A008578 -id:A008578 - cp's OEIS frontend

A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.

1, 3, 4, 6, 6, 11, 8, 12, 12, 17, 12, 22, 14, 23, 23, 24, 18, 33, 20, 34, 31, 35, 24, 44, 30, 41, 36, 46, 30, 61, 32, 48, 47, 53, 47, 66, 38, 59, 55, 68, 42, 83, 44, 70, 69, 71, 48, 88, 56, 85, 71, 82, 54, 99, 71, 92, 79, 89, 60, 122, 62, 95, 93, 96, 83, 127

Offset: 1

Jaroslav Krizek, Oct 25 2013

nonn

			For n = 6: a(6) = 6 * (1/1 + 1/2 + 1/3) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A080339, A000027, A008578, A329039.

Coincides with A129283 on squarefree numbers, A005117.

a[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; Array[a, 100] (* Amiram Eldar, Nov 12 2021 *)

A230593(n) = sumdiv(n,d,((1==d)||isprime(d))*(n/d)); \\ Antti Karttunen, Nov 12 2021

For n > 1, a(n) = n + n * Sum_(p|n) 1 / p, where p are primes dividing n.
a(n) = A069359(n) + n.
a(n) = Sum_{d|n}  A080339(d) * A000027(n/d).
a(n) = A080339(n) * A000027(n), where operation * denotes Dirichlet convolution, i.e. convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d).
For p, q = distinct primes, a(p) = p + 1, a(pq) = pq - 1.
From Antti Karttunen, Nov 12 2021: (Start)
a(n) = A129283(n) - A329039(n).
a(A005117(n)) = A129283(A005117(n)), for all n >= 1.
(End)
For p prime, k>=1, a(p^k) = p^(k-1) * (p+1). - Bernard Schott, Nov 12 2021

A075527 a(n) = A008578(n+3) - A008578(n+1).

Original entry on oeis.org

2, 3, 4, 6, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12, 10, 10, 14, 12, 12, 18, 12

Offset: 0

Reinhard Zumkeller, Sep 22 2002

nonn

For n>0: a(n) = A031131(n) and a(n) - a(n-1) = A075526(n).

Cf. A000040, A076272, A076273.

Cf. A008578, A031131, A075526.

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A249340 Position of the first occurrence of n-th noncomposite number, A008578(n), in A249336; positions of records in A249338.

Original entry on oeis.org

1, 3, 5, 13, 20, 39, 50, 70, 80, 97, 131, 142, 193, 240, 257, 296, 322, 379, 397, 435, 478, 490, 542, 569, 629, 736, 758, 764, 828, 835, 872, 1067, 1100, 1209, 1214, 1369, 1414, 1468, 1514, 1549, 1606, 1681, 1700, 1853, 1871, 1903, 1931, 2116, 2244, 2293, 2303, 2343

Offset: 1

Antti Karttunen, Oct 26 2014

nonn

Cf. A249336, A249338, A249339.

a(n) = A249339(n+1) - 1.
Other identities. For all n >= 1:
A249336(a(n)) = A008578(n).
A249338(a(n)) = n-1.

A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

Original entry on oeis.org

3, 6, 1, 10, 1, 12, 1, 14, 1, 4, 1, 28, 1, 1, 1, 22, 1, 12, 1, 20, 1, 4, 1, 102, 1, 1, 1, 4, 1, 4, 1, 26, 1, 1, 1, 66, 1, 1, 1, 104, 1, 12, 1, 6, 1, 4, 1, 92, 1, 1, 1, 4, 1, 4, 1, 6, 1, 4, 1, 132, 1, 1, 1, 34, 1, 4, 1, 1, 1, 4, 1, 1240, 1, 1, 1, 1, 1, 4, 1, 57, 1, 4, 1, 21, 1, 1, 1, 28, 1, 1, 1, 6, 1, 1, 1, 492, 1, 1, 1, 12, 1, 4, 1, 6, 1

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A014197, A320000, A322311 (rgs-transform).

