A305300 -id:A305300 - cp's OEIS frontend

A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4

Offset: 1

Jaroslav Krizek, Sep 29 2010

Sequence is cardinal and not fractal. Cardinal sequence is sequence with infinitely many times occurring all natural numbers. Fractal sequence is sequence such that when the first instance of each number in the sequence is erased, the original sequence remains.

Ordinal transform of the nextprime function, A151800(1..) = 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, ..., also ordinal transform of A304106. - Antti Karttunen, Jun 09 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000720, A007917, A008578, A151799, A151800, A305300.
Cf. A065358 for another way of visualizing prime gaps.
Cf. A304106 (ordinal transform of this sequence).
Cf. A049711.

a[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n)); \\ Antti Karttunen, Mar 04 2018

a(1) = 1, a(n) = n - A007917(n) + 1 for n >= 2. a(1) = 1, a(2) = 1, a(n) = n - A151799(n+1) + 1 for n >= 3.
a(n) = Sum_{i=1..n} floor(pi(i)/pi(n)), for n>1 with pi(n) = A000720(n). - Ridouane Oudra, Jun 24 2024
a(n) = A049711(n+1), for n>1. - Ridouane Oudra, Jul 16 2024

A305437 Running index for the lexicographically least irreducible factor when (0,1)-polynomial obtained from the binary expansion of n is factored over Q.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 3, 2, 6, 2, 7, 2, 3, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 13, 2, 14, 2, 3, 2, 5, 2, 15, 2, 3, 2, 16, 2, 17, 2, 3, 2, 18, 2, 5, 2, 3, 2, 19, 2, 20, 2, 3, 2, 21, 2, 22, 2, 3, 2, 4, 2, 23, 2, 24, 2, 25, 2, 26, 2, 3, 2, 27, 2, 28, 2, 29, 2, 30, 2, 4, 2, 31, 2, 32, 2, 33, 2, 10, 2, 4, 2, 34, 2, 3, 2, 35, 2, 36, 2, 3

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

			Numbers 1 .. 6 encode the following (0,1)-polynomials by their binary representation:
  1 -> 1       [Empty factorization]
  2 -> x       [Irreducible, the only and thus also the least factor is x]
  3 -> x + 1   [Irreducible, the least factor is (x+1)]
  4 -> x^2      = (x)(x)   [The least factor (x) occurred already for the first time at n=2, thus a(4) = 2.]
  5 -> x^2 + 1 [Irreducible, the least factor is (x^2 + 1)]
  6 -> x^2 + x  = (x)(x+1) [The least factor (x) occurred already at n=2, thus a(6) = 2.]
Binary representation of 7 is "111", encoding (0,1)-polynomial x^2 + x + 1, which is irreducible over Q, so it is the first time that polynomial occurs as a "smallest" (lexicographically least) irreducible factor, while before it, already four different kinds of "smallest" factors have occurred, thus a(7) = 5.
The second time the same factor occurs as the smallest one is for n=35, whose binary representation "100011" encodes polynomial x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1) , thus a(35) = 5 also.
The third time the same factor occurs as the smallest one is for n=49, whose binary representation "110001" encodes polynomial x^5 + x^4 + 1 = (x^2 + x + 1)(x^3 - x + 1), thus a(49) = 5 also.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206074, A305300.
Cf. A305438 (ordinal transform).
Cf. also A055396, A302786.

allocatemem(2^30);
default(parisizemax,2^31);
up_to = 65537;
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; };
pollexcmp(a,b) = { my(ad = poldegree(a), bd = poldegree(b),e); if(ad != bd, return(sign(ad-bd))); for(i=0,ad,e = polcoeff(a,ad-i) - polcoeff(b,ad-i); if(0!=e, return(sign(e)))); (0); };
lexleastpolfactor(n) = if(1==n,0,my(fs = factor(Pol(binary(n)))[,1]~); vecsort(fs,pollexcmp)[1]);
v305437 = rgs_transform(vector(up_to,n,lexleastpolfactor(n)));
A305437(n) = v305437[n];

cp's OEIS Frontend

A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

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