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 1-3 of 3 results.

A371906 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 13, 12, 14, 11, 27, 24, 56, 49, 55, 54, 118, 117, 245, 240, 250, 235, 491, 488, 492, 461, 463, 454, 966, 961, 1985, 1984, 2002, 1939, 1951, 1948, 3996, 3869, 3903, 3898, 7994, 7985, 16177, 16160, 16166, 15911, 32295, 32292, 32300, 32297, 32363
Offset: 1

Views

Author

Michael De Vlieger, Apr 15 2024

Keywords

Comments

The only powers of 2 in the sequence are likely 1 and 2.

Examples

			a(1) = 0 since n = 1 is the empty product.
a(2) = 1 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 2^(1-1) = 1.
a(3) = 3 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3.
a(4) = 2 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 2^(2-1) = 2.
a(5) = 6 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 2^(2-1) + 2^(3-1) = 2 + 4 = 6, etc.
Table relating a(n) with b(n), diagramming powers of 2 with "x" that sum to a(n), or prime factors with "x" that produce b(n), where b(n) = A372000(n).
             Power of 2
   n   a(n)  01234567      b(n)
  ----------------------------
   1     0   .               1
   2     1   x               2
   3     3   xx              6
   4     2   .x              3
   5     6   .xx            15
   6     5   x.x            10
   7    13   x.xx           70
   8    12   ..xx           35
   9    14   .xxx          105
  10    11   xx.x           42
  11    27   xx.xx         462
  12    24   ...xx          77
  13    56   ...xxx       1001
  14    49   x...xx        286
  15    55   xxx.xx       4290
  16    54   .xx.xx       2145
  17   118   .xx.xxx     36465
  18   117   x.x.xxx     24310
  19   245   x.x.xxxx   461890
  20   240   ....xxxx    46189
  ----------------------------
                 1111
             23571379
             Prime factor

Links

Crossrefs

Cf. A008336, A260850, A372000.

Programs

  • Mathematica
    Table[Total[2^(-1 + Select[Range@ PrimePi[n], OddQ@ Quotient[n, Prime[#]] &])], {n, 50}]
  • PARI
    a(n) = sum(k=1, primepi(n), if (n\prime(k) % 2, 2^(k-1))); \\ Michel Marcus, Apr 16 2024

A371907 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is even.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 1, 4, 4, 7, 7, 14, 8, 9, 9, 10, 10, 15, 5, 20, 20, 23, 19, 50, 48, 57, 57, 62, 62, 63, 45, 108, 96, 99, 99, 226, 192, 197, 197, 206, 206, 223, 217, 472, 472, 475, 467, 470, 404, 437, 437, 438, 418, 427, 297, 808, 808, 815, 815, 1838, 1828
Offset: 1

Views

Author

Michael De Vlieger, Apr 17 2024

Keywords

Comments

This is a transform of A372007(n) = s(n). Write the prime indices of k factors prime(k) | s(n) instead as 2^(k-1) and take the sum for all primes p | s(n). Hence, s(14) = 105 = 3*5*7 becomes a(14) = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14.

Examples

			a(1) = 0 since n = 1 is the empty product.
a(2) = 0 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 0.
a(3) = 0 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 0.
a(4) = 1 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(4) = 2^(1-1) = 1.
a(8) = 1 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3, etc.
Table relating a(n) with b(n), s(n), and t(n), diagramming powers of 2 with "x" that sum to a(n) or b(n), or prime factors with "x" that produce s(n) or t(n). Sequences s(n) = A372007(n), t(n) = A372000(n), c(n) = A034386(n), b(n) = A371906(n), and c(n) = A357215(n) = a(n) + b(n). Column A (at top) shows powers of 2 that sum to a(n), with B same for b(n), while column S represents prime factors of s(n), T same of t(n).
      [A] 2^k     [B] 2^k
   n   0123  a(n)  012345  b(n)   c(n)   s(n)   t(n)  v(n)
  --------------------------------------------------------
   1   .       0   .         0   2^0-1     1      1   P(0)
   2   .       0   x         1   2^1-1     1      2   P(1)
   3   .       0   xx        3   2^2-1     1      6   P(2)
   4   x       1   .x        2   2^2-1     2      3   P(2)
   5   x       1   .xx       6   2^3-1     2     15   P(3)
   6   .x      2   x.x       5   2^3-1     3     10   P(3)
   7   .x      2   x.xx     13   2^4-1     3     70   P(4)
   8   xx      3   ..xx     12   2^4-1     6     35   P(4)
   9   x       1   .xxx     14   2^4-1     2    105   P(4)
  10   ..x     4   xx.x     11   2^4-1     5     42   P(4)
  11   ..x     4   xx.xx    27   2^5-1     5    462   P(5)
  12   xxx     7   ...xx    24   2^5-1    30     77   P(5)
  13   xxx     7   ...xxx   56   2^6-1    30   1001   P(6)
  14   .xxx   14   x...xx   49   2^6-1   105    286   P(6)
  15   ...x    8   xxx.xx   55   2^6-1     7   4290   P(6)
  16   x..x    9   .xx.xx   54   2^6-1    14   2145   P(6)
  --------------------------------------------------------
       2357        [T] 11
       [S]         235713

Links

Crossrefs

Cf. A357215, A371906, A372007.

