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 1-6 of 6 results.

A339816 Numbers k for which A339814(k) >= A339822(k).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 12, 14, 16, 18, 21, 32, 64, 65, 68, 72, 84, 128, 129, 132, 136, 138, 141, 145, 159, 170, 192, 204, 208, 256, 258, 261, 324, 385, 448, 462, 512, 513, 515, 516, 520, 536, 576, 578, 581, 640, 705, 723, 776, 908, 912, 1024, 1036, 1040, 1049, 1160, 1172, 1280, 1352, 1537, 1600, 1609, 1666, 1732, 1795, 2048
Offset: 1

Views

Author

Antti Karttunen, Dec 19 2020

Keywords

Comments

Terms of (1/2)*A048675(A339817(i)), for i = 2.., sorted into ascending order.
First occurrences of terms with binary weight (A000120) w = 1..19 are at n=1, 4, 8, 23, 50, 25, 125, 136, 176, 502, 749, 1142, 791, 1882, 2913, 4327, 17979, 16991, 12441. The terms themselves are: 1, 5, 14, 141, 908, 159, 8921, 9948, 18390, 175449, 400237, 1223389, 441805, 3234271, 28743379, 53892047, 1631024969, 1331412056, 725475951.

Links

Crossrefs

Positions of zeros and negative terms in A339815.
Cf. A000010, A000120, A007814, A019565, A048675, A339814, A339817, A339822.
Cf. A000079 (a subsequence).

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339816(n) = { my(x=A019565(2*n)); (valuation(eulerphi(x),2)<=valuation(x-1,2)); };

A339821 a(n) = phi(A019565(2n)), where phi is Euler totient function.

Original entry on oeis.org

1, 2, 4, 8, 6, 12, 24, 48, 10, 20, 40, 80, 60, 120, 240, 480, 12, 24, 48, 96, 72, 144, 288, 576, 120, 240, 480, 960, 720, 1440, 2880, 5760, 16, 32, 64, 128, 96, 192, 384, 768, 160, 320, 640, 1280, 960, 1920, 3840, 7680, 192, 384, 768, 1536, 1152, 2304, 4608, 9216, 1920, 3840, 7680, 15360, 11520, 23040, 46080, 92160
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Crossrefs

Bisection of A339820.
Cf. A000010, A003961, A003972, A006093, A019565, A339822 (2-adic valuation).
Cf. also A324651.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339821(n) = eulerphi(A019565(n+n));
  • PARI
    A339821(n) = { my(m=1, p=2); while(n>0, p = nextprime(1+p); if(n%2, m *= (p-1)); n >>= 1); (m); };

Formula

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A006093(e1) * A006093(e2) * ... * A006093(ek).
a(n) = A339820(2n) = A000010(A019565(2n)) = A000010(A019565(2n+1)).
a(n) = A003972(A019565(n)) = A000010(A003961(A019565(n))).

A339971 Odd part of A339821(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 26 2020

Keywords

Crossrefs

Cf. A000010, A000265, A003961, A019565, A053575, A057023, A336466, A339821, A339822, A339970.
Cf. A339972 (quadrisection).

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A339971(n) = { my(m=1, p=2); while(n>0, p = nextprime(1+p); if(n%2, m *= A000265(p-1)); n >>= 1); (m); };

Formula

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A057023(e1) * A057023(e2) * ... * A057023(ek).
a(n) = A053575(A019565(2n)) = A336466(A003961(A019565(n))) = A000265(A339821(n)).
a(n) = A339821(n) / A000079(A339822(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).

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).

A339902 Number of prime divisors of A339821(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 5, 6, 7, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 4, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 7, 8, 9, 10, 9, 10, 11, 12, 9, 10, 11, 12, 11, 12, 13, 14, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 6, 7, 8, 9, 8, 9, 10, 11, 8, 9, 10, 11, 10, 11, 12, 13, 7
Offset: 0

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000010, A001222, A003972, A019565, A023508, A339821, A339822, A339906.

Programs

  • PARI
    A339902(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += bigomega(p-1)); n >>= 1); (s); };

Formula

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A023508(e1) + A023508(e2) + ... + A023508(ek).
a(n) = A001222(A339821(n)).
a(n) = A001222(A003972(A019565(n))) = A001222(A000010(A019565(2n))).
a(n) >= A339822(n).
Showing 1-6 of 6 results.

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