Cf. also A322312.

A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ From A320000
A322310(n) = if(1==n,3,my(m=1); fordiv(n,d, my(s, p=d+1, q=n/d); if(isprime(p) && (s = sum(i=0,valuation(n, p), A320000sq(q/(p^i),p))), m *= prime(s))); (m));

a(n) = Product_{d|n} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)])^A010051(1+d).

For all n, A056239(a(n)) = A014197(n).

A327167 a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 8, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 12, 1, 1, 1, 3, 6, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 8, 1, 1, 1, 1, 48, 1, 2, 1, 1, 2, 7, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 128

Offset: 1

Antti Karttunen, Sep 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008578, A276086, A286561, A327168.

Cf. also A293514, A322312, A323155, A327154, A327155, A327156.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327167(n) = { my(m=1,v); fordiv(A276086(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)).
Other identities. For all n >= 1:
1+A001222(a(n)) = A327168(n).

A138607 List first A008578(1) odd numbers, then first A008578(2) even numbers, then the next A008578(3) odd numbers, then the next A008578(4) even numbers, etc.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 14, 9, 11, 13, 15, 17, 19, 21, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73

Offset: 1

Ctibor O. Zizka, May 14 2008

A permutation of numbers.

			Let
  S1={1}
  S2={2,4}
  S3={3,5,7}
  S4={6,8,10,12,14}
  S5={9,11,13,15,17,19,21}
  S6={16,18,20,22,24,26,28,30,32,34,36}
  ...
then S1, S2, S3, S4, S5, S6,... gives this sequence.

Index entries for sequences that are permutations of the natural numbers

Inverse: A166014. Cf. A138606, A138608, A138609, A138612.

If n < 3, a(n) = n. If n-2 = A007504(A083375(n-2)), then a(n) = a(n-1-A000040(A083375(n-2)))+2, otherwise a(n) = a(n-1)+2. - Antti Karttunen, Oct 05 2009.

Edited, extended, and offset changed from 0 to 1 by Antti Karttunen, Oct 05 2009

A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 9, 10, 13, 18, 19, 24, 27, 28, 31, 36, 41, 42, 47, 50, 51, 56, 59, 64, 71, 74, 75, 78, 79, 82, 95, 98, 103, 104, 113, 114, 119, 124, 127, 132, 137, 138, 147, 148, 151, 152, 163, 174, 177, 178, 181, 186, 187, 196, 201, 206, 211, 212, 217, 220, 221

Offset: 1

Jaroslav Krizek, Jun 27 2009

Essentially the same as A065890.

a(n) = number of terms of A073169(n) less than n.

			A008578(6) = 11, and there are 5 composites smaller than 11, viz. 4, 6, 8, 9, 10, hence a(6) = 5.

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

Cf. A002808 (composites), A008578 (1 and the primes), A065890, A073169.

T:=[0,1] cat PrimesUpTo(300); [ T[n+1]-n: n in [1..#T-1] ]; // Klaus Brockhaus, Sep 08 2009

Join[{0},Module[{nn=300,cmps},cmps=Accumulate[Table[If[CompositeQ[n],1,0],{n,nn}]];Table[cmps[[p]],{p,Prime[ Range[ PrimePi[ nn]]]}]]] (* Harvey P. Dale, Nov 11 2024 *)

a(n) = A008578(n) - n = A158611(n+1) -n.

a(n) = A065890(n-1) for n > 1.

Edited and extended by Klaus Brockhaus, Sep 09 2009

A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Oct 27 2013

nonn

Dirichlet convolution of A080339(n) with itself, where A080339 = characteristic function of noncomposite numbers (A008578).
Dirichlet convolution of functions b(n) and c(n) is function a(n) = Sum_{d|n} b(d) * c(n/d).
a(n) = 0, 1 or 2. a(n) = 0 for numbers n from A033942 (numbers with least 3 prime factors (counted with multiplicity)); a(n) = 1 for n = p^2, p = prime; a(n) = 2 for numbers n from A167171 (A006881 union A000040).

