A045920 -id:A045920 - cp's OEIS frontend

A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Nov 13 2002

nonn

If a(n) > 0 then a(n+1) = a(n)-1.

			33=3*11 has only two successors also with two factors: 34=2*17 and 35=5*7 (whereas 33+3=36=2*2*3*3), therefore a(33)=2.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A077656, A045920, A045939, A045940, A045941, A045942, A077657.

snpf[n_]:=Module[{f=PrimeOmega[n],k=0},While[f==PrimeOmega[n+k],k++];k]; Array[snpf,110]-1 (* Harvey P. Dale, Aug 01 2021 *)

A077655(n) = { my(k=n+1,w=bigomega(n)); while(bigomega(k)==w,k++); (k-n)-1; }; \\ Antti Karttunen, Jan 22 2020

A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 26, 29, 31, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 134, 137, 139, 143, 146, 149

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

			49 = 7*7 has two prime factors, while 50 = 2*5*5 has three, so 49 belongs to the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001222, A006049, A045920, A294277, A294278, A302242, A322837, A322838, A322839.

Select[Range[100],PrimeOmega[#]?(#[[1]]< #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 23 2021 *)

isok(n) = bigomega(n) < bigomega(n+1); \\ Michel Marcus, Dec 29 2018

A369139 Numbers k such that Omega(k) = 1 + Omega(k + 1).

Original entry on oeis.org

4, 6, 8, 10, 20, 22, 45, 46, 50, 58, 68, 76, 80, 82, 92, 104, 105, 106, 110, 114, 117, 152, 154, 165, 166, 178, 182, 186, 189, 212, 226, 236, 246, 258, 260, 261, 262, 266, 273, 286, 290, 315, 318, 322, 325, 333, 338, 342, 344, 345, 346, 354, 357, 358, 370, 382, 385, 402, 406, 410, 412, 424, 426

Offset: 1

Zak Seidov and Robert Israel, Jan 14 2024

nonn

Numbers k that have one more prime divisor (counted by multiplicity) than k + 1.

			a(3) = 8 is a term because 8 = 2^3 has 3 prime divisors (counted by multiplicity) and 8 + 1 = 9 = 3^2 has 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A045920, A076156. Contains A077065.

N:= 1000: # for terms <= N
V:= map(numtheory:-bigomega, [$1..N+1]):
select(t -> V[t] = 1 + V[t+1], [$1..N]);

s = {}; Do[If[PrimeOmega[k] == 1 + PrimeOmega[k + 1], AppendTo[s, k]], {k, 500}]; s

A077656 Numbers having a different number of prime factors as their successors (counted with multiplicity).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Reinhard Zumkeller, Nov 13 2002

nonn

A077655(a(n))=0; A001222(a(n))<>A001222(a(n)+1).

			20=2*2*5 and 20+1=21=3*7: A001222(20)<>A001222(21), therefore 20 is a term (A077655(20)=0).

Cf. A001222, A045920.

Position[Partition[PrimeOmega[Range[100]],2,1],?(#[[1]]!=#[[2]]&),{1}, Heads->False]//Flatten (* _Harvey P. Dale, Sep 19 2016 *)

A237929 Numbers n such that (i) the sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1, and (ii) n and n+1 have the same number of prime divisors (with repetition).

Original entry on oeis.org

2, 9, 98, 170, 1274, 4233, 4345, 7105, 7625, 14905, 21385, 30457, 34945, 66585, 69874, 77314, 82946, 98841, 175354, 177122, 233090, 236282, 238017, 263145, 265225, 295274, 298082, 322234, 335793, 336106

Offset: 1

Abhiram R Devesh, Feb 16 2014

nonn

The first term a(1)=2 is the only prime number in this sequence.

			For n=98: prime factors = 2,7,7; sum of prime factors = 16; number of prime divisors = 3
For n+1=99: prime factors = 3,3,11; sum of prime factors = 17; number of prime divisors=3.

Abhiram R Devesh, Table of n, a(n) for n = 1..96049

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
Cf. A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.
Cf. A045920 Numbers n such that factorizations of n and n+1 have same number of primes (including multiplicities).

