A282291 -id:A282291 - cp's OEIS frontend

A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

1, 2, 6, 3, 24, 12, 4, 8, 120, 60, 20, 5, 40, 10, 30, 15, 840, 420, 140, 35, 7, 280, 70, 14, 210, 105, 21, 168, 84, 28, 56, 7560, 42, 1890, 945, 315, 63, 9, 3780, 1260, 252, 36, 2520, 630, 126, 18, 1512, 756, 189, 27, 378, 54, 1080, 540, 180, 45, 360, 90, 270, 135, 83160, 504, 72, 216, 108, 41580, 13860, 3465, 693, 99, 11, 27720, 6930, 1386, 198, 22, 20790

Offset: 0

Antti Karttunen, May 06 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A304086 (inverse).
Cf. A304083, A304087.
Cf. also A064736, A113552, A207901, A281978, A282291, A302350, A302781, A302783, A303751, A303771 for similar permutations.

up_to_e = 16; \\ Good for computing up to n = (2^16)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A304085(n) = A052330(A304083(n)); \\ Needs also code from A304083

a(n) = A052330(A304083(n)).

A304755 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the second smallest divisor of a(n-1) not already present in sequence, or the smallest divisor if it is the only one not yet used, or (if all divisors have been already encountered), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 36, 18, 9, 72, 24, 8, 216, 54, 27, 108, 540, 10, 5, 20, 60, 30, 15, 120, 40, 360, 90, 45, 180, 1260, 14, 7, 28, 84, 42, 21, 168, 56, 504, 126, 63, 252, 6300, 35, 70, 210, 105, 420, 140, 3780, 189, 378, 1890, 270, 135, 1080, 7560, 315, 630, 6930, 22, 11, 44, 132, 66, 33, 264, 88, 792, 198, 99, 396, 1980, 110, 55

Offset: 1

Antti Karttunen, May 20 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences that are permutations of the natural numbers

Cf. A304756 (inverse).
Cf. A053669, A304757.
Cf. A303751, also A282291, A304531 for variants.

up_to = 2^16;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v304755 = vector(up_to);
m304756 = Map();
find_kth_unused_divisor(k,n,m_inverses) = { my(pd=0); fordiv(n,d,if(!mapisdefined(m_inverses,d),pd=d;k--); if((!k || (d == n)), return(pd))); };
prev=1; for(n=1,up_to, if((try = find_kth_unused_divisor(2,prev,m304756))!=0, mapput(m304756,v304755[n] = try,n), p = A053669(prev); while(mapisdefined(m304756,prev), prev *= p); v304755[n] = prev; mapput(m304756,prev,n)); prev = v304755[n]);
A304755(n) = v304755[n];
A304756(n) = mapget(m304756,n);

A298480 Lexicographically earliest sequence of distinct positive terms such that the Fermi-Dirac factorizations of two consecutive terms differ by exactly one factor.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 8, 24, 120, 30, 10, 5, 15, 60, 20, 40, 280, 56, 14, 7, 21, 42, 168, 84, 28, 140, 35, 70, 210, 105, 420, 840, 7560, 1080, 216, 54, 18, 9, 27, 108, 36, 72, 360, 90, 45, 135, 270, 1890, 378, 126, 63, 189, 756, 252, 504, 1512, 16632, 1848, 264

Offset: 1

Rémy Sigrist, Jul 21 2018

nonn

For Fermi-Dirac representation of n see A182979. - N. J. A. Sloane, Jul 21 2018
For any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) belongs to A050376.
This sequence has similarities with A282291; in both sequences, each pair of consecutive terms contains a term that divides the other.

			The first terms, alongside a(n+1)/a(n), are:
  n   a(n)  a(n+1)/a(n)
  --  ----  -----------
   1     1        2
   2     2        3
   3     6      1/2
   4     3        2^2
   5    12      1/3
   6     4        2
   7     8        3
   8    24        5
   9   120      1/2^2
  10    30      1/3
  11    10      1/2
  12     5        3
  13    15        2^2
  14    60      1/3
  15    20        2
  16    40        7
  17   280      1/5
  18    56      1/2^2
  19    14      1/2
  20     7        3

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A298480

Cf. A000120, A050376, A052330, A052331, A282291, A182979.