			For n = 6: a(6) = Sum_(d|6) A080339(d) * A080339(6/d) = 1*0 + 1*1 + 1*1 + 0*1 = 2.

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

Cf. A080339, A033942, A167171, A006881, A000040, A230595.

a[n_] := Switch[FactorInteger[n][[;;, 2]], {2}, 1, {1}, 2, {1, 1}, 2, , 0]; a[1] = 1; Array[a, 100] (* _Amiram Eldar, Aug 29 2023 *)

A080339(n) = ((1==n)||isprime(n));
A230594(n) = sumdiv(n,d,(A080339(d)*A080339(n/d))); \\ Antti Karttunen, Jul 27 2017

from sympy import factorint
def A230594(n): return 0 if sum(f:=factorint(n).values())>2 else (1 if n==1 or max(f)==2 else 2) # Chai Wah Wu, Jul 23 2024

a(n) = Sum_{d|n} A080339(d) * A080339(n/d).

A249342 Position of the first occurrence of n-th noncomposite number, A008578(n), in A249337; positions of records in A249072.

Original entry on oeis.org

1, 2, 6, 13, 20, 39, 45, 70, 80, 97, 121, 145, 193, 240, 257, 296, 322, 379, 400, 432, 481, 490, 532, 566, 632, 715, 760, 766, 807, 835, 878, 1067, 1104, 1209, 1215, 1369, 1414, 1468, 1514, 1540, 1605, 1683, 1706, 1853, 1871, 1905, 1935, 2116, 2276, 2293, 2304, 2343

Offset: 1

Antti Karttunen, Oct 26 2014

nonn

After the initial term, one less than A249341.

Cf. A008578, A249337, A249340, A249072.

For all n >= 1:
A249337(a(n)) = A008578(n).
A249072(a(n)) = n-1.
For all n >= 2:
a(n) = A249341(n) - 1.

A255324 Difference between the n-th Ludic number and the n-th noncomposite number: a(n) = A003309(n) - A008578(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 6, 4, 2, 4, 6, 8, 8, 10, 10, 12, 16, 12, 14, 18, 18, 18, 18, 20, 22, 30, 22, 26, 24, 34, 26, 28, 24, 30, 42, 38, 42, 42, 36, 40, 38, 40, 36, 34, 38, 48, 50, 48, 60, 56, 56, 66, 62, 66, 64, 72, 76, 68, 70, 72, 76, 80, 76, 78, 72, 72, 78, 74, 70, 72, 84, 84, 86, 84, 92, 88, 84, 88, 86, 94, 96, 98, 104

Offset: 1

Antti Karttunen, Feb 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..14000

Cf. A255325 (the same terms halved).

Cf. A032600, A003309, A008578, A255421, A255422.

t = Range[2, 1000];
A003309 = {1};
While[Length[t]>0, k = t[[1]]; AppendTo[A003309, k]; t = Drop[t, {1, -1, k}]];
a[n_] := If[n == 1, 0, A003309[[n]] - Prime[n - 1]];
Array[a, Length[A003309]] (* Jean-François Alcover, Jan 08 2022, after Ray Chandler in A003309 *)

(define (A255324 n) (- (A003309 n) (A008578 n)))

a(n) = A003309(n) - A008578(n).

a(n) = 2*A255325(n).

cp's OEIS Frontend

A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.

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A075527 a(n) = A008578(n+3) - A008578(n+1).

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A249340 Position of the first occurrence of n-th noncomposite number, A008578(n), in A249336; positions of records in A249338.

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A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

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A327167 a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A138607 List first A008578(1) odd numbers, then first A008578(2) even numbers, then the next A008578(3) odd numbers, then the next A008578(4) even numbers, etc.

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A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

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A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

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A249342 Position of the first occurrence of n-th noncomposite number, A008578(n), in A249337; positions of records in A249072.