A076191 -id:A076191 - cp's OEIS frontend

A305077 Partial sums of absolute values of A076191.

1, 1, 2, 3, 4, 5, 7, 8, 8, 9, 11, 13, 14, 14, 16, 19, 21, 23, 25, 26, 26, 27, 30, 32, 32, 33, 33, 35, 37, 39, 43, 46, 46, 46, 48, 51, 52, 52, 54, 57, 59, 61, 63, 63, 64, 65, 69, 72, 73, 74, 75, 77, 80, 82, 84, 86, 86, 87, 90, 93, 94, 95, 98, 102, 103, 105, 107, 108, 109, 111, 115, 119, 120, 121

Offset: 1

Robert Israel, May 24 2018

nonn

Math Overflow, Asymptotic formula for absolute difference of number of prime factors between consecutive integers

Cf. A001222, A076191.

L:= map(numtheory:-bigomega, [$1..200]):
ListTools:-PartialSums(map(abs, L[2..-1]-L[1..-2]));

a(n) = Sum_{k=1..n} |A001222(n+1) - A001222(n)|.

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

A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

Original entry on oeis.org

1, 4, 6, 8, 8, 12, 12, 13, 13, 15, 17, 20, 19, 22, 21, 22, 21, 24, 25, 26, 25, 31, 30, 32, 30, 31, 33, 33, 32, 36, 34, 36, 34, 36, 36, 37, 36, 40, 40, 42, 40, 45, 48, 49, 49, 51, 50, 52, 49, 50, 50, 53, 50, 56, 53, 53, 53, 55, 58, 60, 59, 62, 60, 60, 55, 58, 59, 61, 60, 65, 62, 65, 63, 66, 69, 68

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A022559, A023978, A076191, A248707.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n+1, n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / ((n-1)!*n!*(n+1)!)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A248707(n)).
a(n) = A022559(3*n) - A022559(n-1) - A022559(n) - A022559(n+1) = A022559(3*n) - 3*A022559(n) - A001222(n+1) + A001222(n) = A023978(n) - A076191(n). (End)

A023835 Sum of exponents in prime-power factorization of C(4n,2n-1).

Original entry on oeis.org

2, 4, 6, 7, 7, 9, 11, 12, 13, 13, 13, 15, 14, 16, 17, 18, 16, 18, 18, 20, 22, 22, 23, 25, 24, 24, 25, 24, 24, 28, 27, 29, 27, 30, 30, 32, 31, 31, 34, 32, 31, 33, 34, 37, 36, 35, 36, 40, 37, 37, 38, 39, 39, 42, 41, 42, 42, 42, 42, 43, 44, 46, 47, 48, 45, 46, 44, 45, 48, 49, 47, 50, 47, 49, 53, 52, 51

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A023826, A023827, A023828, A023829, A023830, A023831, A023832, A023834, A023836, A076191.

Table[PrimeOmega[Binomial[4 n, 2 n - 1]], {n, 78}] (* Ivan Neretin, Nov 08 2017 *)

a(n) = bigomega(binomial(4*n,2*n-1)); \\ Amiram Eldar, Jun 13 2025

a(n) = A023832(n) - A001222(2*n+1) + A001222(2*n) = A023832(n) - A076191(2*n). - Amiram Eldar, Jun 13 2025

a(74)-a(77) corrected by Ivan Neretin, Nov 08 2017

A023817 Sum of exponents in prime-power factorization of C(2n,n-1).

Original entry on oeis.org

0, 2, 2, 4, 4, 6, 4, 7, 6, 7, 6, 9, 8, 11, 9, 12, 9, 13, 10, 13, 12, 13, 12, 15, 14, 14, 14, 16, 14, 17, 12, 18, 16, 16, 15, 18, 15, 18, 17, 20, 19, 22, 20, 22, 22, 23, 20, 25, 21, 24, 22, 24, 22, 25, 21, 24, 22, 24, 23, 28, 26, 27, 26, 29, 25, 27, 25, 30, 28, 30, 26, 32, 29, 31, 31, 31, 31, 34, 29, 32

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A001791, A023816, A023818, A076191.

Join[{0},Total[FactorInteger[#][[All,2]]]&/@Table[Binomial[2n,n-1], {n,2,80}]] (* Harvey P. Dale, Sep 14 2016 *)
Table[PrimeOmega[Binomial[2 n, n + 1]], {n, 1, 200}] (* Clark Kimberling, May 04 2025 *)

a(n) = bigomega(binomial(2*n, n-1)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001791(n)).
a(n) = A023816(n) - A076191(n). (End)

A079057 a(n) = Sum_{k=1..n} bigomega(tau(k)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 21, 22, 24, 25, 27, 29, 31, 32, 35, 36, 38, 40, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 60, 62, 65, 66, 69, 70, 72, 74, 76, 77, 79, 80, 82, 84, 86, 87, 90, 92, 95, 97, 99, 100, 103, 104, 106, 108, 109, 111, 114, 115, 117, 119

Offset: 1

Benoit Cloitre, Feb 02 2003

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
E. Heppner, Über die Iteration von Teilerfunktionen, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.
G. J. Rieger, Über einige arithmetische Summen, Manuscripta Mathematica, Vol. 7 (1972), pp. 23-34.

