A276379 Write a "1" for each distinct prime divisor p of n in the (pi(p) - 1)-th place, ignoring multiplicity.

0, 1, 10, 1, 100, 11, 1000, 1, 10, 101, 10000, 11, 100000, 1001, 110, 1, 1000000, 11, 10000000, 101, 1010, 10001, 100000000, 11, 100, 100001, 10, 1001, 1000000000, 111, 10000000000, 1, 10010, 1000001, 1100, 11, 100000000000, 10000001, 100010, 101, 1000000000000, 1011, 10000000000000, 10001, 110

Offset: 1

Author

Michael De Vlieger, Sep 02 2016

Keywords

• easy
• nonn
• base

Comments

a(n) notes the distinct prime divisors p of n by writing "1" in the (pi(n)-1)-th place. Zeros hold the places of primes q less than the greatest prime divisor p that do not divide n. Thus a(n) consists of 1's and 0's like a binary number where each bit value, instead of representing 2^k, represents prime(k + 1).

a(n) = A054841(n) with all nonzero digits converted to 1's.

a(n) = a(A007947(n)), that is, a number n shares a value of a(n) with the largest squarefree divisor A007947(n). Thus a(18) = a(6) = 11.

a(p) = 1 in the leftmost place followed by (pi(p)-1) zeros.

This function is akin to A054841(n) except we don't note the multiplicity e of p in n, rather merely note "1" if e > 0.

Unlike A054841(1024) = 10, there are no overflows in a(n) into the next place that encodes prime(p+1) due to "carry". 1024 = 2^10, thus a(1024) = a(2^e) = 1, with e >= 1 = 1.

Examples

			a(1) = 0 since 1 is the empty product. a(0) is undefined.
a(6) = a(12) = 11, since 6 and 12 are products of the 1st and 2nd primes (i.e., 2 and 3). Thus we write 1's in the corresponding places. Any number n that is the product only of powers e >= 1 of 2 and 3 (e.g., 24, 96, 144, etc.) has a(n) = 11.
a(42) = 1011, since the prime divisors of 42 are 2, 3 and 7. Any number n that is the product only of powers e >= 1 of all of 2, 3 and 7 has a(n) = 1011.
a(70) = 1101, since its prime divisors are 2, 5 and 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A027748, A054841 (write multiplicity instead of 1 in the (pi(p)-1)th place), A079067 (reverse 0's and 1's in a(n) and convert to decimal), A087207 (a(n) interpreted as a binary number), A273258 (a(n) reversed and converted to decimal).

Sequence A087207 shown in base-2.

Programs

Maple

a:= n-> add(10^numtheory[pi](i[1]), i=ifactors(n)[2])/10:
seq(a(n), n=1..53);  # Alois P. Heinz, Feb 10 2020

Mathematica

f[n_] := If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> 1 &, k]]@ FactorInteger@ n]; Table[FromDigits@ Reverse@ f@ n, {n, 45}] (* or *)
FromDigits[IntegerDigits[#, 2]] & /@ Table[Floor@ Total[2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)], {n, 45}] (* latter program after Jean-François Alcover at A087207 *)

Formula

a(n) = A054841(A007947(n)) = A007088(A087207(n)). - Antti Karttunen, Jun 18 2017

G.f.: Sum_{k>=1} 10^(k-1) * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 10 2020