A175851 -id:A175851 - cp's OEIS frontend

A096500 Let f(n) = smallest prime > n; a(n) = f(n+1) - f(n).

1, 2, 0, 2, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0

Offset: 1

Labos Elemer, Jul 09 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000720, A001223, A010051, A096501, A175851.
First differences of A151800.
Cf. also A109578.

seq(nextprime(n+1)-nextprime(n),n=1..250); # Muniru A Asiru, Jan 03 2019

Mathematica
```
<Michael De Vlieger, Jan 03 2019 *)
```

A151800(n) = nextprime(1+n);
A096500(n) = (A151800(1+n)-A151800(n)); \\ Antti Karttunen, Jan 03 2019

From Antti Karttunen, Jan 03 2019: (Start)
a(n) = A151800(n+1) - A151800(n).
a(n) = A010051(1+n) * A001223(A000720(1+n)).
(End)

A096501 Difference between primes preceding n+1 and n.

Original entry on oeis.org

0, 4, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 2, 0, 0

Offset: 1

Labos Elemer, Jul 09 2004

nonn

Values a(1) = 0 and a(2) = 4 are based on convention in Mathematica-language that PreviousPrime(1) = PreviousPrime(2) = -2. - Antti Karttunen, Jan 03 2019

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000720, A001223, A010051, A096500, A136548, A151799, A175851, A007917.

0,4,seq(prevprime(n+1)-prevprime(n),n=3..150); # Muniru A Asiru, Jan 03 2019

Table[NextPrime[n+1, -1] - NextPrime[n, -1], {n, 1, 256}]
Differences[NextPrime[Range[110],-1]] (* Harvey P. Dale, Jul 09 2017 *)

A136548(n) = if(n<3,1,precprime(n-1));
A096501(n) = if(2==n,4,A136548(1+n)-A136548(n)); \\ Antti Karttunen, Jan 03 2019

For n > 2, a(n) = A010051(n) * A001223(A000720(n)-1) = A136548(1+n)-A136548(n). - Antti Karttunen, Jan 03 2019

a(n) = A007917(n) - A007917(n-1), for n > 2. - Ridouane Oudra, Oct 05 2024

A301295 Smallest distance from n to a prime power (as defined in A000961).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 24 2018

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

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2).
Identical to A080732 except here a(1)=0.
Cf. also A051699, A175851.

A301295(n) = if(1==n,0,my(k=0);while(!isprimepower(n+k) && !isprimepower(n-k), k++); (k)); \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A305300 Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

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

Cf. A206074 (gives the positions of other 1's after the initial one).
Cf. A257000, A305429, A305430, A305437, A305438.
Cf. also A175851.

binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];
ip[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];
A305430[n_] := Module[{k = n + 1}, While[ip[k] == 0, k++]; k];
b[_] = 0;
a[n_] := a[n] = With[{t = A305430[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3,1, my(k=n-1); while(k>1 && !A257000(k),k--); (k));
A305300(n) = if((1==n)||(1==A257000(n)),1,1+(n-A305429(n)));

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A305430(n) = { my(k=1+n); while(!A257000(k),k++); (k); };
v305300 = ordinal_transform(vector(up_to,n,A305430(n)));
A305300(n) = v305300[n];

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], a(n) = 1, otherwise a(n) = 1 + n - A305429(n).

cp's OEIS Frontend

A096500 Let f(n) = smallest prime > n; a(n) = f(n+1) - f(n).

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A096501 Difference between primes preceding n+1 and n.

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A301295 Smallest distance from n to a prime power (as defined in A000961).

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PARI

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A305300 Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.

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Author

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Crossrefs

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Mathematica

PARI

PARI

Formula