, 135, 45, 135, 9, 27, 27, 27, 3, 81, 81, 81, 135, 27, 45, 135, 405
Offset: 0

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Comments

Compare also to the scatter plot of A339898.

Links

Crossrefs

Cf. A000265, A053575, A019565, A160595, A339809, A339821, A339971, A339898, A339899, A340075, A340085.
Cf. A339973 (positions of ones).

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339901(n) = { my(x=A019565(2*n), y=A000265(eulerphi(x))); y/gcd((x-1),y); };

Formula

a(n) = A339971(n) / A339899(n).
a(n) = A000265(A160595(A019565(2*n))).
a(n) = A340075(A019565(n)) = A340085(A019565(2*n)).

A339812 Number of prime divisors of (A019565(n) - 1), counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 4, 2, 2, 2, 5, 2, 4, 1, 3, 2, 3, 3, 3, 1, 8, 1, 2, 1, 3, 2, 2, 2, 6, 2, 2, 1, 4, 1, 5, 2, 2, 3, 4, 1, 2, 3, 3, 1, 4, 1, 6, 1, 6, 3, 3, 2, 5, 1, 2, 1, 4, 2, 3, 1, 4, 2, 2, 1, 2, 2, 3, 2, 5, 2, 4, 2, 3, 1, 6, 2, 2, 3, 3, 2, 4, 1, 3, 1, 5, 2, 2, 2, 4, 4, 2, 3, 6, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Links

Crossrefs

Cf. A001222, A019565, A339809, A339810.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339812(n) = bigomega(A019565(n)-1);

Formula

a(n) = A001222(A339809(n)) = A001222(A019565(n)-1).
a(n) = A001222(A339810(n)).

A339814 The exponent of the highest power of 2 dividing (A019565(2n) - 1).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 1, 5, 1, 2, 2, 1, 7, 1, 2, 1, 6, 1, 1, 4, 1, 2, 1, 2, 1, 5, 3, 1, 2, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 1, 4, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 3, 1, 4, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 6, 1, 2, 1, 2, 1, 4, 1, 1, 5, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 5, 1
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Comments

The 2-adic valuation of A339809(2n).

Links

Crossrefs

Bisection of A339813.
Cf. A003961, A007814, A019565, A339809, A339815, A339822.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339814(n) = valuation((A019565(2*n)-1),2);

Formula

a(n) = A007814(A339809(2*n)) = A007814(A019565(2*n)-1).
a(n) = A007814(A003961(A019565(n))-1).

A339898 a(n) = A019565(2n)-1 mod A000265(phi(A019565(2n))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 4, 4, 1, 5, 9, 14, 0, 2, 1, 2, 0, 2, 4, 5, 7, 8, 9, 14, 10, 32, 9, 29, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 4, 4, 3, 11, 4, 14, 1, 2, 0, 2, 7, 5, 3, 2, 0, 2, 4, 14, 6, 20, 34, 14, 0, 2, 4, 5, 24, 20, 16, 23, 28, 41, 9, 29, 112, 68, 24, 74, 3, 11, 19, 5, 27, 2, 58, 14, 16, 50, 84, 119, 388, 356
Offset: 0

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A053575, A019565, A339809, A339821, A339971, A339899, A339901.
Cf. A339973 (positions of zeros).

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339898(n) = { my(x=A019565(2*n)); ((x-1)%A000265(eulerphi(x))); };

Formula

a(n) = A339809(2*n) modulo A339971(n), where A339971(n) = A053575(A019565(2n)).

A339810 a(n) = A046523(A019565(n) - 1).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 2, 6, 2, 12, 2, 6, 6, 24, 6, 6, 6, 32, 6, 24, 2, 12, 6, 12, 12, 30, 2, 384, 2, 6, 2, 12, 4, 6, 6, 64, 6, 6, 2, 60, 2, 48, 6, 6, 12, 60, 2, 6, 30, 12, 2, 210, 2, 96, 2, 216, 30, 30, 6, 180, 2, 6, 2, 16, 6, 12, 2, 60, 4, 6, 2, 6, 6, 12, 6, 120, 6, 24, 6, 30, 2, 240, 6, 6, 30, 12, 6, 60, 2, 30, 2, 48
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Links

Crossrefs

Cf. A019565, A046523, A339809, A339811 (rgs-transform), A339812.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A339810(n) = A046523(A019565(n)-1);

Formula

a(n) = A046523(A339809(n)) = A046523(A019565(n) - 1).

A339815 Let x = A019565(2*n); a(n) is the difference between 2-adic valuations of phi(x) and (x-1).

