A099774 -id:A099774 - cp's OEIS frontend

A285702 a(n) = A000010(A064216(n)).

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A064216, A064989, A099774, A151799, A285703.

Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285702 n) (A000010 (A064216 n)))

a(n) = A000010(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A359211 a(n) = tau(3*n-1)/2, where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 2, 3, 1, 2, 2, 5, 1, 2, 1, 3, 2, 3, 1, 4, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 1, 6, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 5, 1, 4, 2, 3, 1, 2, 1, 6, 2, 2, 2, 3, 2, 2, 2, 6, 1, 4, 1, 3, 1, 3, 3, 4, 1, 2, 1, 6, 1, 4, 1

Offset: 1

Seiichi Manyama, Dec 21 2022

Also number of divisors of 3*n-1 of form 3*k+1 (or 3*k+2).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A078703, A099774.

Cf. A000005, A001620, A001817, A001822, A359212.

a[n_] := DivisorSigma[0, 3*n-1]/2; Array[a, 100] (* Amiram Eldar, Dec 21 2022 *)

PARI
```
a(n) = numdiv(3*n-1)/2;
```
PARI
```
a(n) = sumdiv(3*n-1, d, d%3==1);
```
PARI
```
a(n) = sumdiv(3*n-1, d, d%3==2);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(3*k-1))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(3*k-2))))

G.f.: Sum_{k>0} x^k/(1 - x^(3*k-1)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(3*k-2)).
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 2*log(3))*n/3 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 26 2022

A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 1, 2, 2, 2, 4, 1, 1, 3, 1, 1, 2, 2, 2, 3, 2

Offset: 1

Andrew S. Plewe, Feb 20 2004

Omega(n) of the odd integers follows a pattern similar to A001222, with 4 maxima instead of 2 - i.e. between 2^n and (2^(n+1) - 1) there are two numbers with exactly n factors (2^n and 2^(n-1) * 3) while the odd integers have 4 maxima (3^n, 3^(n-1) * 5, 3^(n-1) * 7, 5^2*3^(n-2)) between 3^n and 3^(n+1) - 1.

			Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).
For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - _Antti Karttunen_, May 31 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A285716 (after the initial term).
Cf. A001222, A000120, A092523, A099774, A099990, A244153, A278223, A286582.
Cf. A006254 (positions of ones).

a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

a(n) = bigomega(2*n-1) \\ Michel Marcus, Jul 26 2013, edited to reflect the changed starting offset by Antti Karttunen, May 31 2017

a(n) = Omega(2n-1). [Odd bisection of A001222.]
From Antti Karttunen, May 31 2017: (Start)
For n >= 1, a(n) = A000120(A244153(n)).
For n >= 2, a(n) = 1+A285716(n).
(End)

Starting offset changed to 1 and the definition modified respectively. Also values of the initial term and of term a(92) (= 2, previously a(91) = 1) corrected by Antti Karttunen, May 31 2017

A228441 G.f.: Sum_{k>0} -(-x)^k / (1 + x^k).

Original entry on oeis.org

1, -2, 2, -1, 2, -4, 2, 0, 3, -4, 2, -2, 2, -4, 4, 1, 2, -6, 2, -2, 4, -4, 2, 0, 3, -4, 4, -2, 2, -8, 2, 2, 4, -4, 4, -3, 2, -4, 4, 0, 2, -8, 2, -2, 6, -4, 2, 2, 3, -6, 4, -2, 2, -8, 4, 0, 4, -4, 2, -4, 2, -4, 6, 3, 4, -8, 2, -2, 4, -8, 2, 0, 2, -4, 6, -2, 4

Offset: 1

Michael Somos, Nov 02 2013

			G.f. = x - 2*x^2 + 2*x^3 - x^4 + 2*x^5 - 4*x^6 + 2*x^7 + 3*x^9 - 4*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Peter Bala, A signed Dirichlet product of arithmetical functions, 2019.
