A072084 -id:A072084 - cp's OEIS frontend

A072085 a(n) = A072084(A072084(n)).

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

Offset: 1

Reinhard Zumkeller, Jun 14 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072086.

a072085 = a072084 . a072084  -- Reinhard Zumkeller, Feb 10 2013

b[n_] := Times @@ Power @@@ (FactorInteger[n] /. {p_Integer, e_} :> {DigitCount[p, 2, 1], e}); a[n_] := b[b[n]]; Array[a, 100] (* Jean-François Alcover, Feb 09 2018 *)

A072086 Number of steps to reach 1, starting with n and applying the A072084-map repeatedly.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 14 2002

nonn

			n=21: '11'*'111' -> '10'*'11' -> '11'*'1' -> '1'; i.e., 21=3*7 -> 6=2*3 -> 2*1 -> 1*1=1, therefore a(21)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072085.

a072086 n = fst $
   until ((== 1) . snd) (\(i, x) -> (i + 1, a072084 x)) (0, n)
-- Reinhard Zumkeller, Feb 10 2013

A072086 := proc(n) local c,i,j; c :=0; i := n;
while i > 1 do i:=A072084(i); c:=c+1 od; c end:
# Note that this gives A072086(0)=0 if desired
# without any additional case discrimination.
# Peter Luschny, Jan 16 2010

b[1] = 1; b[n_] := Times @@ Power @@@ (FactorInteger[n] /. {p_Integer, e_} :> {DigitCount[p, 2, 1], e}); a[n_] := Length[FixedPointList[b, n]] - 2; Array[a, 100] (* Jean-François Alcover, Feb 09 2018 *)

A072087 Least k such that A072084(k) = n.

Original entry on oeis.org

1, 3, 7, 9, 31, 21, 127, 27, 49, 93, 3583, 63, 8191, 381, 217, 81, 131071, 147, 524287, 279, 889, 10749, 14680063, 189, 961, 24573, 343, 1143, 1073479679, 651, 2147483647, 243, 25081, 393213, 3937, 441, 266287972351, 1572861, 57337, 837

Offset: 1

Reinhard Zumkeller, Jun 14 2002

If p is a Mersenne prime then a(p) = 2^p - 1 (A000120(2^n-1)=n), for other primes p: a(p) > 2^p - 1.

Amiram Eldar, Table of n, a(n) for n = 1..3322
Index to divisibility sequences.

Cf. A000120, A000043, A000668, A027746, A061712, A072084.

a072087 1 = 1
a072087 n = product $ map a061712 $ a027746_row n
-- Reinhard Zumkeller, Feb 10 2013

s[n_] := s[n] = Module[{p = 2}, While[DigitCount[p, 2, 1] != n, p = NextPrime[p]]; p]; f[p_, e_] := s[p]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 22] (* Amiram Eldar, Nov 02 2023 *)

Completely multiplicative with a(p) = A061712(p). - David W. Wilson, Aug 03 2005

Sum_{n>=1} 1/a(n) = Product_{p prime} 1/(1 - 1/A061712(p)) = 1.82343415954263449963... . - Amiram Eldar, Nov 02 2023

More terms from David W. Wilson, Aug 03 2005

A014499 Number of 1's in binary representation of n-th prime.

Original entry on oeis.org

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

Offset: 1

Ingemar Assarsjo (ingemar(AT)binomen.se)

a(n) is the rank of prime(n) in the base-2 dominance order on the natural numbers. - Tom Edgar, Mar 25 2014

			From _M. F. Hasler_, Mar 03 2023: (Start)
a(n) = 1 only for p(n = 1) = 2, the only prime equal to a power of 2.
a(n) = 2 for n in A159611 = A000720(A019434) = {2, 3, 7, 55, 6543} (probably complete), the Fermat primes F[k] = 2^2^k + 1 with k = 0, 1, 2, 3, 4. (On the graph one can distinctly see a(6543) = 2 corresponding to F[4] = 65537.)
a(n) = 3 for n in A000720(A081091) = (4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, ...). (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016.
Index entries for sequences related to binary expansion of n

Cf. A035103, A035100, A004676, A090455.
Cf. A027697, A027699
Cf. A180024. - Reinhard Zumkeller, Aug 08 2010
Cf. A072084.
Cf. A159611 (indices of 2s), A000720(A081091) (indices of 3s). - M. F. Hasler, Mar 03 2023

a014499 = a000120 . a000040  -- Reinhard Zumkeller, Feb 10 2013

[&+Intseq(NthPrime(n), 2): n in [1..100] ]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 2], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

