A046660 -id:A046660 - cp's OEIS frontend

A297167 a(1) = 0, for n > 1, a(n) = -1 + the excess of n (A046660) + the index of the largest prime factor (A061395).

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 2, 3, 6, 2, 7, 3, 3, 4, 8, 3, 3, 5, 3, 4, 9, 2, 10, 4, 4, 6, 3, 3, 11, 7, 5, 4, 12, 3, 13, 5, 3, 8, 14, 4, 4, 3, 6, 6, 15, 3, 4, 5, 7, 9, 16, 3, 17, 10, 4, 5, 5, 4, 18, 7, 8, 3, 19, 4, 20, 11, 3, 8, 4, 5, 21, 5, 4, 12, 22, 4, 6, 13, 9, 6, 23, 3, 5, 9, 10, 14, 7, 5, 24, 4, 5, 4, 25, 6, 26, 7, 3

Offset: 1

Antti Karttunen, Feb 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A033265, A046660, A061395, A156552, A252464, A297113, A297168, A300827.

Array[-1 + PrimeOmega@ # - PrimeNu@ # + PrimePi[FactorInteger[#][[-1, 1]]] /. k_ /; k < 0 -> 0 &, 105] (* or, slightly faster *)
Array[-1 + Length@ # - Length@ Union@ # + PrimePi@ Last@ # /. k_ /; k < 0 -> 0 &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #] &@ FactorInteger[#] &, 105] (* Michael De Vlieger, Mar 13 2018 *)

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A252464(n) = if(1==n, 0, (bigomega(n) + A061395(n) - 1));
A297167(n) = (A252464(n) - omega(n));
\\ Or just as:
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
\\ Antti Karttunen, Mar 13 2018

from sympy import factorint, primepi
def A297167(n): return primepi(max(f:=factorint(n)))+sum(e-1 for e in f.values())-1 if n>1 else 0 # Chai Wah Wu, Jul 29 2023

(define (A297167 n) (- (A252464 n) (A001221 n)))

a(n) = A252464(n) - A001221(n).
For n > 1, a(n) = A033265(A156552(n)) = A297113(n) - 1.
For n > 1, a(n) = A046660(n) + A061395(n) - 1. - Antti Karttunen, Mar 13 2018

Name changed, original equivalent definition is the first entry in the Formula section - Antti Karttunen, Mar 13 2018

A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

Original entry on oeis.org

1, 4, 9, 8, 24, 27, 16, 48, 72, 80, 32, 96, 144, 160, 216, 64, 192, 288, 320, 432, 448, 128, 384, 576, 640, 864, 896, 1296, 256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312

Offset: 0

Reinhard Zumkeller, Nov 29 2015

At the suggestion of Michel Marcus's remark in Carlos Eduardo Olivieri's A261256.

			0:    1
1:    4     9
2:    8    24      27
3:   16    48      72    80
4:   32    96     144   160     216
5:   64   192     288   320     432   448
6:  128   384     576   640     864   896    1296
7:  256   768    1152  1280    1728  1792    2592   2816
8:  512  1536    2304  2560    3456  3584    5184   5632    6400
--  ------------------------------------------------------------
0:  1
1:  2^2   3^2
2:  2^3 2^3*3     3^3
3:  2^4 2^4*3 2^3*3^2 2^4*5
4:  2^5 2^5*3 2^4*3^2 2^5*5 2^3*3^3
5:  2^6 2^6*3 2^5*3^2 2^6*5 2^4*3^3 2^6*7
6:  2^7 2^7*3 2^6*3^2 2^7*5 2^5*3^3 2^7*7 2^4*3^4
7:  2^8 2^8*3 2^7*3^2 2^8*5 2^6*3^3 2^8*7 2^5*3^4 2^8*11
8:  2^9 2^9*3 2^8*3^2 2^9*5 2^7*3^3 2^9*7 2^6*3^4 2^9*11 2^8*5^2

Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Cf. A046660, A151821, A261256.

Cf. A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069.

a257851 n k = a257851_tabl !! n !! k
a257851_row n = a257851_tabl !! n
a257851_tabl = map
   (\x -> take (x + 1) $ filter ((== x) . a046660) [1..]) [0..]