Subhash Chand Bhoria, Pramod Eyyunni, and Bibekananda Maji, Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities, arXiv:2011.07767 [math.NT], 2020.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A007814, A016825, A046897, A051062, A099774, A321558.

a[ n_] := SeriesCoefficient[ Sum[ -(-x)^k / (1 + x^k), {k, 1, n}], {x, 0, n}];
a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(# + n/#) &]]; (* Michael Somos, Jan 08 2015 *)
a[n_] := Module[{e = IntegerExponent[n, 2]}, DivisorSigma[0, n] * If[e == 0, 1, (e-3)/(e+1)]]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

{a(n) = if( n<1, 0, sumdiv(n, k, (-1)^(k + n/k)))};

{a(n) = if( n<1, 0, numdiv(n) - 4 * sumdiv( n, k, k%4 == 2))};

{a(n) = my(e); if( n<1, 0, e = valuation( n, 2); numdiv( n/2^e) * if( e>0, e-3, 1))};

a(n)=direuler(p=1,n,if(p==2,(1-2*X)^2/(1-X)^2,1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

a(n) = number of divisors of n minus 4 times number of divisors of n of the form 4*k+2.
a(n) = Sum_{d|n} (-1)^(d+n/d). - N. J. A. Sloane, Nov 23 2018
Multiplicative with a(2^e) = e-3 if e>0, a(p^e) = e+1 if p>2.
Moebius transform is period 4 sequence [1, -3, 1, 1, ...].
G.f.: Sum_{k>0} x^k / (1 - x^k) - 4 * x^(4*k + 2) / (1 - x^(4*k + 2)).
a(2*n - 1) = A099774(n).
Dirichlet g.f.: zeta(s)^2*(1-2^(-s+1))^2 = eta^2(s) (the Dirichlet eta). - Ralf Stephan, Mar 27 2015
a(16n+8) = a(A051062(n)) = 0. - Michel Marcus, Mar 27 2015
O.g.f.: Sum_{n >= 1} (-1)^(n*(n+1))*x^(n^2)*(1 - x^n)/(1 + x^n). - Peter Bala, Mar 11 2019
Conjecture: a(n) = (7 - 2*(-1)^n)*tau(n) - 4*tau(2*n) = 5*tau(n) - (3 + (-1)^n)*tau(2*n), where tau = A000005. - Velin Yanev, Dec 17 2019
The proof of the above conjecture easily follows from the fact that both a(n) and tau(n) are multiplicative arithmetical functions and tau(p^e) = e + 1 for prime p. - Peter Bala, Jan 28 2022
a(n) = A000005(n) if n is odd, and A000005(n) * (A007814(n)-3)/(A007814(n)+1) if n is even. - Amiram Eldar, Sep 18 2023

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

a(A001605(n)) = 2; a(A105802(n)) = n.

It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */

{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Corrected and extended by Sascha Kurz, Jan 26 2003

Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.

A285703 a(n) = A000203(A064216(n)).

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 12, 14, 18, 18, 20, 13, 15, 24, 30, 24, 24, 32, 36, 38, 42, 28, 44, 31, 42, 48, 32, 54, 54, 60, 42, 48, 62, 60, 68, 72, 39, 48, 74, 31, 80, 56, 72, 84, 72, 90, 72, 90, 56, 98, 102, 72, 104, 108, 96, 110, 80, 84, 84, 57, 114, 40, 114, 126, 128, 108, 60, 132, 138, 132, 96, 96, 93, 140, 150, 98, 120, 152, 144, 120, 158, 96, 164, 133, 126

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A064216, A099774, A151799, A285702, A285704, A285705.

Table[DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 86}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285703 n) (A000203 (A064216 n)))

a(n) = A000203(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.8168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A318734 a(n) = Sum_{k=1..n} (-1)^(k + 1) * d(2*k - 1), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, -1, 1, -1, 2, 0, 2, -2, 0, -2, 2, 0, 3, -1, 1, -1, 3, -1, 1, -3, -1, -3, 3, 1, 4, 0, 2, -2, 2, 0, 2, -4, 0, -2, 2, 0, 2, -4, 0, -2, 3, 1, 5, 1, 3, -1, 3, -1, 1, -5, -3, -5, 3, 1, 3, -1, 1, -3, 3, -1, 2, -2, 2, 0, 4, 2, 6, -2, 0, -2, 2, -2

