A297108 -id:A297108 - cp's OEIS frontend

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1

Offset: 1

Marc LeBrun

There are arbitrarily long runs of ones (Sierpiński). - Franz Vrabec, Sep 26 2005
a(n) is the smallest positive integer such that n divides Product_{k=1..n} a(k), for all positive integers n. - Leroy Quet, May 01 2007
For n>1, resultant of the n-th cyclotomic polynomial with the 1st cyclotomic polynomial x-1. - Ralf Stephan, Aug 14 2013
A368749(n) is the smallest prime p such that the interval [a(p), a(q)] contains n 1's; q = nextprime(p), n >= 0. - David James Sycamore, Mar 21 2024

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
I. Vardi, Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153 and 249, 1991.

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
Carl McTague, On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q), arXiv:1510.06696 [math.CO], 2015.
A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
W. Sierpiński, On the numbers [1,2,...n], (Polish) Wiadom. Mat. (2) 9 1966 9-10.
Eric Weisstein's World of Mathematics, Mangoldt Function.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
Index entries for sequences related to lcm's

Apart from initial 1, same as A020500. With ones replaced by zeros, equal to A120007.
Cf. A003418, A007947, A008683, A008472, A008578, A048671 (= n/a(n)), A072107 (partial sums), A081386, A081387, A099636, A100994, A100995, A140255 (inverse Mobius transform), A140254 (Mobius transform), A297108, A297109, A340675, A000027, A348846, A368749.
First column of A140256. Row sums of triangle A140581.
Cf. also A140579, A140580 (= n*a(n)).

a014963 1 = 1
a014963 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 1
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 09 2011

a := n -> if n < 2 then 1 else numtheory[factorset](n); if 1 < nops(%) then 1 else op(%) fi fi; # Peter Luschny, Jun 23 2009
A014963 := n -> n/ilcm(op(numtheory[divisors](n) minus {1,n}));
seq(A014963(i), i=1..69); # Peter Luschny, Mar 23 2011
# The following is Nowicki's LCM-Transform - N. J. A. Sloane, Jan 09 2024
LCMXFM:=proc(a)  local p,q,b,i,k,n:
if whattype(a) <> list then RETURN([]); fi:
n:=nops(a):
b:=[a[1]]: p:=[a[1]];
for i from 2 to n do q:=[op(p),a[i]]; k := lcm(op(q))/lcm(op(p));
b:=[op(b),k]; p:=q;; od:
RETURN(b); end:
# Alternative, to be called by 'seq' as shown, not for a single n.
a := proc(n) option remember; local i; global f; f := ifelse(n=1, 1, f*n);
iquo(f, mul(a(i)^iquo(n, i), i=1..n-1)) end: seq(a(n), n=1..95); # Peter Luschny, Apr 05 2025

a[n_?PrimeQ] := n; a[n_/;Length[FactorInteger[n]] == 1] := FactorInteger[n][[1]][[1]]; a[n_] := 1; Table[a[n], {n, 95}] (* Alonso del Arte, Jan 16 2011 *)
a[n_] := Exp[ MangoldtLambda[n]]; Table[a[n], {n, 95}] (* Jean-François Alcover, Jul 29 2013 *)
Ratios[LCM @@ # & /@ Table[Range[n], {n, 100}]] (* Horst H. Manninger, Mar 08 2024 *)
Table[Which[PrimeQ[n],n,PrimePowerQ[n],Surd[n,FactorInteger[n][[-1,2]]],True,1],{n,100}] (* Harvey P. Dale, Mar 01 2025 *)