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 1, 0, -3, 2, 2, 0, 2, -3, 4, 0, 2, -2, 4, 2, 0, 4, 4, 2, 2, 4, 1, 1, 4, 4, 6, 0, 4, 4, 6, 4, 4, 6, 5, 4, 2, 6, 6, 4, 6, 4, 8, 4, 6, 4, 8, 6, 3, 8, 8, 6, 6, 8, 6, 5, 8, 8, 10, 0, -1, 2, 2, 0, 2, 1, 4, -2, 2, 2, 4, 2, 2, 4, 3, 2, 2, 4, 3, -2, 4, 4, 6, 2, 4, 2, 6, 4, 1, 6, 6, 4, 3, 6, 6, 4, 6, 5, 8, 1, 6
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Links

Crossrefs

Cf. A000010, A007814, A019565, A339809, A339814, A339821, A339822.
Cf. A339816 (indices of terms < 1).

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339815(n) = { my(x=A019565(2*n)); valuation(eulerphi(x),2)-valuation(x-1,2); };

Formula

a(n) = A339822(n) - A339814(n).
a(n) = A007814(A000010(A019565(2n))) - A007814(A019565(2n)-1).

A339899 a(n) = gcd(A019565(2n)-1, A000265(phi(A019565(2n)))).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 15, 1, 1, 1, 3, 5, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 27, 1, 1, 1, 1, 5, 3, 1, 1, 1, 81, 1, 1, 1, 3, 1, 1, 1, 9, 1, 3, 1
Offset: 0

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A000010, A000265, A049559, A053575, A019565, A339809, A339821, A339971, A339898, A339901.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339899(n) = { my(x=A019565(2*n)); gcd((x-1),A000265(eulerphi(x))); };

Formula

a(n) = gcd(A339809(2*n), A339971(n)), where A339971(n) = A053575(A019565(2n)).
a(n) = gcd(A339971(n), A339898(n)).
a(n) = A339971(n) / A339901(n).
a(n) = A000265(A049559(A019565(2*n))).

A339973 Numbers k for which A019565(2k)-1 is a multiple of A000265(phi(A019565(2k))).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 16, 20, 32, 33, 34, 35, 38, 41, 50, 56, 64, 128, 176, 256, 259, 290, 512, 1024, 2048, 2056, 2081, 2089, 2096, 2180, 4096, 4130, 8192, 9218, 16384, 18436, 32768, 65536, 131072, 131140, 262144, 279552, 524288, 524308, 524546, 1048576, 1048736, 2097152, 4194304, 4194352, 4194420, 4196656, 4202499, 8388608
Offset: 1

Views

Author

Antti Karttunen, Dec 26 2020

Keywords

Comments

Numbers k such that A339971(k) divides A339809(2k).
Union of {0}, A000079 and the terms of (1/2)*A048675(A339880(i)), for i >= 1, sorted into ascending order.

Links

Crossrefs

Positions of zeros in A339898, and of ones in A339901.
Cf. A000010, A000265, A048675, A339821, A339880, A339809, A339971, A339974.
Cf. A000079 (subsequence).
Cf. also A339816, A339906.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339971(n) = { my(x=A019565(2*n)); !((x-1)%A000265(eulerphi(x))); };

A339811 Lexicographically earliest infinite sequence such that a(i) = a(j) => A339810(i) = A339810(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Links

Crossrefs

Cf. A019565, A046523, A339809, A339810, A339812.

Programs

  • PARI
    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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A339810(n) = A046523(A019565(n)-1);
v339811 = rgs_transform(vector(up_to, n, A339810(n)));
A339811(n) = v339811[n];

A339820 a(n) = phi(A019565(n)), where phi is Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 6, 6, 12, 12, 24, 24, 48, 48, 10, 10, 20, 20, 40, 40, 80, 80, 60, 60, 120, 120, 240, 240, 480, 480, 12, 12, 24, 24, 48, 48, 96, 96, 72, 72, 144, 144, 288, 288, 576, 576, 120, 120, 240, 240, 480, 480, 960, 960, 720, 720, 1440, 1440, 2880, 2880, 5760, 5760, 16, 16, 32, 32, 64, 64, 128, 128
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Crossrefs

Cf. A000010, A019565, A339821 (bisection).
Cf. also A324650, A339809.

Programs

  • PARI
    A339820(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= (p-1)); n >>= 1); (m); };

Formula

If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A006093(e1) * A006093(e2) * ... * A006093(ek).
a(n) = A000010(A019565(n)).
Showing 1-10 of 11 results. Next

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