Partial sums of A058061.

Cf. A000005, A001222, A076191, A077761.

Accumulate[PrimeOmega[DivisorSigma[0,Range[70]]]] (* Harvey P. Dale, Dec 05 2013 *)

PARI
```
a(n)=sum(i=1,n,bigomega(numdiv(i)))
```

a(n) = n*log(log(n)) + O(n).

a(n) = b * n * log(log(n)) + Sum_{k=0..floor(sqrt(n))} b_k * n/log(n)^k + O(n * exp(-c*sqrt(log(n)))), where b, b_k and c are constants (Heppner, 1974). b = 1 and b_0 = B + C, where B is Mertens's constant (A077761), C = Sum_{k>=2} A076191(k)*P(k) = 0.12861146810484151346..., and P(s) is the prime zeta function. - Amiram Eldar, Jan 15 2024 and Feb 11 2024

A285787 Least number k such that the absolute value of the difference between the number of prime factors, with multiplicity, of k and k-1 is equal to n.

Original entry on oeis.org

3, 2, 8, 17, 32, 97, 128, 257, 769, 2048, 4097, 6144, 8192, 40961, 73728, 65537, 131072, 524289, 524288, 3145728, 6291456, 8388608, 18874368, 50331648, 113246209, 167772161, 268435457, 805306368, 1610612737, 2147483649, 2147483648, 17179869184, 21474836480

Offset: 0

Paolo P. Lava, Apr 26 2017

nonn

a(n) <= A051900(n), with equality for n=3,5,7,8,13,15. - Robert Israel, Apr 26 2017

			a(9) = 2048 because 2047 = 23 * 89, 2048 = 2^11 and 11 - 2 = 9.

Giovanni Resta, Table of n, a(n) for n = 0..40

Cf. A001222, A051900, A076191, A285457.

with(numtheory): P:=proc(q) local a,b,k,v; v:=array(0..100);
for k from 0 to 100 do v[k]:=0; od; a:=0;
for k from 2 to q do b:=bigomega(k); if v[abs(b-a)]=0 then v[abs(b-a)]:=k; fi; a:=b; od; k:=0;
while v[k]>0 do print(v[k]); k:=k+1; od; print(); end: P(10^6);

s = PrimeOmega@ Range[10^6]; 1 + First /@ Values@ KeySort@ PositionIndex@ Flatten@ Map[Abs@ Differences@ # &, Partition[s, 2, 1]] (* Michael De Vlieger, Apr 26 2017, Version 10 *)

Least solutions of the equation abs(A001222(k) - A001222(k-1)) = n.

a(24)-a(32) from Giovanni Resta, Apr 26 2017

A248211 First differences of omega(n), the number of distinct prime factors function (A001221).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Oct 04 2014

First instance of abs(a(n)) > 2 is for n = 210. - Alonso del Arte, Oct 05 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions. II, Acta Math. Hungar. 49 (1987), 251-259.

Cf. A001221 (omega).
Cf. A053222: first differences of sigma(n) = A000203.
Cf. A076191: first differences of bigomega(n) = A001222.
Cf. A127440: first differences of mobius(n) = A008683.

with(numtheory): A248211:=n->nops(factorset(n+1))-nops(factorset(n)): seq(A248211(n), n=1..100);

Table[PrimeNu[n + 1] - PrimeNu[n], {n, 100}] (* Hurt *)
Differences[PrimeNu[Range[100]]] (* Alonso del Arte, Oct 04 2014 *)

a(n) = omega(n+1) - omega(n); \\ Michel Marcus, Dec 29 2022

a(n) = omega(n+1) - omega(n) = A001221(n+1) - A001221(n).

G.f.: (1 - x)*Sum_{k>=1} x^(prime(k)-1)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 15 2017

cp's OEIS Frontend

A305077 Partial sums of absolute values of A076191.

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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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A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

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A023835 Sum of exponents in prime-power factorization of C(4n,2n-1).

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A023817 Sum of exponents in prime-power factorization of C(2n,n-1).

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A079057 a(n) = Sum_{k=1..n} bigomega(tau(k)).

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A285787 Least number k such that the absolute value of the difference between the number of prime factors, with multiplicity, of k and k-1 is equal to n.

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Maple

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A248211 First differences of omega(n), the number of distinct prime factors function (A001221).