A003965 -id:A003965 - cp's OEIS frontend

A324905 a(n) = A007895(A003965(n)).

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

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17711
Index entries for sequences computed from indices in prime factorization

Cf. A003965, A007895, A300837, A324888, A324907.

A003965(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = fibonacci(2+primepi(f[i, 1]))); factorback(f); };
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A324905(n) = A007895(A003965(n));

a(n) = A007895(A003965(n)).

A003980 Möbius transform of A003965.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 4, 6, 4, 12, 4, 20, 7, 8, 8, 33, 6, 54, 8, 14, 12, 88, 8, 20, 20, 18, 14, 143, 8, 232, 16, 24, 33, 28, 12, 376, 54, 40, 16, 609, 14, 986, 24, 24, 88, 1596, 16, 56, 20, 66, 40, 2583, 18, 48, 28, 108, 143, 4180, 16, 6764, 232, 42, 32, 80, 24, 10945, 66

Offset: 1

Marc LeBrun

N. J. A. Sloane, Transforms.

Cf. A000045, A000720, A003965.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; q = Fibonacci[PrimePi[p] + 2]; (q-1) q^(e-1), {pe, FactorInteger[n]}]];
Array[a, 100] (* Jean-François Alcover, Sep 29 2020 *)

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], q); prod(i = 1, #p, q = fibonacci(primepi(p[i])+2); (q-1)*q^(e[i]-1));} \\ Amiram Eldar, Sep 14 2023

Multiplicative with a(p^e) = (q-1)q^(e-1) were q = Fibonacci(pi(p)+2) = A000045(A000720(p)+2). - David W. Wilson, Sep 01 2001

Sum_{n>=1} 1/a(n) = Product_{k>=3} (1 + Fibonacci(k)/(Fibonacci(k)-1)^2) = 9.955734312016908009501... . - Amiram Eldar, Sep 14 2023

More terms from David W. Wilson, Aug 29 2001

A003981 Inverse Möbius transform of A003965.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 9, 15, 13, 18, 14, 28, 22, 27, 24, 31, 35, 39, 56, 42, 36, 42, 90, 60, 31, 66, 40, 63, 145, 72, 234, 63, 56, 105, 54, 91, 378, 168, 88, 90, 611, 108, 988, 98, 78, 270, 1598, 124, 73, 93, 140, 154, 2585, 120, 84, 135, 224, 435, 4182, 168, 6766, 702

Offset: 1

Marc LeBrun

N. J. A. Sloane, Transforms.

Cf. A000045, A000720, A003965.

f[p_, e_] := Module[{q = Fibonacci[PrimePi[p] + 2]}, (q^(e+1)-1)/(q-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = Fibonacci(pi(p)+1) = A000045(A000720(p)+2). - David W. Wilson, Sep 01 2001

More terms from David W. Wilson, Aug 29 2001

A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 2, 3, 8, 5, 12, 8, 24, 18, 20, 13, 36, 21, 30, 30, 64, 34, 54, 55, 60, 45, 48, 89, 96, 50, 68, 81, 88, 144, 90, 233, 160, 72, 102, 75, 144, 377, 148, 102, 160, 610, 132, 987, 140, 135, 224, 1597, 240, 112, 150, 153, 188, 2584, 216, 120, 232, 222, 346, 4181, 240, 6765, 528, 198, 384, 170, 210, 10946, 272, 336, 220, 17711, 360

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Cf. A000040, A000045, A000720, A003965, A007895, A328848.
Cf. also A003415, A258851, A328768, A328769, A328845 for other arithmetic derivatives, and also A371192 for another PrimePi-Fibonacci variant.
Cf. A374035 [= gcd(a(n), A328845(n))], A374048 (antiparity of this sequence), A374049 (indices of even terms), A374050 (of odd terms).

A328846(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(2+primepi(f[i,1]))/f[i, 1]));

a(n) = n * Sum e_j * A000045(2+A000720(p_j))/p_j for n = Product p_j^e_j.
a(A000040(n)) = A000045(2+n).
A007895(a(n)) = A328848(n).

A248692 Fully multiplicative with a(prime(i)) = 2^i; If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime A000040(k) and c_k >= 0 then a(n) = Product_{k >= 1} 2^(k*c_k).

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 16, 8, 16, 16, 32, 16, 64, 32, 32, 16, 128, 32, 256, 32, 64, 64, 512, 32, 64, 128, 64, 64, 1024, 64, 2048, 32, 128, 256, 128, 64, 4096, 512, 256, 64, 8192, 128, 16384, 128, 128, 1024, 32768, 64, 256, 128, 512, 256, 65536, 128, 256, 128, 1024, 2048, 131072, 128, 262144, 4096, 256, 64

Offset: 1

Antti Karttunen, Oct 11 2014

Equally, if n = p_i * p_j * ... * p_k, where p_i, p_j, ..., p_k are the primes A000040(i), A000040(j), ..., A000040(k) in the prime factorization of n (indices i, j, ..., k not necessarily distinct), then a(n) = 2^i * 2^j * 2^k.
a(1) = 1 (empty product).
Fully multiplicative with a(prime(i)) = 2^i.