A014963(n)=
{
    local(r);
    if( isprime(n), return(n));
    if( ispower(n,,&r) && isprime(r), return(r) );
    return(1);
}  \\ Joerg Arndt, Jan 16 2011

a(n)=ispower(n,,&n);if(isprime(n),n,1) \\ Charles R Greathouse IV, Jun 10 2011

from sympy import factorint
def A014963(n):
    y = factorint(n)
    return list(y.keys())[0] if len(y) == 1 else 1
print([A014963(n) for n in range(1, 71)]) # Chai Wah Wu, Sep 04 2014

def A014963(n) : return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))
[A014963(n) for n in (1..50)]  # Peter Luschny, Feb 02 2012

def a(n):
    if n == 1: return 1
    return prod(1 - E(n)**k for k in ZZ(n).coprime_integers(n+1))
[a(n) for n in range(1, 14)] # F. Chapoton, Mar 17 2020

a(n) = A003418(n) / A003418(n-1) = lcm {1..n} / lcm {1..n-1}. [This is equivalent to saying that this sequence is the LCM-transform (as defined by Nowicki, 2013) of the positive integers. - David James Sycamore, Jan 09 2024.]
a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - Vladeta Jovovic, Jan 24 2002
a(n) = gcd( C(n+1,1), C(n+2,2), ..., C(2n,n) ) where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003
a(n) = gcd(C(n,1), C(n+1,2), C(n+2,3), ...., C(2n-2,n-1)), where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003; corrected by Ant King, Dec 27 2005
Note: a(n) != gcd(A008472(n), A007947(n)) = A099636(n), GCD of rad(n) and sopf(n) (this fails for the first time at n=30), since a(30) = 1 but gcd(rad(30), sopf(30)) = gcd(30,10) = 10.
a(n)^A100995(n) = A100994(n). - N. J. A. Sloane, Feb 20 2005
a(n) = Product_{k=1..n-1, if(gcd(n, k)=1, 1-exp(2*Pi*i*k/n), 1)}, i=sqrt(-1); a(n) = n/A048671(n). - Paul Barry, Apr 15 2005
Sum_{n>=1} (log(a(n))-1)/n = -2*A001620 [Bateman Manuscript Project Vol III, ed. by Erdelyi et al.]. - R. J. Mathar, Mar 09 2008
n*a(n) = A140580(n) = n^2/A048671(n) = A140579 * [1,2,3,...]. - Gary W. Adamson, May 17 2008
a(n) = (2*Pi)^phi(n) / Product_{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). - Peter Luschny, Aug 08 2009
a(n) = A166140(n) / A166142(n). - Mats Granvik, Oct 08 2009
a(n) = GCD of rows in A167990. - Mats Granvik, Nov 16 2009
a(n) = A010055(n)*(A007947(n) - 1) + 1. - Reinhard Zumkeller, Mar 26 2010
a(n) = 1 + (A007947(n)-1) * floor(1/A001221(n)), for n>1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = Product_{k=1..n-1} if(gcd(k,n)=1, 2*sin(Pi*k/n), 1). - Peter Luschny, Jun 09 2011
a(n) = exp(Sum_{k>=1} A191898(n,k)/k) for n>1 (conjecture). - Mats Granvik, Jun 19 2011
Dirichlet g.f.: Sum_{n>0} e^Lambda(n)/n^s = Zeta(s) + Sum_{p prime} Sum_{k>0} (p-1)/p^(k*s) = Zeta(s) - ppzeta(s) + Sum(p prime, p/(p^s-1)); for a ppzeta definition see A010055. - Enrique Pérez Herrero, Jan 19 2013
a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} moebius(d)/d^(s-1)) for n>1. - Mats Granvik, Jul 31 2013
a(n) = gcd_{k=1..n-1} binomial(n,k) for n > 1, see A014410. - Michel Marcus, Dec 08 2015 [Corrected by Jinyuan Wang, Mar 20 2020]
a(n) = 1 + Sum_{k=2..n} (k-1)*A010051(k)*(floor(k^n/n) - floor((k^n - 1)/n)). - Anthony Browne, Jun 16 2016
The Dirichlet series for log(a(n)) = Lambda(n) is given by the logarithmic derivative of the zeta function -zeta'(s)/zeta(s). - Mats Granvik, Oct 30 2016
a(n) = A008578(1+A297109(n)), For all n >= 1, Product_{d|n} a(d) = n. - Antti Karttunen, Feb 01 2021
Product_{k=1..floor(n/2)} Product_{j=1..floor(n/k)} a(j) = n!. - Ammar Khatab, Jan 28 2025