T[n_] := Reap[For[m = 1; k = 1, k <= n+1, If[PrimeOmega[m] - PrimeNu[m] == n, Sow[m]; k++]; m++]][[2, 1]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Sep 17 2021 *)

T(n,0) = A151821(n+1);
T(n,n-1) = A261256(n) for n > 0;
T(n,n) = A264959(n).
T(0,0) = A005117(1);
T(1,k) = A060687(k+1), k = 0..1;
T(2,k) = A195086(k+1), k = 0..2;
T(3,k) = A195087(k+1), k = 0..3;
T(4,k) = A195088(k+1), k = 0..4;
T(5,k) = A195089(k+1), k = 0..5;
T(6,k) = A195090(k+1), k = 0..6;
T(7,k) = A195091(k+1), k = 0..7;
T(8,k) = A195092(k+1), k = 0..8;
T(9,k) = A195093(k+1), k = 0..9;
T(10,k) = A195069(k+1), k = 0..10.

A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

Original entry on oeis.org

1, 2, 3, 6, 5, 18, 7, 30, 15, 50, 11, 270, 13, 98, 75, 210, 17, 450, 19, 1050, 147, 242, 23, 9450, 35, 338, 105, 3234, 29, 11250, 31, 2310, 363, 578, 245, 47250, 37, 722, 507, 57750, 41, 43218, 43, 9438, 2625, 1058, 47, 727650, 77, 2450, 867, 17238, 53, 22050, 605, 210210, 1083, 1682, 59, 8268750, 61, 1922, 8085, 30030

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A046660, A046523, A048675, A061395, A097248, A156552, A297113, A324202, A332462.

A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A332461(n) = if(1==n,1, my(m=1); fordiv(n,d,if(d>1, m *= prime(A297113(d)))); (m));

a(n) = Product_{d|n, d>1} A000040(A297113(d)).
a(p) = p for all primes p.
For all n >= 0, a(2^n) = A002110(n).
For all n >= 1:
A046523(a(n)) = A324202(n).
A048675(a(n)) = A156552(n).
A097248(a(n)) = A332462(n).

A358817 Numbers k such that A046660(k) = A046660(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

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

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

A340818 Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

Original entry on oeis.org

5, 7, 41, 3, 197, 229, 5827, 277, 1157, 8382, 268049, 94175911, 964941119, 1929224113, 31529606831, 835346466959, 3398377571053, 52665885581009, 119955940157647877, 34063199364211668943, 315047077264055066629, 199089493729235251718903, 47411489829747180146759

Offset: 1

Amiram Eldar, Jan 22 2021

Let Omega_n(k) be the number of prime divisors of k not exceeding prime(n) counted with multiplicity, and omega_n(k) the number of distinct prime divisors of k not exceeding prime(n). Then, f(n) = a(n)/A340819(n) is the asymptotic density of numbers k such that Omega_n(k) == omega_n(k) (mod 2).

Equivalently, f(n) is the asymptotic density of numbers k such that A046660(d_n(k)) is even, where d_n(k) is the largest prime(n)-smooth divisor of k.

			The sequence of fractions begins with 5/6, 7/9, 41/54, 3/4, 197/264, 229/308, 5827/7854, 277/374, 1157/1564, 8382/11339, ...
For n=1, Omega_2(k)-omega_2(k) is even for either odd k (A005408), or even k whose binary representation ends in an odd number of zeros (A036554). The disjoint union of these 2 sequences has an asymptotic density 1/2 + 1/3 = 5/6.

Amiram Eldar, Table of n, a(n) for n = 1..462
Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, Computing the rho constant, 2016.
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Cf. A005408, A036554, A046660, A162644, A340819 (denominators), A340820.

d[p_] := 1/(p*(p + 1)); delta[n_] := delta[n] = d[Prime[n]]; f[0] = 1; f[n_] := f[n] = f[n - 1] * (1 - delta[n]) + (1 - f[n - 1]) * delta[n]; Numerator @ Array[f, 30]

Let delta(n) = 1/(prime(n)*(prime(n)+1)) be the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. Let f(0) = 1. Then, f(n) = f(n-1)*(1 - delta(n)) + (1 - f(n))*delta(n).

Limit_{n->oo} f(n) = 0.73584... (A340820).

A340819 Denominators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

Original entry on oeis.org

6, 9, 54, 4, 264, 308, 7854, 374, 1564, 11339, 362848, 127541072, 1307295988, 2614591976, 42742894912, 1132686715168, 4608863185856, 71437379380768, 162734350229389504, 46216555465146619136, 427503138052606227008, 270181983249247135469056, 64347502466822129824768

Offset: 1

Amiram Eldar, Jan 22 2021

See A340818 for details.