PARI
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See Links section.
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A000120(A052331(a(n)) XOR A052331(a(n+1))) = 1 for any n > 0 (where XOR denotes the bitwise XOR operator).

Apparently, a(n) = A052330(A163252(n-1)) for any n > 0.

A302854 Inverse of A302853: if A302853(k) = n, a(n) = k, or -1 if n does not occur in A302853.

Original entry on oeis.org

0, 1, 3, 2, 5, 25, 4, 26, 7, 8, 10, 9, 6, 28, 11, 27, 13, 14, 16, 15, 18, 30, 17, 31, 20, 21, 23, 22, 19, 29, 12, 24, 65, 66, 7483, 7484, 70, 90, 7488, 7508, 68, 67, 7486, 7485, 69, 91, 7487, 7509, 72, 73, 7490, 7491, 71, 93, 7489, 7511, 75, 74, 7493, 7492, 76, 92, 7494, 7510, 33, 34, 36, 35, 38, 58, 37, 59, 40, 41, 43, 42, 39

Offset: 0

Antti Karttunen, May 17 2018

nonn

This is a left inverse of A302853, and also the right inverse if A282291 (and thus also A302853) is surjective (a permutation of natural numbers), in which case the fallback-clause is unnecessary.

Antti Karttunen, Table of n, a(n) for n = 0..259

Cf. A302853 (inverse).

\\ For other needed code, follow A050376 and A282291:
A052330(n) = { my(p=1, i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A302854(n) = (A304090(A052330(n))-1);

For all n >= 0, a(A302853(n)) = n.

A304097 Restricted growth sequence transform of A278222(A302853(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 17 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..60822

Cf. A278222, A302791, A282291, A302853, A304098.

Compare also the scatter-plot to that of A304535.

\\ Needs also code from A282291 and A302853:
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
write_to_bfile(0,rgs_transform(vector(60823,n,A278222(A302853(n-1)))),"b304097.txt");

A338221 Square spiral of distinct positive integers built by greedy algorithm such that each new value (except the initial one) is a divisor or a multiple of some earlier horizontally or vertically adjacent value.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 8, 16, 12, 20, 40, 24, 9, 18, 36, 30, 15, 45, 90, 50, 14, 28, 32, 64, 48, 60, 80, 120, 240, 160, 72, 27, 54, 108, 216, 144, 150, 25, 75, 180, 360, 270, 100, 42, 21, 63, 126, 252, 84, 96, 192, 320, 480, 720, 1440, 288, 576, 432, 81, 162

Offset: 0

Rémy Sigrist, Jan 30 2021

This sequence has similarities with A113552 and A282291, as each term is adjacent to one of its divisors or multiples.

			The spiral begins:
      216--108---54---27---72--160--240
        |                             |
      144   36---18----9---24---40  120
        |    |                   |    |
      150   30    6----3----4   20   80
        |    |    |         |    |    |
       25   15    5    1----2   12   60
        |    |    |              |    |
       75   45   10----7----8---16   48
        |    |                        |
      180   90---50---14---28---32---64
        |
      360--270--100---42---21---63--126

Rémy Sigrist, Table of n, a(n) for n = 0..10200
Rémy Sigrist, Colored representation of the spiral for -500 <= x, y <= 500 (where the color is function of a(n))
Rémy Sigrist, Colored representation of the spiral for -500 <= x, y <= 500 (where the color is function of A006530(a(n)))
Rémy Sigrist, PARI program for A338221

Cf. A006530, A113552, A282291.

PARI
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See Links section.
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cp's OEIS Frontend

A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

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A298480 Lexicographically earliest sequence of distinct positive terms such that the Fermi-Dirac factorizations of two consecutive terms differ by exactly one factor.

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A302854 Inverse of A302853: if A302853(k) = n, a(n) = k, or -1 if n does not occur in A302853.

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A304097 Restricted growth sequence transform of A278222(A302853(n)).

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A338221 Square spiral of distinct positive integers built by greedy algorithm such that each new value (except the initial one) is a divisor or a multiple of some earlier horizontally or vertically adjacent value.

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Crossrefs

Programs

PARI