Additional reference from Eric W. Weisstein, Jun 29 2008

A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 5, 8, 20, 24, 40, 48, 80, 88, 120, 120, 160, 168, 240, 240, 320, 288, 312, 360, 480, 384, 408, 528, 640, 616, 520, 648, 960, 840, 816, 960, 1280, 1072, 1440, 1248, 1248, 1368, 1224, 1360, 1920, 1680, 1920, 1848, 1632, 1872, 2640, 2208, 2560, 2384, 3016

Offset: 1

Rémy Sigrist, Nov 29 2017

This sequence is a variant of A256739; both sequences have nice graphical features.
Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).
For any sequence f of nonnegative integers with positive indices:
- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,
- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),
- also, x_A178910 = A000027 and x_A055895 = A000079,
- see the links section for a gallery of x_f plots for some classic f functions,
- x_f(1) = f(1),
- x_f(p) = f(1) XOR f(p) for any prime p,
- x_A063524 = A008966,
- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,
- the function x: f -> x_f is a bijection,
- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),
- for any sequence f, x_{2*f} = 2 * x_f,
- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.
From Antti Karttunen, Dec 29 2017: (Start)
The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..16383
Rémy Sigrist, Colored scatterplot of the first 2^17-1 terms (where the color is function of A087207(n) % 8)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000203 (sigma)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)

Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.

Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).

a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)

a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.

A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 15, 16, 31, 12, 34, 16, 25, 20, 41, 16, 39, 24, 34, 24, 53, 16, 56, 32, 42, 32, 49, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 63, 48, 101, 36, 79, 48, 72, 56, 112, 32, 116, 60, 69, 64, 98, 40, 130, 64, 90, 48, 137, 48, 142, 72, 81, 72, 121, 48, 152, 64

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - Antti Karttunen & M. F. Hasler, Mar 10 2018

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

Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

a(n) = Sum_{d|n} A005187(d)*A008683(n/d).
a(n) = n + A297114(n).
From Antti Karttunen, Mar 11 2018: (Start)
Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]
a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).
a(n) = A005187(n) - A300244(n).
a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).
(End)

A297106 Xor-Moebius transform of A156552.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 8, 4, 4, 12, 16, 12, 32, 24, 12, 8, 64, 12, 128, 24, 24, 48, 256, 24, 8, 96, 8, 48, 512, 20, 1024, 16, 48, 192, 24, 24, 2048, 384, 96, 48, 4096, 40, 8192, 96, 24, 768, 16384, 48, 16, 24, 192, 192, 32768, 24, 48, 96, 384, 1536, 65536, 40, 131072, 3072, 48, 32, 96, 80, 262144, 384, 768, 40, 524288, 48

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A156552(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of the Xor-Moebius transform.
The ordinary Möbius transform of A156552 is given in A297112.
It seems that A091629 gives the fixed points of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A064989, A091629, A156552, A295901, A297108, A297112.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297106(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A156552(d)))); (v); } \\ after code in A295901.

A297109 If n is prime(k)^e, e >= 1, then a(n) = k, otherwise 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 0, 4, 1, 2, 0, 5, 0, 6, 0, 0, 1, 7, 0, 8, 0, 0, 0, 9, 0, 3, 0, 2, 0, 10, 0, 11, 1, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 4, 0, 0, 0, 16, 0, 0, 0, 0, 0, 17, 0, 18, 0, 0, 1, 0, 0, 19, 0, 0, 0, 20, 0, 21, 0, 0, 0, 0, 0, 22, 0, 2, 0, 23, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 26, 0, 27, 0, 0

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000720, A000961, A001221, A001511, A055396, A100995, A297108.

A297109(n) = if(1==omega(n),primepi(factor(n)[1,1]),0);

If A001221(n) = 1 [when n is in A000961], then a(n) = A055396(n) = A001511(A297108(n)), otherwise 0.

