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.

Previous Showing 11-14 of 14 results.

A286533 Restricted growth sequence of A278533 (prime-signature of A253563).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 2, 6, 7, 7, 5, 4, 5, 3, 2, 8, 9, 10, 7, 9, 11, 7, 5, 6, 7, 7, 5, 4, 5, 3, 2, 12, 13, 14, 9, 14, 15, 10, 7, 13, 15, 15, 11, 9, 11, 7, 5, 8, 9, 10, 7, 9, 11, 7, 5, 6, 7, 7, 5, 4, 5, 3, 2, 16, 17, 18, 13, 19, 20, 14, 9, 18, 21, 21, 15, 14, 15, 10, 7, 17, 20, 21, 15, 20, 22, 15, 11, 13, 15, 15, 11, 9, 11, 7, 5, 12, 13, 14, 9, 14, 15, 10, 7
Offset: 0

Views

Author

Antti Karttunen, May 17 2017

Keywords

Links

Crossrefs

Cf. A046523, A253563, A286531, A286535, A286622.

Programs

  • PARI
    rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253563(n) = if(n<2,(1+n),if(!(n%2),A253560(A253563(n/2)),A253550(A253563((n-1)/2)))); \\ Would be better if memoized!
A278533(n) = A046523(A253563(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278533(n-1))),"b286533.txt");

A286535 Restricted growth sequence of A278535 (prime-signature of A253565).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 6, 6, 7, 2, 3, 4, 5, 4, 6, 6, 7, 4, 6, 8, 9, 6, 10, 9, 11, 2, 3, 4, 5, 4, 6, 6, 7, 4, 6, 8, 9, 6, 10, 9, 11, 4, 6, 8, 9, 8, 12, 12, 13, 6, 10, 12, 14, 9, 14, 13, 15, 2, 3, 4, 5, 4, 6, 6, 7, 4, 6, 8, 9, 6, 10, 9, 11, 4, 6, 8, 9, 8, 12, 12, 13, 6, 10, 12, 14, 9, 14, 13, 15, 4, 6, 8, 9, 8, 12, 12, 13, 8, 12, 16, 17, 12, 18
Offset: 0

Views

Author

Antti Karttunen, May 17 2017

Keywords

Links

Crossrefs

Cf. A046523, A253565, A286531, A286533, A286622.

Programs

  • PARI
    rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253565(n) = if(n<2,(1+n),if(!(n%2),A253550(A253565(n/2)),A253560(A253565((n-1)/2)))); \\ Would be better if memoized!
A278535(n) = A046523(A253565(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278535(n-1))),"b286535.txt");

A277331 a(n) = A253563(A003714(n)).

Original entry on oeis.org

1, 2, 4, 8, 6, 16, 12, 18, 32, 24, 36, 54, 30, 64, 48, 72, 108, 60, 162, 90, 150, 128, 96, 144, 216, 120, 324, 180, 300, 486, 270, 450, 750, 210, 256, 192, 288, 432, 240, 648, 360, 600, 972, 540, 900, 1500, 420, 1458, 810, 1350, 2250, 630, 3750, 1050, 1470, 512, 384, 576, 864, 480, 1296, 720, 1200, 1944, 1080, 1800, 3000, 840
Offset: 0

Views

Author

Antti Karttunen, Oct 12 2016

Keywords

Comments

After the initial terms 1, 2 and 4, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 4, but with such a restriction that A253550 may not be applied twice in succession.
A permutation of A055932.

Links

Crossrefs

Cf. A003714, A055932 (same sequence sorted into ascending order), A253550, A253560, A253563, A122111.
Cf. also A277006, A277332.

Programs

Formula

a(n) = A253563(A003714(n)).
a(n) = A122111(A277006(n)).

A381500 a(n) = A019565(A187769(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 195, 182, 273, 455, 286, 429, 715, 1001, 390, 546, 910, 1365, 858, 1430, 2145, 2002, 3003
Offset: 0

Views

Author

Michael De Vlieger, Peter Munn, and Antti Karttunen, Feb 27 2025

Keywords

Comments

The squarefree numbers, ordered first by largest prime factor (dividing the sequence into rows), then by number of prime factors, then lexicographically by their prime factors (written in descending order).
We index (a(n)) from offset 0, matching the choice for A019565 and similar sequences.

Examples

			Table begins:
  Row 0:  1;
  Row 1:  2;
  Row 2:  3,  6;
  Row 3:  5, 10, 15, 30;
  Row 4:  7, 14, 21, 35, 42, 70, 105, 210;
  Row 5: 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310;
  ...
Table of a(n) for n = 0..31, demonstrating relationship of this sequence with s = A187769:
          <-factors                    <-factors
   n  a(n)  2 3 5 7  s(n)  |   n   a(n)  2 3 5 7 11 s(n)
  -------------------------|----------------------------
   0    1   .          0   |  16    11   . . . . x   16
   1    2   x          1   |  17    22   x . . . x   17
   2    3   . x        2   |  18    33   . x . . x   18
   3    6   x x        3   |  19    55   . . x . x   20
   4    5   . . x      4   |  20    77   . . . x x   24
   5   10   x . x      5   |  21    66   x x . . x   19
   6   15   . x x      6   |  22   110   x . x . x   21
   7   30   x x x      7   |  23   165   . x x . x   22
   8    7   . . . x    8   |  24   154   x . . x x   25
   9   14   x . . x    9   |  25   231   . x . x x   26
  10   21   . x . x   10   |  26   385   . . x x x   28
  11   35   . . x x   12   |  27   330   x x x . x   23
  12   42   x x . x   11   |  28   462   x x . x x   27
  13   70   x . x x   13   |  29   770   x . x x x   29
  14  105   . x x x   14   |  30  1155   . x x x x   30
  15  210   x x x x   15   |  31  2310   x x x x x   31
  -------------------------|----------------------------
            1 2 4 8  s(n)  |             1 2 4 8 16 s(n)
             bits->                         bits->

Links

Crossrefs

Cf. A019565, A119416, A163866, A187769, A253550, A294648, A344085.
A002110, A098012 are subsequences.

Programs

  • Mathematica
    a187769 = {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten; a019565[x_] := Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[x, 2]; Map[a019565, a187769]

Formula

a(n) = A019565(A187769(n)).
As an irregular triangle T(n,k), where row 0 = {1}:
For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.)
S(-1,-1) = (1).
For n >= 0:
S(n-1, n) = (); S(n, -1) = ();
for 0 <= m <= n, S(n,m) = ( A253550'(S(n-1, m)), A119416'(S(n-1, m-1)) ), where Axxx'((i_1, i_2, ..., i_j)) denotes Axxx(i_1), Axxx(i_2), ..., Axxx(i_j).
a(A163866(n)) = A098012(n).
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.