A078349 -id:A078349 - cp's OEIS frontend

A292258 a(1) = 1; for n > 1, a(n) = prime(A101296(n)-1) * a(floor(n/2)).

1, 2, 2, 6, 4, 10, 4, 42, 18, 20, 8, 110, 20, 20, 20, 546, 84, 198, 36, 220, 100, 40, 16, 1870, 330, 100, 140, 220, 40, 380, 40, 12558, 2730, 420, 420, 5742, 396, 180, 180, 3740, 440, 1900, 200, 440, 440, 80, 32, 57970, 5610, 3630, 1650, 1100, 200, 2380, 700, 3740, 1100, 200, 80, 14060, 760, 200, 440, 514878

Offset: 1

Antti Karttunen, Sep 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A000040, A000523, A004526, A007814, A078349, A101296, A292259 (rgs-version of this filter).

With[{nn = 64}, Block[{s = Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}], a}, a[n_] := a[n] = If[n == 1, 1, Prime[s[[n]] - 1]*a[Floor[n/2]]]; Array[a, nn]]] (* Michael De Vlieger, Sep 22 2017 *)

up_to = 8191
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A292258(n) = if(1==n,n,prime(A101296(n)-1) * A292258(n\2));

a(1) = 1; for n > 1, a(n) = A000040(A101296(n)-1) * a(A004526(n)).
Other identities. For all n >= 1:
A001222(a(n)) = A000523(n).
A007814(a(n)) = A078349(n).

A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

Records occur at positions 1, 2, 5, 11, 23, 47, 2879, ... (A292937).

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

Cf. A007814, A078349, A292599, A292937.
Cf. A000040, A005385, A066179, A157358, A157359 (positions of terms that are > k, for k = 0..4).
Cf. A059456 (positions of ones).

A292936 := proc(n)
    for k from 0 do
        if not isprime(floor(n/2^k)) then
            return k;
        end if;
    end do:
end proc:
seq(A292936(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

Table[SelectFirst[Range[0, 10], ! PrimeQ@ Floor[n/(2^#)] &], {n, 105}] (* Michael De Vlieger, Sep 29 2017 *)

A292936(n) = { my(k=0); while(isprime(n), n >>= 1; k++); k; };

(define (A292936 n) (A007814 (1+ (A292599 n))))

a(n) = A007814(1+A292599(n)).
For n >= 1, a(n) <= A078349(n).
For n > 47, a(n) <= A007814(1+n).

A292259 Restricted growth sequence transform of A292258; filter constructed from the prime signatures of the sequence [n, floor(n/2), floor(n/4), ..., 1].

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 4, 6, 7, 8, 9, 10, 8, 8, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 16, 21, 15, 17, 22, 17, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 30, 30, 33, 34, 35, 36, 37, 38, 39, 32, 40, 41, 29, 39, 32, 33, 42, 43, 32, 30, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 65, 66, 62, 67, 61, 62

Offset: 1

Antti Karttunen, Sep 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65535

Cf. A292258.

Cf. A078349 (one of the matching sequences).

up_to = 65535
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A292258(n) = if(1==n,n,prime(A101296(n)-1) * A292258(n\2));
write_to_bfile(1,rgs_transform(vector(up_to,n,A292258(n))),"b292259.txt");

A292598 a(n) is the number of odd primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 1, 2, 2, 2, 0, 1, 0, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 4, 2, 3, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 4, 4, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

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

Cf. A010051, A065091, A078349, A292596.

a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + a(floor(n/2)).
a(n) = A000120(A292596(n)).
For all n >= 1, a(n) <= A078349(n).

A292599 a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 6, 7, 4, 5, 6, 6, 8, 9, 8, 9, 12, 12, 14, 15, 8, 8, 10, 10, 12, 13, 12, 13, 16, 16, 18, 18, 16, 17, 18, 18, 24, 25, 24, 25, 28, 28, 30, 31, 16, 16, 16, 16, 20, 21, 20, 20, 24, 24, 26, 27, 24, 25, 26, 26, 32, 32, 32, 33, 36, 36, 36, 37, 32, 33, 34, 34, 36, 36, 36, 37, 48, 48, 50, 51, 48, 48, 50, 50, 56, 57, 56, 56, 60, 60, 62, 62, 32

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010051, A078349, A292258, A292259, A292936.

Cf. also A292596 (variant for odd primes).

A292599 := proc(n)
    option remember;
    if n = 1 then
        0 ;
    else
        A010051(n) + 2*procname(floor(n/2)) ;
    end if;
end proc:
seq(A292599(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

a[1] = 0; a[n_] := a[n] = Boole[PrimeQ[n]] + 2*a[Floor[n/2]]; Array[a, 96] (* Jean-François Alcover, Sep 29 2017 *)

a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).
Other identities. For all n >= 1:
A000120(a(n)) = A078349(n).
A007814(1+a(n)) = A292936(n).

cp's OEIS Frontend

A292258 a(1) = 1; for n > 1, a(n) = prime(A101296(n)-1) * a(floor(n/2)).

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A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

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A292259 Restricted growth sequence transform of A292258; filter constructed from the prime signatures of the sequence [n, floor(n/2), floor(n/4), ..., 1].

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PARI

A292598 a(n) is the number of odd primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

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Formula

A292599 a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).

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Keywords

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Links

Crossrefs

Programs

Maple

Mathematica

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