A278291 -id:A278291 - cp's OEIS frontend

A278293 a(n) is the number of prime factors of A278291(n) (with multiplicity).

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 2, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 3, 2

Offset: 1

Ely Golden, Nov 16 2016

nonn

a(n) is also the number of prime factors in A045920(n), by definition.

Empirically, it seems as though there are relatively fewer instances of a(n)=x as x tends toward positive infinity (with the exception of a(n)=1, of which there is exactly one instance due to 2 and 3 being the only consecutive primes). For example, in the first 10000 terms, 2391 are 2, 5046 are 3, 2126 are 4, 381 are 5, 51 are 6, 3 are 7, and only one is 8, with no terms in the first 10000 greater than 8.

			a(2)=2, as A278291(2)=10, which has 2 prime factors.
a(6)=3, as A278291(6)=28, which has 3 prime factors.

Ely Golden, Table of n, a(n) for n = 1..10000

Cf. A045920(n).

public class A278293{
public static void main(String[] args)throws Exception{
    long dim0=numberOfPrimeFactors(2);//note that this method must be manually implemented by the user
    long dim1;
    long counter=3;
    long index=1;
    while(index<=10000){
      dim1=numberOfPrimeFactors(counter);
      if(dim1==dim0){System.out.println(index+" "+dim1);index++;}
      dim0=dim1;
      counter++;
    }
  }
}

def bigomega(x):
    s=0;
    f=list(factor(x));
    for c in range(len(f)):
        s+=f[c][1]
    return s;
dim0=bigomega(2);
counter=3
index=1
while(index<=10000):
    dim1=bigomega(counter);
    if(dim1==dim0):
        print(str(index)+" "+str(dim1))
        index+=1;
    dim0=dim1;
    counter+=1;

a(n) = A001222(A278291(n)) = A001222(A045920(n))

A278294 a(n) = first term in A278291 with n prime factors.

Original entry on oeis.org

3, 10, 28, 136, 945, 5265, 29889, 50625, 203392, 3290625, 6082048, 32536000, 326481921, 3274208001, 6929459200, 72523096065, 37694578689, 471672487936, 11557226700801, 54386217385984, 50624737509376, 275892612890625, 4870020829413376, 68091093855502336, 2280241934368768

Offset: 1

Ely Golden, Nov 16 2016

nonn

a(n)-1 is the first term in A115186 with n prime factors, by definition.

If f(n) is the index of the first occurrence of n in A278293, a(n)=A278291(f(n)), by definition.

			a(2)=10, as it is the first term of A278291 with 2 prime factors.
a(3)=28, as it is the first term of A278291 with 3 prime factors.

Ely Golden, Table of n, a(n) for n = 1..25

Cf. A115186.

import java.math.BigInteger;
public class A278294{
public static void main(String[] args)throws Exception{
    BigInteger dim0=numberOfPrimeFactors(BigInteger.valueOf(2));//note that this method must be manually implemented by the user
    BigInteger dim1;
    BigInteger maxdim=BigInteger.ONE;
    BigInteger counter=BigInteger.valueOf(3);
    long index=1;
    while(index<=20){
      dim1=numberOfPrimeFactors(counter);
      if(dim1.equals(dim0)&&maxdim.testBit(dim1.intValue())==false){System.out.println(dim1+" "+counter);maxdim=maxdim.setBit(dim1.intValue());index++;}
      dim0=dim1;
      counter=counter.add(BigInteger.ONE);
    }
  }
}

Function[w, Flatten@ Map[w[[#]] &, #] &@ Map[First, DeleteCases[#, w_ /; Length@ w == 0]] &@ Map[Position[w, k_ /; PrimeOmega@k == #] &, Range@ 9]]@ Select[Range[10^6], Equal @@ Map[PrimeOmega, {# - 1, #}] &] (* Michael De Vlieger, Dec 01 2016 *)

def bigomega(x):
    s=0;
    f=list(factor(x));
    for c in range(len(f)):
        s+=f[c][1]
    return s;
dim0=bigomega(2);
maxdim=1
counter=3
index=1
while(index<=20):
    dim1=bigomega(counter);
    if((dim1==dim0)&((maxdim&(1<

