A077218 -id:A077218 - cp's OEIS frontend

A307467 The number of points, corresponding to the first n primes, and placed on the unit circle according to an algorithm using the data from A077218 (in the spirit of Ulam's spiral, and described in the COMMENTS section below), which lie on the closed arc of the unit circle from 0 to 45 degrees.

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 6, 7

Offset: 1

Bernard X. Russo, Apr 09 2019

Algorithm:
Place a dot, representing p_1, on the unit circle at (1,0), or equivalently, at angle 0 degrees, and label this point as the complex number lambda_1 = 1 = e^(i*0).
Place a dot, representing p_2, on the unit circle at (-1,0), or equivalently, at angle 180 degrees, and label this point as the complex number lambda_2 = -1 = e^(i*Pi).
For n > 2, supposing that dots representing p_1, ..., p_n have already been placed on the unit circle, place a dot, representing p_{n+1}, on the unit circle as follows: Starting at the dot representing p_n, proceed counterclockwise around the circle until you reach a dot representing one of p_1, p_2, ..., p_{n-1}.
Starting at this latter dot, proceed counterclockwise until you reach another dot representing one of p_1, p_2, ..., p_n.
Continue this process until you have traversed a(n) (of A077218) successive arcs connecting dots.
Proceed in a counterclockwise direction one more time to the next dot and place the dot which will represent p_{n+1} at the midpoint of the arc just traversed. Label this point as the complex number lambda_{n+1} = e^(i*theta_{n+1}).
The sequence is of interest since the points seem to cluster in the mentioned arc as well as in the closed arc from 180 degrees to 225 degrees. The sequences have been constructed by James Farrington up to n=256, which suggests the clustering. The precise clustering properties are formulated as questions in a file which is referenced below under LINKS.

			Following this procedure, we would place a dot representing p_3 on the unit circle at (0,-1), or equivalently, at angle 270 degrees, and we would place a dot representing p_4 on the unit circle at the midpoint of the arc connecting the point (-1,0) to the point (0,-1), that is, at (-1/root2,-1/root2), or equivalently, at angle 225 degrees.
Thus lambda_3 = -i = e^(3*Pi/2) and lambda_4 = e^(5*Pi/4).

Bernard X. Russo, Clustering of primes relative to factors of composites

Uses A077218 in algorithm.

A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Feb 09 2000

nonn

From Lei Zhou, Mar 18 2014: (Start)
This is also the number of primes such that the (n+1)-th prime (mod i-th prime) is smaller than the (n+1)-th prime (mod n-th prime) for 1 <= i < n.
Proof: We denote the n-th prime number as P_n. Suppose P_(n+1) mod P_i = k; we can write P_(n+1) = m*P_i + k. Setting l = P_(n+1) - P_n, the composite numbers between P_n and P_(n+1) will be consecutively m*P_i + C, where C = k-l+1, k-l+2, ..., k-1. If k < l, there must be a value at which C equals zero since k-1 > 0 and k-l+1 <= 0, so P_i is a factor of a composite number between P_n and P_(n+1). If k >= l, all C values are greater than zero, thus P_i cannot be a factor of a composite number between P_n and P_(n+1). (End)

			n=30, p(30)=113, the next prime is 127. Between them are 13 composites: {114, 115, ..., 126}. Factorizing all and collecting prime factors, the set {2,3,5,7,11,13,17,19,23,29,31,41,59,61} is obtained, consisting of 14 primes, so a(30)=14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A052180, A052248, A061214, A077218.

Length[Union[Flatten[Table[Transpose[FactorInteger[n]][[1]],{n, First[#]+ 1, Last[#]-1}]]]]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, Jan 19 2012 *)

A308487 a(n) is the least prime p such that the total number of prime factors, with multiplicity, of the numbers between p and the next prime is n.