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

Cf. A000040, A000079, A003961, A003965, A048675, A056239, A061142, A065446, A122111, A297109.

a:= n-> mul((2^numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..64);  # Alois P. Heinz, Jan 14 2021

a[n_] := Product[{p, e} = pe; (2^PrimePi[p])^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 03 2022 *)

A248692(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = 2^primepi(f[i,1])); factorback(f)); \\ Antti Karttunen, Feb 01 2021

a(n) = 2^A056239(n) = A000079(A056239(n)).
Other identities. For all n >= 1:
a(A122111(n)) = a(n).
a(A000040(n)) = A000079(n).
For all n >= 0:
a(A000079(n)) = A000079(n).
a(n) = Product_{d|n} 2^A297109(d). - Antti Karttunen, Feb 01 2021
Sum_{n>=1} 1/a(n) = A065446. - Amiram Eldar, Dec 24 2022

A324900 Fully multiplicative with a(prime(k)) = Lucas(2*(k+1)) for k-th prime p, where Lucas(n) = A000032(n).

Original entry on oeis.org

1, 7, 18, 49, 47, 126, 123, 343, 324, 329, 322, 882, 843, 861, 846, 2401, 2207, 2268, 5778, 2303, 2214, 2254, 15127, 6174, 2209, 5901, 5832, 6027, 39603, 5922, 103682, 16807, 5796, 15449, 5781, 15876, 271443, 40446, 15174, 16121, 710647, 15498, 1860498, 15778, 15228, 105889, 4870847, 43218, 15129, 15463, 39726, 41307, 12752043

Offset: 1

Antti Karttunen, Apr 15 2019

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences computed from indices in prime factorization.

Cf. A000032, A000045, A000720, A003965, A324901.

f[p_, e_] := LucasL[2*(PrimePi[p]+1)]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

A000032(n) = (fibonacci(n+1)+fibonacci(n-1));
A324900(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = A000032(2*(1+primepi(f[i, 1])))); factorback(f); };

Fully multiplicative with a(prime(k)) = A000032(2*(k+1)) = A000045(2k+1) + A000045(2k+3).

Sum_{n>=1} 1/a(n) = 1 / Product_{k>=1} (1 - 1/Lucas(2*k+2)) = 1.278911382005... . - Amiram Eldar, Aug 28 2023

A227324 Result of changing both the prime indices and the exponents in the prime factorization of n: increment odd values, decrement even values.

Original entry on oeis.org

1, 9, 4, 3, 49, 36, 25, 81, 2, 441, 169, 12, 121, 225, 196, 27, 361, 18, 289, 147, 100, 1521, 841, 324, 7, 1089, 16, 75, 529, 1764, 1369, 729, 676, 3249, 1225, 6, 961, 2601, 484, 3969, 1849, 900, 1681, 507, 98, 7569, 2809, 108, 5, 63, 1444, 363, 2209, 144

Offset: 1

Alex Ratushnyak, Jul 07 2013

A self-inverse permutation on the positive integers: a(a(n)) = n.

			n = 2^3 => a(n) = 3^4 = 81.
n = 3^2 => a(n) = 2^1 = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A003958-A003965, A011262, A011264, A045965-A045973, A151799, A151800.

a:= n-> mul(ithprime(i[1])^i[2], i=map(x->map(y->y-1+2*irem(y, 2),
        [numtheory[pi](x[1]), x[2]]), ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 17 2013

a[n_] := If[n == 1, 1, Product[{p, e} = pe; Prime[BitXor[PrimePi[p] - 1, 1] + 1]^(BitXor[e - 1, 1] + 1), {pe, FactorInteger[n]}]];
Array[a, 100] (* Jean-François Alcover, May 31 2019, after Andrew Howroyd *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); prime( bitxor( primepi(p)-1, 1)+1)^(bitxor(e-1, 1)+1))} \\ Andrew Howroyd, Jul 23 2018

primes = [2]*2
primes[1] = 3
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5, 1000000, 6):
  addPrime(n)
  addPrime(n+2)
for n in range(1,99):
  p = 1
  j = n
  i = 0
  while j>1:
    e = 0
    while j % primes[i] == 0:
      j /= primes[i]
      e+=1
    if e:
      e = ((e-1)^1) + 1
      p*= primes[i^1]**e
    i += 1
  print(str(p), end=', ')

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p-1)*(p^6 + q(p) +(p^3-1)*q(p)^2))/(p^7 - p*q(p)^2) = 0.3120270364..., where q(p) = nextprime(p) = A151800(p) if p has an odd index, and q(p) = prevprime(p) = A151799(p) otherwise. - Amiram Eldar, Sep 17 2023

Keyword:mult added by Andrew Howroyd, Jul 23 2018

cp's OEIS Frontend

A324905 a(n) = A007895(A003965(n)).

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A003980 Möbius transform of A003965.

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A003981 Inverse Möbius transform of A003965.

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A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.

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A248692 Fully multiplicative with a(prime(i)) = 2^i; If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime A000040(k) and c_k >= 0 then a(n) = Product_{k >= 1} 2^(k*c_k).

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A324900 Fully multiplicative with a(prime(k)) = Lucas(2*(k+1)) for k-th prime p, where Lucas(n) = A000032(n).

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A227324 Result of changing both the prime indices and the exponents in the prime factorization of n: increment odd values, decrement even values.

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A328846 The second Fibonacci based variant of arithmetic derivative: a(p) = A000045(2+A000720(p)) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0. Also called PrimePi-Fibonacci variant of the arithmetic derivative.