A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

Original entry on oeis.org

2, 9, 14, 21, 25, 27, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 121, 122, 124, 133, 135, 141, 142, 145, 147, 153, 158, 164, 170, 171, 174, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 245, 253, 284, 285, 296, 298, 301, 302, 326, 332, 334, 350, 356, 361

Offset: 1

Felice Russo

A115186 is a subsequence: A001222(A115186(n)) = A001222(A115186(n)+1) = n. - Reinhard Zumkeller, Jan 16 2006
Indices k such that A076191(k) = 0. - Ray Chandler, Dec 10 2008
A045939 is a subsequence. - Zak Seidov, Jul 02 2020
This sequence is infinite (Heath-Brown, 1984). - Amiram Eldar, Jul 11 2020

C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 250.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, A parity problem from sieve theory, Mathematika, Vol. 29, No. 1 (1982), pp. 1-6.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika, Vol. 31, No. 1 (1984), pp. 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific journal of mathematics, Vol. 129, No. 2 (1987), pp. 307-319.

Numbers m through m+k have the same number of prime divisors (with multiplicity): this sequence (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

import Data.List (elemIndices)
a045920 n = a045920_list !! (n-1)
a045920_list = map (+ 1) $ elemIndices 0 a076191_list
-- Reinhard Zumkeller, Mar 23 2012, Oct 11 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};Do[If[f[n]==f[n+1],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Transpose[Transpose[Select[Partition[Table[{n,PrimeOmega[n]},{n,400}], 2,1], #[[1,2]]==#[[2,2]]&]][[1]]][[1]] (* Harvey P. Dale, Feb 21 2012 *)
Position[Differences[PrimeOmega[Range[400]]], 0] // Flatten (* Zak Seidov, Oct 30 2012 *)

is(n)=bigomega(n)==bigomega(n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A278291(n) - 1. - Zak Seidov, Nov 17 2018

More terms from David W. Wilson

A278311 Numbers n such that n-1 and n+1 have the same number of prime factors as n (with multiplicity).

Original entry on oeis.org

34, 86, 94, 122, 142, 171, 202, 214, 218, 245, 285, 302, 394, 429, 435, 446, 507, 603, 604, 605, 634, 638, 698, 842, 922, 963, 1042, 1075, 1084, 1085, 1131, 1138, 1245, 1262, 1275, 1310, 1346, 1402, 1413, 1431, 1435, 1449, 1491, 1533, 1557, 1587, 1605, 1635, 1642, 1676, 1762, 1772, 1838, 1886, 1894, 1925, 1942

Offset: 1

Ely Golden, Nov 17 2016

nonn

			a(1) = 34, as 33, 34, and 35 all have 2 prime factors.
a(2) = 86, as 85, 86, and 87 all have 2 prime factors.

Ely Golden, Table of n, a(n) for n = 1..10000

Intersection of A045920 and A278291.

a(n) = A045939(n) + 1.

public class A278311{
public static void main(String[] args)throws Exception{
    long dim0=numberOfPrimeFactors(2);//note that this method must be manually implemented by the user
    long dim1=numberOfPrimeFactors(3);
    long dim2;
    long counter=4;
    long index=1;
    while(index<=10000){
      dim2=numberOfPrimeFactors(counter);
      if(dim2==dim1&&dim1==dim0){System.out.println(index+" "+(counter-1));index++;}
      dim0=dim1;
      dim1=dim2;
      counter++;
    }
  }
}

isok(n) = (bigomega(n-1) == bigomega(n)) && (bigomega(n) == bigomega(n+1)); \\ Michel Marcus, Nov 17 2016

def bigomega(x):
    s=0;
    f=list(factor(x));
    for c in range(len(f)):
        s+=f[c][1]
    return s;
dim0=bigomega(2);
dim1=bigomega(3);
counter=4
index=1
while(index<=10000):
    dim2=bigomega(counter);
    if(dim2==dim1&dim1==dim0):
        print(str(index)+" "+str(counter-1))
        index+=1;
    dim0=dim1;
    dim1=dim2;
    counter+=1;

cp's OEIS Frontend

A278293 a(n) is the number of prime factors of A278291(n) (with multiplicity).

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A278294 a(n) = first term in A278291 with n prime factors.

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A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

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A278311 Numbers n such that n-1 and n+1 have the same number of prime factors as n (with multiplicity).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Java

PARI

SageMath