Original entry on oeis.org

3, 11, 59, 71, 239, 7, 13, 103, 97, 79, 127, 73, 23, 31, 61, 157, 373, 383, 251, 89, 359, 401, 683, 701, 139, 337, 283, 241, 211, 631, 1471, 199, 1399, 661, 113, 619, 1511, 509, 293, 953, 317, 773, 1583, 863, 2423, 1831, 2251, 1933, 1381, 4057, 2803, 523, 1069, 2861, 1259, 1759, 3803, 4159, 4703

Offset: 2

J. M. Bergot and Robert Israel, May 31 2019

nonn

a(n) <= A164291(n).

			a(8) = 13 because between 13 and the next prime, 17, are 14 with 2 prime factors, 15 with 2, 16 with 4 (counted with multiplicity), for a total of 2+2+4=8, and this is the first prime for which the total of 8 occurs.

Robert Israel, Table of n, a(n) for n = 2..923

Cf. A000720, A001222, A077218, A164291.

N:= 100: # to get a(2)..a(N)
V:= Array(2..N): count:= 0:
q:= 3:
while count < N-1 do
  p:= q;
  q:= nextprime(q);
  v:= add(numtheory:-bigomega(t),t=p+1..q-1);
  if v > N or V[v] > 0 then next fi;
  V[v]:= p; count:= count+1;
od:
convert(V,list);

Module[{nn=60,pfm},pfm=Table[{p,Total[PrimeOmega[Range[Prime[p]+1,Prime[ p+1]-1]]]},{p,2,1000}];Prime[#]&/@Table[SelectFirst[pfm,#[[2]]==n&],{n,2,nn}]][[All,1]] (* Harvey P. Dale, Aug 25 2022 *)

count(start, end) = my(i=0); for(k=start+1, end-1, i+=bigomega(k)); i
a(n) = forprime(p=1, , if(count(p, nextprime(p+1))==n, return(p))) \\ Felix Fröhlich, May 31 2019

A077218(A000720(a(n))) = n.

A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

0, 2, 5, 10, 5, 17, 5, 28, 30, 10, 45, 42, 12, 44, 47, 76, 10, 72, 57, 5, 97, 51, 117, 150, 28, 22, 83, 5, 65, 321, 66, 131, 28, 298, 10, 108, 172, 145, 109, 205, 10, 276, 5, 127, 16, 441, 582, 130, 24, 80, 232, 10, 276, 195, 270, 256, 10, 218, 187, 52, 388, 701, 162

Offset: 1

Karl-Heinz Hofmann, Mar 26 2023

			a(6): 6th prime = 13 and the (6+1)th prime = 17; the composites between are {14,15,16} and the distinct prime factors of this set are {2,7,3,5} (no duplicates allowed); so a(6) = 2 + 7 + 3 + 5 = 17.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A052297, A077218, A008472, A061214.

a[n_] := Plus @@ Union@ (Join @@ (FactorInteger[#][[;; , 1]] & /@ Range[Prime[n] + 1, Prime[n + 1] - 1])); Array[a, 65] (* Amiram Eldar, Mar 27 2023 *)

a(n) = my(list=List()); for(i=prime(n)+1, prime(n+1)-1, my(f=factor(i)[,1]); for (k=1, #f, listput(list, f[k]))); vecsum(Set(list)); \\ Michel Marcus, Mar 27 2023

from sympy import primefactors, sieve
def A361806(n):
    primeset = []
    for composites in range (sieve[n]+1, sieve[n+1]):
        for p in primefactors(composites): primeset.append(p)
    return(sum(set(primeset)))

a(n) = A008472(A061214(n)).

cp's OEIS Frontend

A307467 The number of points, corresponding to the first n primes, and placed on the unit circle according to an algorithm using the data from A077218 (in the spirit of Ulam's spiral, and described in the COMMENTS section below), which lie on the closed arc of the unit circle from 0 to 45 degrees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A308487 a(n) is the least prime p such that the total number of prime factors, with multiplicity, of the numbers between p and the next prime is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula