A083399 -id:A083399 - cp's OEIS frontend

A205745 a(n) = card { d | d*p = n, d odd, p prime }.

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

Offset: 1

Peter Luschny, Jan 30 2012

nonn

Equivalently, a(n) is the number of prime divisors p|n such that n/p is odd. - Gus Wiseman, Jun 06 2018

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

Cf. A000005, A000607, A001221, A008683, A010051, A068050, A077761, A083399, A088705, A106404, A305614.

a205745 n = sum $ map ((`mod` 2) . (n `div`))
   [p | p <- takeWhile (<= n) a000040_list, n `mod` p == 0]
-- Reinhard Zumkeller, Jan 31 2012

a[n_] := Sum[ Boole[ OddQ[d] && PrimeQ[n/d] ], {d, Divisors[n]} ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 27 2013 *)

a(n)=if(n%2,omega(n),n%4/2) \\ Charles R Greathouse IV, Jan 30 2012

def A205745(n):
    return sum((n//d) % 2 for d in divisors(n) if is_prime(d))
[A205745(n) for n in (1..105)]

O.g.f.: Sum_{p prime} x^p/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

Sum_{k=1..n} a(k) = (n/2) * (log(log(n)) + B) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A292587 Compound filter: a(n) = P(A001221(n), A292582(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 7, 1, 5, 1, 5, 3, 3, 1, 8, 2, 3, 4, 5, 1, 6, 1, 11, 3, 3, 3, 23, 1, 3, 3, 8, 1, 6, 1, 5, 5, 3, 1, 12, 2, 5, 3, 5, 1, 8, 3, 8, 3, 3, 1, 9, 1, 3, 5, 22, 3, 6, 1, 5, 3, 6, 1, 38, 1, 3, 5, 5, 3, 6, 1, 12, 7, 3, 1, 9, 3, 3, 3, 8, 1, 9, 3, 5, 3, 3, 3, 17, 1, 5, 5, 23, 1, 6, 1, 8, 6

Offset: 1

Antti Karttunen, Sep 26 2017

This is essentially also a filter constructed from the runlengths of numbers of the form 4k+0 and the runlengths of numbers of the form 4k+2 encountered in trajectories of A005940-tree. See comments in A083399 and A292586.
For all i, j: A291757(i) = A291757(j) => a(i) = a(j), that is, this filter matches to a subset of the sequences matched by filter A291757.
Moreover, for all i, j: a(i) = a(j) <=> A101296(i) = A101296(j), thus the subset is exactly the sequences matched by A101296 (A046523). This follows because the prime signature of n can be recovered from the two components as A046523(n) = A046523(A003557(n)) * A292586(n) and also vice versa as A046523(A003557(n)) = A003557(A046523(n)).

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

Cf. A000027, A001221, A003557, A005940, A046523, A101296, A291757, A292582, A292584, A292586, A292588.

a(n) = (1/2)*(2 + ((A001221(n) + A292582(n))^2) - A001221(n) - 3*A292582(n)).

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 19 2021

nonn

First differs from A096825 at a(525) = 3, A096825(525) = 4.
First differs from A345926 at a(90) = 4, A345926(90) = 3.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.

			The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.

The half-length submultisets are counted by A114921.
Including all multisets of prime factors gives A305611(n) + 1.
The strict rounded version appears to be counted by A342343.
The version for prime indices instead of prime factors is A345926.
A000005 counts divisors, which add up to A000203.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 and A299701 count positive subset-sums of partitions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A334968 counts subsequence-sums of standard compositions.
Cf. A008549, A032443, A083399, A096825, A344609, A345957.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A343943(n):
    fs = factorint(n)
    return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A082767 Number of edges in the prime graph.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 14, 16, 18, 21, 23, 26, 28, 31, 34, 36, 38, 41, 43, 46, 49, 52, 54, 57, 59, 62, 64, 67, 69, 73, 75, 77, 80, 83, 86, 89, 91, 94, 97, 100, 102, 106, 108, 111, 114, 117, 119, 122, 124, 127, 130, 133, 135, 138, 141, 144, 147, 150, 152, 156, 158, 161, 164, 166, 169, 173

Offset: 1

Jon Perry, May 24 2003

nonn

The prime graph is defined to be the graph formed by writing the integers 0 to n in a straight line as vertices and then connecting i and j (i > j) iff i-j=1 or i=j+p, where p is a prime factor of i. It can be visualized as the Sieve of Eratosthenes, with each integer connected to its neighbors and the striking out process as a wave forming the remaining edges.

			a(1) = 1.
a(2) = a(1) + 1 + omega(2) = 1 + 1 + 1 = 3.
a(6) = a(5) + 1 + omega(6) = 9 + 1 + 2 = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Partial sums of A083399.

I:=[1]; [n le 1 select I[n] else Self(n-1)+1+#PrimeDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jun 10 2017

Accumulate[PrimeNu[Range[120]] + 1] (* Vincenzo Librandi, Jun 10 2017 *)

a=1; c=2; while (c<50,print1(a","); a=a+1+omega(c); c++)

a(n) = a(n-1) + 1 + omega(n) if n > 1, with a(1) = 1, where omega(n) is the number of distinct prime factors of n.

a(n) = Sum_{p is 1 or a prime, p <= n} floor(n/p); e.g., a(12) = floor(12/1) + floor(12/2) + floor(12/30) + floor(12/5) + floor(12/7) + floor(12/11) = 12 + 6 + 4 + 2 + 1 + 1 = 26. - Amarnath Murthy, Jul 06 2005

Corrected by T. D. Noe, Oct 25 2006

A262095 Number of non-semiprime divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 3, 4, 2, 5, 2, 5, 3, 3, 3, 6, 2, 3, 3, 6, 2, 5, 2, 4, 4, 3, 2, 8, 2, 4, 3, 4, 2, 6, 3, 6, 3, 3, 2, 8, 2, 3, 4, 6, 3, 5, 2, 4, 3, 5, 2, 9, 2, 3, 4, 4, 3, 5, 2, 8, 4, 3, 2, 8, 3, 3, 3, 6, 2, 8, 3, 4, 3, 3, 3, 10, 2

Offset: 1

Juri-Stepan Gerasimov, Sep 11 2015

			(1, 2, 3, 4, 6, 8, 12, 24) are the divisors of n = 24: 1, 2, 3, 8, 12, and 24 are non-semiprimes, therefore a(24) = 6.

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

Cf. A000005, A083399, A086971, A100959.

Cf. A064911, A027750.

a262095 = sum . map ((1 -) . a064911) . a027750_row
-- Reinhard Zumkeller, Sep 14 2015

Table[Count[Divisors@ n, x_ /; PrimeOmega@ x != 2], {n, 97}] (* Michael De Vlieger, Sep 14 2015 *)

a(n) = sumdiv(n, d, bigomega(d)!=2); \\ Michel Marcus, Sep 11 2015

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1) - sum(i=1,#f,f[i]>1) - #f*(#f-1)/2 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A000005(n) - A086971(n).
A083399(n) <= a(n) <= A000005(n).
a(n) = Sum_{k=1..A000005(n)} (1 - A064911(A027750(n,k))). - Reinhard Zumkeller, Sep 14 2015

A275693 Lexicographically earliest increasing sequence such that the a(n)th term of the sequence has n noncomposite divisors.

Original entry on oeis.org

1, 2, 4, 6, 7, 30, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 2310, 2311, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2321, 2322, 2323, 2324, 2325, 2326, 2327, 2328, 2329, 2330

Offset: 1

Jaroslav Krizek, Aug 05 2016

nonn

We let tau_nc(n) = number of noncomposite divisors of n = A083399(n) = A001221(n) + 1 = omega(n) + 1.

Primorial numbers from A002110 are terms.

			a(1)=1 because tau_nc(1)=1; a(2)=2 because tau_nc(2)=2; a(3) cannot be 3 because tau_nc(3)=2, a(3)=4 (4 is the smallest number x>3); if a(3)=4, a(4) must be the smallest number x>a(3) with 3 noncomposite divisors, a(4)=6; a(6) must be number with 4 noncomposite divisors and must keep increase of the sequence, a(6)=30; a(5)=7 because 7>a(4); a(7) must be the smallest number with 5 noncomposite divisors because a(5)=7, a(7)=210; if a(6)=30, a(30) must be the smallest number x>a(7) with 6 noncomposite divisors and must keep increase of the sequence, a(30)=2310; a(8)-a(29) are numbers from interval 211-232; etc...

Jaroslav Krizek, Table of n, a(n) for n = 1..250

Cf. A001221, A002110, A083399, A275658.

tau_nc(a(a(n))) = A083399(a(a(n))) = A001221(a(a(n))) + 1 = omega(a(a(n))) + 1 = n.

A330908 a(n+1) = a(n) + (number of divisors of a(n) that are not divisors of other divisors of a(n)) for n>1; a(1)=1.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 13, 15, 18, 21, 24, 27, 29, 31, 33, 36, 39, 42, 46, 49, 51, 54, 57, 60, 64, 66, 70, 74, 77, 80, 83, 85, 88, 91, 94, 97, 99, 102, 106, 109, 111, 114, 118, 121, 123, 126, 130, 134, 137, 139, 141, 144, 147, 150, 154, 158, 161, 164, 167, 169

Offset: 1

Samuel Frankmartin-Sebastian Wiethüchter, May 01 2020

nonn

The sequence is similar built like A094222 but includes 1 as divisor or adds 1 to the number of distinct primes dividing a(n).

			For n = 2 calculate a(2)= a(2-1) + A083399(a(2-1))= 1 + 1 = 2;
For n = 3 a(3)=a(2) + A083399(a(2))= 2 + 2 = 4;
For n = 4 a(4)=a(3) + A083399(a(3))= 4 + 2 = 6;
For n = 5 a(5)=a(4) + A083399(a(4))= 6 + 3 = 9;

Samuel Frankmartin-Sebastian Wiethüchter, Table of n, a(n) for n = 1..5000

Cf. A094222.

A330908 := proc(n) option remember;
    if n < 2 then
        n
    else
        procname(n-1)+A083399(procname(n-1))
    end if;
end proc:
seq(A330908(n), n=1..30);

a[1] = 1; a[n_] := a[n] = a[n - 1] + PrimeNu[a[n - 1]] + 1; Array[a, 60] (* Amiram Eldar, May 01 2020 *)

f(n) = omega(n) + 1; \\ A083399
lista(nn) = {my(a=1, va = List(a)); for (n=2, nn, a = a+f(a); listput(va, a);); Vec(va);} \\ Michel Marcus, May 03 2020

a(n) = a(n-1) + A083399(a(n-1)) for n>1.

A332687 a(n) = Sum_{k=1..n} ceiling(n/prime(k)).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 27, 29, 32, 35, 37, 39, 42, 44, 47, 50, 53, 55, 58, 60, 63, 65, 68, 70, 74, 76, 78, 81, 84, 87, 90, 92, 95, 98, 101, 103, 107, 109, 112, 115, 118, 120, 123, 125, 128, 131, 134, 136, 139, 142, 145, 148, 151, 153

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Cf. A000720, A001221, A006590, A013939, A082767, A083399.

Table[Sum[Ceiling[n/Prime[k]], {k, 1, n}], {n, 1, 60}]
Table[n + Sum[PrimeNu[k], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x)^2 + (x/(1 - x)) Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
With[{nmax = 100}, Range[nmax] + Join[{0}, Accumulate[Table[PrimeNu[k], {k, 1, nmax - 1}]]]] (* Amiram Eldar, Sep 21 2024 *)

a(n) = sum(k=1, n, ceil(n/prime(k))); \\ Michel Marcus, Feb 21 2020

lista(nmax) = my(s = 1); for(n = 2, nmax, print1(s, ", "); s += omega(n-1) + 1); \\ Amiram Eldar, Sep 21 2024

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} x^prime(k) / (1 - x^prime(k)).
a(n) = n + Sum_{k=1..n-1} omega(k), where omega = A001221.
a(n) = n - omega(n) + Sum_{k=1..n} pi(floor(n/k)), where pi = A000720.
a(n) = n + A013939(n-1) for n >= 2. - Amiram Eldar, Sep 21 2024

cp's OEIS Frontend

A205745 a(n) = card { d | d*p = n, d odd, p prime }.

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A292587 Compound filter: a(n) = P(A001221(n), A292582(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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A082767 Number of edges in the prime graph.

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A262095 Number of non-semiprime divisors of n.

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A275693 Lexicographically earliest increasing sequence such that the a(n)th term of the sequence has n noncomposite divisors.

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A330908 a(n+1) = a(n) + (number of divisors of a(n) that are not divisors of other divisors of a(n)) for n>1; a(1)=1.

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