Amiram Eldar, Table of n, a(n) for n = 1..462

Cf. A046660, A162644, A340818 (numerators), A340820.

d[p_] := 1/(p*(p + 1)); delta[n_] := delta[n] = d[Prime[n]]; f[0] = 1; f[n_] := f[n] = f[n - 1] * (1 - delta[n]) + (1 - f[n - 1]) * delta[n]; Denominator @ Array[f, 30]

A323253 a(n) is the smallest number k such that factorizations of n consecutive integers starting at k have the same excess of number of primes counted with multiplicity over number of primes counted without multiplicity (A046660).

Original entry on oeis.org

1, 1, 1, 844, 74849, 671346, 8870025

Offset: 1

Ilya Gutkovskiy, Aug 30 2019

Smallest number k such that n or more consecutive integers starting at k have the same number of proper prime power divisors.
a(8) > 10^9. - Vaclav Kotesovec, Sep 01 2019
a(8) <= 254023231417746. - David A. Corneth, Sep 01 2019
a(8) > 10^13. - Giovanni Resta, Sep 05 2019

			671346 = 2 * 3^2 * 13 * 19 * 151,
671347 = 17^2 * 23 * 101,
671348 = 2^2 * 47 * 3571,
671349 = 3 * 7^2 * 4567,
671350 = 2 * 5^2 * 29 * 463,
671351 = 53^2 * 239.
These the first 6 consecutive numbers with the same number of proper prime power divisors, so a(6) = 671346.

Cf. A001221, A001222, A006558, A045983, A045984, A046660, A246547.

Do[find = 0; k = 0; While[find == 0, k++; If[Length[Union[Table[PrimeOmega[j] - PrimeNu[j], {j, k, k + n - 1}]]] == 1, find = 1; Print[k]]], {n, 1, 5}] (* Vaclav Kotesovec, Sep 01 2019 *)
(* faster program *) fak = Table[f = FactorInteger[j]; Total[Transpose[f][[2]]] - Length[f], {j, 1, 10000000}]; m = Max[fak]; Table[Min[Table[SequencePosition[fak, ConstantArray[j, n]], {j, 0, m}]], {n, 1, 7}] (* Vaclav Kotesovec, Sep 01 2019 *)

excess(n) = bigomega(n) - omega(n);
score(n) = my(t=excess(n)); for(k=1, oo, if(excess(n+k) != t, return(k)));
upto(nn) = my(n=1); for(k=1, nn, while(score(k) >= n, print1(k, ", "); n++)); \\ Daniel Suteu, Sep 01 2019

a(7) from Daniel Suteu and Vaclav Kotesovec, Sep 01 2019

A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 64, 9, 1, 1, 12, 1, 1, 1, 4096, 1, 18, 1, 20, 1, 1, 1, 576, 25, 1, 729, 28, 1, 1, 1, 1048576, 1, 1, 1, 1296, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 49, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 60, 1, 1, 63, 1073741824, 1, 1, 1, 68

Offset: 1

Peter Luschny and Michael De Vlieger, Jul 14 2023

nonn

			108 = 2^2 * 3^3 => excess(108) = 5 - 2 => a(108) = 108^3 = 1259712.

Plot2 A363923 vs A205959.
Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k) for n = 1..2^11, with a color function representing e(k) where black represents e(k) = 1, red e(k) = 2, through magenta. The bar at bottom represents a(n) = 1 in black, a(n) a composite prime power in gold, a(n) neither squarefree nor semiprime in blue, and a(n) such that all e(k) > 1 in purple.
Michael De Vlieger, Notes on certain sequences relating BigOmega(n), omega(n), and rad(n), 2023.

Cf. A001221, A001222, A005117, A046660, A053763, A056170, A060687, A013929, A205959, A337945, A363923.

using Nemo
exc(n::fmpz) = sum(e - 1 for (p, e) in factor(n))
A363919(n::fmpz) = n < 2 ? fmpz(1) : n^exc(n)
println([A363919(fmpz(n)) for n in 1:68])

with(NumberTheory):
A363919 := n -> n^(NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct')):
# Alternative:
a := n -> local i: n^add(i[2] - 1, i in ifactors(n)[2]): seq(a(n), n = 1..68);