A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 16, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 16, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 2, 2, 2, 1, 2

Offset: 1

Antti Karttunen and Peter Munn, Feb 01 2021

nonn

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Sequences used in a definition of this sequence: A014963, A048298, A225546, A267116, A297108, A340676.
Cf. A051903, A051144, A331591, A340673.
Positions of 1's: {1} U A340681, 2's: A005117 \ {1}, of 4's: A062503 \ {1}, of 16's: A113849.
Positions of terms > 1: A340682, of terms > 2: A340674.
Sequences used to express relationship between terms of this sequence: A003961, A331590.

A340675(n) = if(1==n,n,if(issquarefree(n), 2, if(!issquare(n), 1, A340675(sqrtint(n))^2)));

a(n) = A225546(A340673(n)) = A225546(A014963(A225546(n))).
a(n) = 2^A048298(A267116(n)).
If A340673(n) = 1, then a(n) = 1, otherwise a(n) = 2^A297108(A340673(n)).
If A340676(n) = 0, then a(n) = 1, otherwise a(n) = 2^(2^(A340676(n)-1)).
If n = s^(2^k), s squarefree >= 2, k >= 0, then a(n) = 2^(2^k), otherwise a(n) = 1.
For n, k > 1, if a(n) = a(k) then a(A331590(n, k)) = a(n), otherwise a(A331590(n, k)) = 1.
a(n^2) = a(n)^2.
a(A003961(n)) = a(n).
a(A051144(n)) = 1.
a(n) = 1 if and only if A331591(n) <> 1, otherwise a(n) = 2^A051903(n).

A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414).

Original entry on oeis.org

0, 2, 3, 2, 5, 0, 7, 2, 3, 0, 11, 0, 13, 0, 0, 2, 17, 0, 19, 0, 0, 0, 23, 0, 5, 0, 3, 0, 29, 0, 31, 2, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 7, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 2, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 3, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Jun 02 2006

nonn

Same as A014963, except this function is zero when n is not a prime power, whereas A014963 is one.

Moreover, this sequence, A014963, A297108 and A297109 partition the natural numbers to identical equivalence classes: For all i, j >= 1, a(i) = a(j) <=> A014963(i) = A014963(j) <=> A297108(i) = A297108(j) <=> A297109(i) = A297109(j). - Antti Karttunen, Feb 01 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factor.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A000040, A001414, A007947, A014963, A010051, A010055, A061397, A070939, A140508 (Möbius transform of this sequence), A297108, A297109.

a120007 1 = 0
a120007 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 0
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 19 2011

Table[If[Length@ # == 1, #[[1, 1]], 0] &@ FactorInteger@ n, {n, 96}] /. 1 -> 0 (* Michael De Vlieger, Jun 19 2016 *)
Table[If[PrimePowerQ[n],FactorInteger[n][[1,1]],0],{n,100}] (* Harvey P. Dale, Jan 25 2020 *)

A120007(n) = { my(v); if(isprimepower(n, &v), v, 0); }; \\ Antti Karttunen, Jan 31 2021

If n is a prime power p^k, k>0, a(n) = p; otherwise a(n) = 0.
Dirichlet g.f. sum_{p prime} p/(p^s-1) = sum_{k>0} primezeta(ks-1).
a(n) = A010055(n) * A007947(n). - Reinhard Zumkeller, Mar 26 2010
a(n) = A061397(A007947(n)). - Reinhard Zumkeller, Sep 19 2011, corrected by Antti Karttunen, Jan 31 2021
a(n) = Sum_{k=2..n} k*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - Anthony Browne, Jun 17 2016
If A297109(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A297109(n)). - Antti Karttunen, Feb 01 2021

cp's OEIS Frontend

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

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A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

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A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

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A297106 Xor-Moebius transform of A156552.

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A297109 If n is prime(k)^e, e >= 1, then a(n) = k, otherwise 0.

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A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

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