A014499(n)=hammingweight(prime(n)) \\ M. F. Hasler, Nov 20 2009, updated Mar 03 2023

from sympy import prime
def A014499(n): return prime(n).bit_count() # Chai Wah Wu, Mar 22 2023

[sum(i.digits(base=2)) for i in primes_first_n(200)] # Tom Edgar, Mar 25 2014

a(n) = A000120(A000040(n)).
a(A049084(A061712(n))) = n. - Reinhard Zumkeller, Feb 10 2013
a(n) = [x^prime(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A349494 a(n) is the maximum of A000120(k)*A000120(n/k) for divisors k of n.

Original entry on oeis.org

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

Offset: 1

Robert Israel, Sep 03 2023

First differs from A072084 at n = 27.

			a(45) = 8 because 45 = 3 * 15 with A072084(3) * A072084(15) = 2 * 4 = 8, and the other factorizations 1 * 45 and 5 * 9 have A072084(k) * A072084(45/k) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A072084.

g:= proc(n) convert(convert(n,base,2),`+`) end proc:
f:= proc(n) local t,r;
      max(seq(g(t)*g(n/t), t = numtheory:-divisors(n)))
    end proc:
map(f, [$1..100]);

a[n_] := Max[(d = DigitCount[Divisors[n], 2, 1]) * Reverse[d]]; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

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

A071785 In prime factorization of n replace each prime with the sum of its decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 12, 4, 14, 15, 16, 8, 18, 10, 20, 21, 4, 5, 24, 25, 8, 27, 28, 11, 30, 4, 32, 6, 16, 35, 36, 10, 20, 12, 40, 5, 42, 7, 8, 45, 10, 11, 48, 49, 50, 24, 16, 8, 54, 10, 56, 30, 22, 14, 60, 7, 8, 63, 64, 20, 12, 13, 32, 15, 70, 8, 72

Offset: 1

Reinhard Zumkeller, Jun 06 2002

			a(143) = a(11*13) = a(11)*a(13) = (1+1)*(1+3) = 2*4 = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002473 (fixed points), A007953, A072084 (binary variant).

a:= n-> mul(add(j, j=convert(i[1], base, 10))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..72);  # Alois P. Heinz, Oct 22 2021

a[n_] := Product[{p, e} = pe; Total[IntegerDigits[p]]^e, {pe, FactorInteger[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 22 2021 *)

a(n, base=10) = my (f=factor(n)); prod(i=1, #f~, sumdigits(f[i,1], base)^f[i,2]) \\ Rémy Sigrist, Feb 19 2019

Multiplicative with a(p) = A007953(p), p prime.

a(n) = n iff n is 7-smooth (A002473).

A324391 Fully multiplicative with a(p^e) = A070939(p)^e, where A070939(p) gives the length of the binary representation of p.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 4, 8, 4, 6, 6, 16, 5, 8, 5, 12, 6, 8, 5, 16, 9, 8, 8, 12, 5, 12, 5, 32, 8, 10, 9, 16, 6, 10, 8, 24, 6, 12, 6, 16, 12, 10, 6, 32, 9, 18, 10, 16, 6, 16, 12, 24, 10, 10, 6, 24, 6, 10, 12, 64, 12, 16, 7, 20, 10, 18, 7, 32, 7, 12, 18, 20, 12, 16, 7, 48, 16, 12, 7, 24, 15, 12, 10, 32, 7, 24, 12, 20, 10, 12

Offset: 1

Antti Karttunen, Mar 05 2019

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

Cf. A000043, A000668, A070939, A072084.

A070939(n) = if(!n,1,#binary(n));
A324391(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = A070939(f[i, 1])); factorback(f); };

For all n >= 1, a(A000668(n)) = A000043(n).

cp's OEIS Frontend

A072085 a(n) = A072084(A072084(n)).

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A072086 Number of steps to reach 1, starting with n and applying the A072084-map repeatedly.

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A072087 Least k such that A072084(k) = n.

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A014499 Number of 1's in binary representation of n-th prime.

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A349494 a(n) is the maximum of A000120(k)*A000120(n/k) for divisors k of n.

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A071785 In prime factorization of n replace each prime with the sum of its decimal digits.

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A324391 Fully multiplicative with a(p^e) = A070939(p)^e, where A070939(p) gives the length of the binary representation of p.

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