Offset: 1

Hugo Pfoertner, Sep 05 2018

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Records of A318734, showing negative and positive records.

Cf. A000005, A099774, A006218, A222068.

Records and their positions: A318735, A318736, A318737, A318738.

a[n_] := Sum[(-1)^(k + 1) DivisorSigma[0, 2 k - 1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Sep 17 2018 *)

s=0;j=-1;forstep(k=1,141,2,j=-j;s=s+j*numdiv(k);print1(s,", "))

a(n) = sum(k=1, n, (-1)^(k+1)*numdiv(2*k-1)); \\ Michel Marcus, Sep 08 2018

A318735 Positive records in A318734.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 21, 23, 25, 30, 31, 32, 34, 39, 45, 47, 48, 51, 52, 56, 60, 62, 71, 76, 78, 83, 84, 88, 91, 103, 108, 119, 127, 129, 132, 142, 143, 151, 166, 168, 171, 178, 181, 183, 189, 197, 215, 237, 241, 244, 266, 270, 274

Offset: 1

Hugo Pfoertner, Sep 05 2018

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..268

Cf. A000005, A099774, A318734, A318736, A318737, A318738.

s=0;smax=0;j=-1;forstep(k=1,20000000,2,j=-j;s=s+j*numdiv(k);if(s>smax,smax=s;print1(s,", ")))

A318736 Absolute value of negative records in A318734.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 21, 22, 23, 25, 30, 32, 36, 40, 42, 44, 46, 50, 53, 55, 56, 57, 59, 65, 67, 69, 73, 79, 87, 96, 98, 100, 104, 108, 110, 113, 115, 118, 122, 126, 134, 139, 151, 161, 167, 169, 172, 178, 180, 182

Offset: 1

Hugo Pfoertner, Sep 05 2018

nonn

1

Hugo Pfoertner, Table of n, a(n) for n = 1..282

Cf. A000005, A099774, A318734, A318735, A318737, A318738.

s=0;j=-1;smin=0;forstep(k=1,5000000,2,j=-j;s=s+j*numdiv(k);if(s

A318737 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 9, 25, 49, 85, 133, 169, 225, 445, 845, 973, 1125, 2205, 2209, 2469, 2829, 7929, 9429, 9945, 23569, 24073, 24645, 26145, 40425, 68153, 71289, 72413, 89517, 112233, 112245, 128973, 162405, 162409, 162429, 297073, 477489, 477493, 502713, 561253

Offset: 1

Hugo Pfoertner, Sep 05 2018

nonn

			a(2) = 9, because s = d(1)-d(3)+d(5)-d(7)+d(9) = 1-2+2-2+3 = 2 exceeds d(1)=1, d(1)-d(3)=-1, d(1)-d(3)+d(5)=1, d(1)-d(3)+d(5)-d(7)=-1.

Hugo Pfoertner, Table of n, a(n) for n = 1..268

Cf. A000005, A099774, A318734, A318735, A318736, A318738.

s=0;smax=0;j=-1;forstep(k=1,600000,2,j=-j;s=s+j*numdiv(k);if(s>smax,smax=s;print1(k,", ")))

cp's OEIS Frontend

A285702 a(n) = A000010(A064216(n)).

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A359211 a(n) = tau(3*n-1)/2, where tau(n) = number of divisors of n, cf. A000005.

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A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

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A228441 G.f.: Sum_{k>0} -(-x)^k / (1 + x^k).

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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

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A285703 a(n) = A000203(A064216(n)).

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A318734 a(n) = Sum_{k=1..n} (-1)^(k + 1) * d(2*k - 1), where d(k) is the number of divisors of k (A000005).

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A318737 Numbers n=2k-1 where Sum_{j=1..k} (-1)^(j+1) d(2*j-1) achieves a new record, with d(n) = number of divisors of n (A000005).