Select[Partition[Table[{n,PrimeOmega[n],Total[Times@@@FactorInteger[n]]},{n,34*10^4}],2,1],#[[1,2]]==#[[2,2]]&&#[[1,3]]+1==#[[2,3]]&][[;;,1,1]] (* Harvey P. Dale, May 03 2024 *)

from sympy import primeomega
def is_A237929(n): return A001414(n) == A001414(n+1)-1 and primeomega(n) == primeomega(n+1) # David Radcliffe, Aug 08 2025

A259504 Numbers n such that n and n+1 are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

27, 44, 75, 98, 116, 124, 147, 153, 164, 170, 171, 174, 230, 244, 245, 284, 285, 332, 356, 369, 387, 425, 428, 429, 434, 435, 474, 506, 507, 530, 548, 555, 574, 595, 602, 603, 604, 605, 609, 627, 637, 638, 645, 651, 657, 710

Offset: 1

Zak Seidov, Nov 08 2015

nonn

Conjecture: this sequence is infinite.
Number of terms < 10^k: 0, 4, 63, 727, 7014, 64556, 585725, 5284711, ... . - Robert G. Wilson v, Nov 09 2015
a(n) = p^3 where p is prime iff p is in intersection of A065508 and A005383. - Altug Alkan, Nov 24 2015
There are 47753279 terms less than 10^9 and 432841730 terms less than 10^10. - Charles R Greathouse IV, Jun 27 2019

			27=3*3*3, 28=2*2*7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A014612 and A045920.

Cf. A067813.

Select[Range[1000], 3 == PrimeOmega[#] == PrimeOmega[# + 1] &]

forcomposite(n=1, 1e3, if(bigomega(n)==3 && bigomega(n+1)==3, print1(n, ", "))); \\ Altug Alkan, Nov 08 2015

list(lim)=my(v=List(),was=1,is); forfactored(n=28,lim\1+1, is=vecsum(n[2][,2])==3; if(is && was, listput(v,n[1]-1)); was=is); Vec(v) \\ Charles R Greathouse IV, Jun 26 2019

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

Original entry on oeis.org

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))

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;

A335652 Numbers k such that Omega(k+1) = Omega(k) + 2, where Omega(k) = A001222(k) is the number of prime factors of k with multiplicity.

Original entry on oeis.org

7, 11, 15, 17, 19, 29, 35, 39, 41, 43, 55, 67, 87, 97, 101, 109, 113, 134, 137, 155, 163, 173, 175, 181, 183, 203, 207, 209, 211, 219, 229, 241, 242, 247, 249, 257, 259, 279, 281, 283, 295, 305, 314, 317, 327, 329, 331, 337, 339, 341, 351, 353, 371, 373, 401, 404, 409, 413, 433, 455

Offset: 1

Zak Seidov, Jun 16 2020

nonn

			7 is prime, Omega(7) = 1, 7 + 1 = 8 = 2*2*2, Omega(8) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Omega(k+1) = Omega(k) + m: A045920 (m = 0), A076156 (m = 1).

Cf. A001222, A335655. Contains A063639.

Omegas:= [seq(numtheory:-bigomega(i),i=1..1001)]:
select(i -> Omegas[i]+2=Omegas[i+1], [$1..1000]); # Robert Israel, Jun 16 2020

m = 2; s = {}; Do[If[PrimeOmega[x + 1] == PrimeOmega[x] + m, AppendTo[s, x]], {x, 500}]; s
Position[Partition[PrimeOmega[Range[500]],2,1],?(#[[1]]==#[[2]]-2&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 04 2021 *)

A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

Original entry on oeis.org

2, 14, 104, 2079, 21735, 3341624, 103488384, 6110171144

Offset: 1

Amiram Eldar, Apr 30 2021

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), a(n) is also the least number k such that k and k+1 both have 2^n infinitary divisors.

a(9) > 2*10^11, if it exists.

			a(1) = 2 since A064547(2) = A064547(3) = 1.
a(2) = 14 since A064547(14) = A064547(15) = 2.

Cf. A064547, A343819.

Similar sequences: A045920, A052215, A075036, A093548, A115186.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, fd1, fd2}, fd1=fd[n]; While[c < m, fd2 = fd[++n]; If[fd1 == fd2 && fd1 <= m && s[[fd1]] == 0, s[[fd1]] = n-1; c++]; fd1=fd2]; s]; seq[5]

cp's OEIS Frontend

A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

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A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

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A369139 Numbers k such that Omega(k) = 1 + Omega(k + 1).

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A077656 Numbers having a different number of prime factors as their successors (counted with multiplicity).

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Mathematica

A237929 Numbers n such that (i) the sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1, and (ii) n and n+1 have the same number of prime divisors (with repetition).

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A259504 Numbers n such that n and n+1 are the product of exactly three (not necessarily distinct) primes.

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

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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).

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