Array[#^(PrimeOmega[#] - PrimeNu[#]) &, 120]

a(n) = my(f=factor(n)[, 2]); n^(vecsum(f)-#f); \\ Michel Marcus, Jul 16 2023

from sympy import factorint
def A363919(n): return n**sum(map(lambda e:e-1,factorint(n).values())) # Chai Wah Wu, Jul 18 2023

def A363919(n):
    if n < 2: return 1
    return n^sum(p[1] - 1 for p in list(factor(n)))
print([A363919(n) for n in srange(1, 69)])

a(n) = n^(Sum_{p in Factors(n)} (mult(p) - 1)), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.
a(n) = A363923(n) / A205959(n).
a(n) = n^A046660(n) = n^(A001222(n) - A001221(n)).
a(n) = 1 or divisible by at least one squared prime.
a(n) = 1 <=> n is squarefree (A005117).
a(n) != 1 <=> A056170(n) != 0.
a(n) = n <=> n = A060687(n - 1) for n >= 2.
a(2^n) = 2^(n*(n - 1)) = A053763(n).
a(n) <= 2^(lb(n)*(lb(n)-1)), where lb(n) = floor(log_{2}(n)).
a(n) is even <=> n = 2*A337945(n).
a(n) > 1 is odd <=> n = A053850(n).
n is prime => a(n) = 1. ('prime' means term of A000040).
n is prime product => a(n) = 1. ('prime product' means term of A144338).
n is proper prime power => a(n) is proper prime power. ('proper prime power' means term of A246547).
Moebius(a(n)) = [a(n) = 1], where [ ] denotes the Iverson bracket.

A358818 a(n) is the least number k such that A046660(k) = A046660(k+1) = n.

Original entry on oeis.org

1, 44, 135, 80, 8991, 29888, 123200, 2316032, 1043199, 24151040, 217713663, 689278976, 11573190656, 76876660736, 311969153024, 2035980763136, 2741258240000, 215189482110975

Offset: 0

Amiram Eldar, Dec 02 2022

a(14) <= 314944159743.

a(18) > 10^14.5; a(19) = 275892612890624; a(20) > 10^14.5. - Martin Ehrenstein, Dec 11 2022

Cf. A046660.
Subsequence of A358817.
Similar sequences: A052215, A059737, A093548, A115186.

e[n_] := PrimeOmega[n] - PrimeNu[n]; a[n_] := Module[{k = 1}, While[e[k] != n || e[k + 1] != n, k++]; k]; Array[a, 10, 0]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
a(n) = {my(k=1); while(e(k) != n || e(k+1) !=n , k++); k};

a(14)-a(16) from Martin Ehrenstein, Dec 04 2022

a(17) from Martin Ehrenstein, Dec 09 2022

A369171 Numbers k such that A000688(k) = A046660(k+1).

Original entry on oeis.org

3, 11, 17, 19, 43, 51, 59, 62, 67, 74, 83, 89, 91, 97, 99, 115, 123, 124, 131, 139, 146, 149, 155, 163, 170, 174, 187, 188, 197, 203, 206, 211, 219, 227, 233, 235, 241, 259, 267, 274, 278, 279, 283, 291, 293, 305, 307, 314, 331, 337, 339, 341, 342, 347, 349, 350

Offset: 1

Amiram Eldar, Jan 15 2024

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 15, 150, 1548, 15499, 154916, 1549105, 15489932, 154901767, 1549014294, ... . From these values the asymptotic density of this sequence, whose existence was proven by Erdős and Ivić (1987) (the constant c in the Formula section), can be empirically evaluated by 0.15490... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, p. 476.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.

Cf. A000688, A046660.

Select[Range[350], FiniteAbelianGroupCount[#] == PrimeOmega[#+1] - PrimeNu[#+1] &]

is(n) = vecprod(apply(numbpart, factor(n)[, 2])) == bigomega(n+1) - omega(n+1);

The number of terms not exceeding x, N(x) = c * x + O(x^(3/4) * log(x)^4), where c > 0 is a constant (Erdős and Ivić, 1987).

cp's OEIS Frontend

A297167 a(1) = 0, for n > 1, a(n) = -1 + the excess of n (A046660) + the index of the largest prime factor (A061395).

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A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

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A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

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A358817 Numbers k such that A046660(k) = A046660(k+1).

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A340818 Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

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A340819 Denominators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

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A323253 a(n) is the smallest number k such that factorizations of n consecutive integers starting at k have the same excess of number of primes counted with multiplicity over number of primes counted without multiplicity (A046660).

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A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

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