A065157 -id:A065157 - cp's OEIS frontend

A345874 a(n) is the number of distinct pairs of positive integers (u, v) such that A065157(u, v) = n.

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

Offset: 1

Rémy Sigrist, Jun 27 2021

If A065157(u, v) = n then u is a divisor of n.

			For n = 10:
- the table A065157(u, v) begins (with terms > 10 replaced by dots):
  u\v|   1   2   3  4  5  6  7  8  9  10
  ---+----------------------------------
    1|   1   2   3  4  5  6  7  8  9  10
    2|   2   4  10  8  .  .  .  .  .  .
    3|   3   6  .   .  .  .  .  .  .  .
    4|   4   8  .   .  .  .  .  .  .  .
    5|   5  10  .   .  .  .  .  .  .  .
    6|   6  .   .   .  .  .  .  .  .  .
    7|   7  .   .   .  .  .  .  .  .  .
    8|   8  .   .   .  .  .  .  .  .  .
    9|   9  .   .   .  .  .  .  .  .  .
   10|  10  .   .   .  .  .  .  .  .  .
- the value 10 appears 4 times,
- so a(10) = 4.

Cf. A000005, A065157.

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a(n) <= A000005(n).
a(n) >= 2 for any n > 1 (as A065157(1, n) = A065157(n, 1) = n).
a(2^k) = k + 1 for any k >= 0.
a(2^k - 1) = A000005(k) for any k > 0.

A065159 Binary string self-substitutions: a(n) is obtained by substituting the binary expansion of n for each 1-bit in the binary expansion of n.

Original entry on oeis.org

0, 1, 4, 15, 16, 85, 108, 511, 64, 585, 660, 5819, 816, 7085, 7644, 65535, 256, 4369, 4644, 78451, 5200, 87381, 91564, 1531639, 6336, 105625, 109876, 1825659, 118384, 1961821, 2029500, 33554431, 1024, 33825, 34884, 1149155, 37008, 1217189, 1250124, 41056743

Offset: 0

Marc LeBrun, Oct 18 2001

			a(5): 5 = 101 -> (101)0(101) = 1010101 = 85.

Paul Tek, Table of n, a(n) for n = 0..10000

Cf. A007088, A065157, A065158, A065160.

bss[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[Flatten[idn2/.{1-> idn2}],2]]; Array[bss,40,0] (* Harvey P. Dale, Aug 15 2017 *)

def a(n): b = bin(n)[2:]; return int(b.replace("1", b), 2)
print([a(n) for n in range(40)]) # Michael S. Branicky, Aug 05 2022

a(0) = 0. a(2^n) = 4^n. a(4n+2) = (4n+2)*(1+a(4n+1)/(4n+1)).
a(n) = A065157(n,n) = A065158(n,n)*n = A065160(n)*n.
a(n) =z(n, n) with z(u, v) = if u=0 then 0 else if u mod 2 = 0 then z(u/2, v)*2 else z([u/2], v)*A062383(v)+v. - Reinhard Zumkeller, Feb 15 2004

Name clarified by Michael S. Branicky, Aug 05 2022

A065158 Table of reduced binary string substitutions: a(i,j) is obtained by substituting i for each 1-bit in j, then dividing by i.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 4, 5, 1, 2, 9, 4, 9, 6, 1, 2, 9, 4, 9, 10, 7, 1, 2, 9, 4, 17, 10, 21, 8, 1, 2, 9, 4, 17, 18, 21, 8, 9, 1, 2, 17, 4, 17, 18, 73, 8, 17, 10, 1, 2, 17, 4, 17, 18, 73, 8, 17, 18, 11, 1, 2, 17, 4, 33, 18, 73, 8, 33, 18, 37, 12, 1, 2, 17, 4, 33, 34, 73, 8, 33

Offset: 1

Marc LeBrun, Oct 18 2001

Table origin is a(1,1). a(1,n)=n. a(n,1)=1. By convention a(0,n)=a(n,0)=0. a(i,j)=A065157(i,j)/i. a(n,n)=A065160(n)=A065159(n)/n.

			a(3,5): 5=101->(3)0(3)=(11)0(11)=11011=27; 27/3=9. a(5,3): 3=11->(5)(5)=(101)(101)=101101=45; 45/5=9

A065157, A065159, A065160.

A065160 Reduced binary string self-substitutions: a(n) is obtained by substituting n for each 1-bit in the binary expansion of n, then dividing by n.

Original entry on oeis.org

1, 2, 5, 4, 17, 18, 73, 8, 65, 66, 529, 68, 545, 546, 4369, 16, 257, 258, 4129, 260, 4161, 4162, 66593, 264, 4225, 4226, 67617, 4228, 67649, 67650, 1082401, 32, 1025, 1026, 32833, 1028, 32897, 32898, 1052737, 1032, 33025, 33026, 1056833, 33028, 1056897

Offset: 1

Marc LeBrun, Oct 18 2001

By convention a(0)=0. a(2^n)=2^n. a(4n+2)=1+a(4n+1). a(n)=A065158(n,n)=A065157(n,n)/n=A065159(n)/n.

			a(5): 5=101->(101)0(101)=1010101=85; 85/5=17.

Cf. A065157, A065158. Equals A065159(n)/n.

a(2^n)=2^n. a(4n+2)=1+a(4n+1). a(n)=A065158(n, n)=A065157(n, n)/n=A065159(n)/n.

a(n)=z(n, A062383(n)) with z(u, v) = if u=0 then 0 else if u mod 2 = 0 then z(u/2, v)*2 else z([u/2], v)*v+1. - Reinhard Zumkeller, Feb 15 2004

cp's OEIS Frontend

A345874 a(n) is the number of distinct pairs of positive integers (u, v) such that A065157(u, v) = n.

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A065159 Binary string self-substitutions: a(n) is obtained by substituting the binary expansion of n for each 1-bit in the binary expansion of n.

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A065158 Table of reduced binary string substitutions: a(i,j) is obtained by substituting i for each 1-bit in j, then dividing by i.

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A065160 Reduced binary string self-substitutions: a(n) is obtained by substituting n for each 1-bit in the binary expansion of n, then dividing by n.

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