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 11-18 of 18 results.

A328478 Divide n by the largest primorial that divides it and repeat until a fixed point is reached; a(n) is the fixed point.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 1, 25, 13, 27, 7, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 1, 73, 37, 75, 19, 77, 13, 79, 5, 81, 41, 83, 7, 85, 43, 87, 11, 89
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Links

Crossrefs

Cf. A007814 (gives the number of iterations to reach a fixed point), A025487 (indices of 1's).
Cf. A002110, A053589, A111701, A328479.
Cf. also A093411 for analogous sequence.

Programs

Formula

If A111701(n) == n, then a(n) = n, otherwise a(n) = a(A111701(n)).
a(n) = n / A328479(n).

Extensions

Definition clarified by N. J. A. Sloane, Jan 19 2021

A328479 a(n) = n/A328478(n), where A328478(n) is obtained by repeatedly dividing n by the largest primorial that divides it until a fixed point is reached.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 30, 1, 4, 1, 2, 1, 96, 1, 2, 1, 4, 1, 6, 1, 8, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Comments

a(n) is the largest term in A025487 that divides n evenly. - Hal M. Switkay, May 04 2021

Links

Crossrefs

Cf. A002110, A053589, A111701, A328478.

Programs

Formula

a(n) = n / A328478(n).

A358218 Number of prime factors (with multiplicity) in A328478(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 3, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 0, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 1, 1, 3, 0, 2, 1, 1, 1, 2, 2, 1, 0, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0, 1, 2, 3, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2022

Keywords

Links

Crossrefs

Cf. A001222, A025487 (positions of zeros), A328478, A355930.
Cf. A358219 (positions where differs from A358217).

Programs

Formula

a(n) = A001222(A328478(n)).
a(n) <= A355930(n).
Apparently, a(n) >= A358217(n) for all n.

A168265 a(n) = A003557(A060735(n)).

Original entry on oeis.org

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

Views

Author

Matthew Vandermast, Nov 23 2009

Keywords

Comments

A060735(n) belongs to A168264 if and only if a(n) belongs to A168267.
Looking at A060735 as an irregular triangle T(n,k) = k*A002110(n) with 1 <= k < prime(n+1), this sequence a(n) = k. - Michael De Vlieger, Jul 26 2016

Links

Crossrefs

Cf. A002110, A006093, A060735.

Programs

  • Mathematica
    Table[Range[Prime[n] - 1], {n, 9}] // Flatten (* or, per title definition: *)
#/Times @@ (FactorInteger[#][[All, 1]]) & /@ Flatten@ Table[Range[Prime[n + 1] - 1] Apply[Times, Prime@ Range@ n], {n, 0, 8}] (* Michael De Vlieger, Jul 26 2016 *)

Formula

Integers 1 to A006093(1) inclusive, followed by integers 1 to A006093(2) inclusive, etc.
a(n) = A111701(A060735(n)).
T(n,k)=k for n >= 1 and 1 <= k < prime(n).

A328399 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328475(i) = A328475(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 4, 5, 1, 6, 2, 7, 7, 8, 4, 9, 5, 10, 10, 11, 7, 12, 8, 13, 13, 14, 10, 15, 11, 16, 16, 17, 16, 18, 17, 19, 19, 20, 1, 21, 2, 22, 22, 23, 4, 24, 5, 25, 25, 26, 7, 27, 8, 28, 28, 29, 10, 30, 11, 31, 31, 32, 31, 33, 32, 34, 34, 35, 16, 36, 17, 37, 37, 38, 19, 39, 20, 40, 40, 41, 22, 42, 23, 43, 43, 44, 25, 45, 26, 46, 46, 47, 46, 48, 47
Offset: 0

Views

Author

Antti Karttunen, Oct 19 2019

Keywords

Comments

Restricted growth sequence transform of A328475, defined as A328475(n) = A111701(A276086(n)).

Links

Crossrefs

Cf. A002110, A053589, A111701, A276086, A143293 (indices of 1's after a(0)=1).
Cf. also A328477.