Programs

  • Mathematica
    Table[Total[2^(-1 + Select[Range@ PrimePi[n], EvenQ@ Quotient[n, Prime[#]] &])], {n, 50}]
  • PARI
    a(n) = my(vp=primes([1, n])); vecsum(apply(x->2^(x-1), Vec(select(x->(((n\x) % 2)==0), vp, 1)))); \\ Michel Marcus, Apr 30 2024

Formula

a(n) = A357215(n) - A371906(n).

A372007 a(n) = product of those prime(k) such that floor(n/prime(k)) is even.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 2, 5, 5, 30, 30, 105, 7, 14, 14, 21, 21, 210, 10, 55, 55, 330, 66, 429, 143, 2002, 2002, 15015, 15015, 30030, 910, 7735, 221, 1326, 1326, 12597, 323, 3230, 3230, 33915, 33915, 746130, 49742, 572033, 572033, 3432198, 490314, 1225785, 24035
Offset: 1

Views

Author

Michael De Vlieger, Apr 17 2024

Keywords

Comments

The only primes in the sequence are 2, 3, 5, and 7.

Examples

			a(1) = 1 since n = 1 is the empty product.
a(2) = 1 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
a(3) = 1 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd.
a(4) = 2 since for n = 4 and p = 2, floor(4/2) = 2 is even, but for p = 3, floor(4/3) = 1 is odd. Therefore, a(4) = 2.
a(5) = 2 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(5) = 2.
a(8) = 6 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2*3 = 6, etc.
Table relating a(n) with b(n), s(n), and t(n), diagramming prime factors with "x" that produce a(n) or b(n), or powers of 2 with "x" that sum to s(n) or t(n). Sequences b(n) = A372000(n), c(n) = A034386(n), s(n) = A371907(n), t(n) = A371906(n), and v(n) = A357215(n) = s(n) + t(n). Column A represents prime factors of a(n), B same of b(n), while column S (at bottom) shows powers of 2 that sum to s(n), with T same for t(n). P(n) = A002110(n).
       [A]          [B] 11
   n   2357   a(n)  235713    b(n)  c(n)  s(n) t(n)   v(n)
  --------------------------------------------------------
   1   .        1   .           1   P(0)    0    0   2^0-1
   2   .        1   x           2   P(1)    0    1   2^1-1
   3   .        1   xx          6   P(2)    0    3   2^2-1
   4   x        2   .x          3   P(2)    1    2   2^2-1
   5   x        2   .xx        15   P(3)    1    6   2^3-1
   6   .x       3   x.x        10   P(3)    2    5   2^3-1
   7   .x       3   x.xx       70   P(4)    2   13   2^4-1
   8   xx       6   ..xx       35   P(4)    3   12   2^4-1
   9   x        2   .xxx      105   P(4)    1   14   2^4-1
  10   ..x      5   xx.x       42   P(4)    4   11   2^4-1
  11   ..x      5   xx.xx     462   P(5)    4   27   2^5-1
  12   xxx     30   ...xx      77   P(5)    7   24   2^5-1
  13   xxx     30   ...xxx   1001   P(6)    7   56   2^6-1
  14   .xxx   105   x...xx    286   P(6)   14   49   2^6-1
  15   ...x     7   xxx.xx   4290   P(6)    8   55   2^6-1
  16   x..x    14   .xx.xx   2145   P(6)    9   54   2^6-1
  --------------------------------------------------------
       0123         012345
     [S] 2^k        [T] 2^k

Links

Crossrefs

Cf. A005117, A034386, A371907, A372000.

Programs

  • Mathematica
    Table[Times @@ Select[Prime@ Range@ PrimePi[n], EvenQ@ Quotient[n, #] &], {n, 51}] (* or *)
Table[Product[Prime[i], {j, PrimePi[n]}, {i, 1 + PrimePi[Floor[n/(2  j + 1)]], PrimePi[Floor[n/(2  j)]]}], {n, 51}]
  • PARI
    a(n) = my(vp=primes([1, n])); vecprod(select(x->(((n\x) % 2)==0), vp)); \\ Michel Marcus, Apr 30 2024

Formula

a(n) = A034386(n) / A372000(n).
a(n) = Product_{k = 1..pi(n)} Product_{j = 1+floor(n/(2*k+1))..floor(n/(2*k))} prime(j), where pi(x) = A000720(n).
Showing 1-3 of 3 results.

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