Programs

  • PARI
    up_to = 32768;
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; };
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328475(n) = A111701(A276086(n));
v328399 = rgs_transform(vector(up_to+1, n, A328475(n-1)));
A328399(n) = v328399[1+n];

A168266 A003557(A168264(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 6, 8, 1, 2, 4, 6, 8, 12, 1, 2, 4, 6, 8, 12, 1, 2, 4, 6, 8, 12, 1, 2, 4, 6, 8, 12, 1, 2, 4, 6, 8, 12, 24, 1, 2, 4, 6, 8, 12, 24, 1, 2, 4, 6, 8, 12, 24, 36, 1, 2, 4, 6, 8, 12, 24, 36, 1, 2, 4, 6, 8, 12, 24, 36, 1, 2, 4, 6, 8, 12, 24, 36, 1, 2, 4, 6, 8, 12, 24, 36, 48
Offset: 1

Views

Author

Matthew Vandermast, Nov 23 2009

Keywords

Crossrefs

For range of values, see A168267.

Formula

Also A111701(A168264(n)).

A330752 Number of values of k, 1 <= k <= n, with A328478(k) = A328478(n), where A328478(n) gives the remainder when all maximal primorial divisors of n (from the largest to smallest) have been divided out.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 30 2019

Keywords

Comments

Ordinal transform of A328478.

Links

Crossrefs

Cf. A111701, A328478, A330751.

Programs

  • Mathematica
    A111701[n_] := A111701[n] = Block[{m = n, k = 1}, While[IntegerQ[m/Prime[k]], m = m/Prime[k]; k++]; m];
A328478[n_] := A328478[n] = If[A111701[n] == n, n, A328478[A111701[n]]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A328478[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 11 2022, after Robert G. Wilson v in A111701 *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A328478(n) = { my(u=A111701(n)); if(u==n, return(n), return(A328478(u))); };
v330752 = ordinal_transform(vector(up_to, n, A328478(n)));
A330752(n) = v330752[n];

A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

Original entry on oeis.org

2, 3, 14, 15, 62, 63, 254, 255, 1022, 1023, 4094, 4095, 16382, 16383, 65534, 65535, 262142, 262143, 1048574, 1048575, 4194302, 4194303, 16777214, 16777215, 67108862, 67108863, 268435454, 268435455, 1073741822, 1073741823, 4294967294, 4294967295, 17179869182, 17179869183
Offset: 0

Views

Author

Paul Curtz, Jan 23 2017

Keywords

Comments

f(n) = (n+1)/A000918(n+2) = 1/2, 2/6, 3/14, 4/30, 5/62, 6/126, 7/254, 8/510, 9/1022, 10/2046, 11/4094, 12/8190, ... .
Partial reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 3/63, 7/254, 4/255, 9/1022, 5/1023, 11/4094, 6/4095, ... = A026741(n+1)/a(n).
Full reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 1/21, 7/254, ... = A111701(n+1)/(2, 3, 14, 15, 62, 21, ... )
A164555(n+1)/A027642(n) = 1/2, 1/6, 0, -1/30, 0, 1/42, ... = f(n) * A198631(n)/A006519(n+1) = 1, 1/2, 0, -1/4, 0, 1/2, ... .).
Via f(n), we go from the second fractional Euler numbers to the second Bernoulli numbers.
a(n) mod 10: periodic sequence of length 4: repeat [2, 3, 4, 5].
a(n) differences table:
. 2, 3, 14, 15, 62, 63, 254, 255, ...
. 1, 11, 1, 47, 1, 191, 1, 767, ... see A198693
. 10, -10, 46, -46, 190, -190, 766, -766, ... see A096045, from Bernoulli(2n).
Extension of a(n): a(-2) = -1, a(-1) = 0.

Links

Crossrefs

Cf. A000027, A000918, A001477, A006519, A026741, A027642, A096045, A111701, A117856, A164555, A198631, A198693, A209308, A267921.

Programs

  • Mathematica
    a[n_] := (3+(-1)^n)*(2^(n+1)-1)/2; (* or *) a[n_] := If[EvenQ[n], 4^(n/2+1)-2, 4^((n+1)/2)-1]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 24 2017 *)
  • PARI
    Vec((2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)) + O(x^50)) \\ Colin Barker, Jan 24 2017

Formula

From Colin Barker, Jan 24 2017: (Start)
G.f.: (2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
From Jean-François Alcover, Jan 24 2017: (Start)
a(n) = (3 + (-1)^n)*(2^(n + 1) - 1)/2.
a(n) = 4^((n + 1 + ((n + 1) mod 2))/2) - 1 - ((n + 1) mod 2). (End)
a(n) = a(n-2) + A117856(n+1) for n>1.
a(2*k) = 4^(k + 1) - 2, a(2*k+1) = a(2*k) + 1 = 4^(k+1) - 1.
a(2*k) + a(2*k+1) = A267921(k+1).
Previous Showing 11-18 of 18 results.

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

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