cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Cino Hilliard

Cino Hilliard's wiki page.

Cino Hilliard has authored 1372 sequences. Here are the ten most recent ones:

A157480 a(n) = least prime p such that p + prime(n) is a square.

Original entry on oeis.org

2, 13, 11, 2, 5, 3, 19, 17, 2, 7, 5, 107, 23, 101, 2, 11, 5, 3, 257, 29, 71, 2, 17, 11, 3, 43, 41, 37, 467, 31, 17, 13, 7, 5, 47, 173, 167, 1601, 2, 23, 17, 719, 5, 3, 59, 701, 113, 2, 29, 347, 23, 17, 83, 5, 67, 61, 131, 53, 47, 43, 41, 31, 17, 13, 11, 7, 569, 239, 53, 227, 47, 2
Offset: 1

Views

Author

Cino Hilliard, Mar 01 2009

Keywords

Examples

			The difference between prime 3 and the square 16 is 13 which is prime and in the sequence.

Links

Programs

  • Mathematica
    Table[p=Prime[n];b=Ceiling[Sqrt[p]];While[!PrimeQ[x=b^2-p],b++];x,{n,72}]
  • PARI
    g(n)= c=0;forprime(x=2,n,for(k=1,n^2,if(issquare(x+k)&&isprime(k),
print1(k",");c++;break)));c

Extensions

Better definition and Mma program from Zak Seidov, Mar 14 2013

A161735 Primes that are the difference between a fourth power and a positive cube.

Original entry on oeis.org

17, 73, 113, 131, 229, 409, 443, 617, 673, 739, 953, 1153, 1171, 1889, 2393, 5087, 6217, 6553, 8669, 9433, 9973, 11321, 11897, 13877, 14633, 14737, 15823, 17377, 18539, 19081, 19441, 20393, 20611, 21841, 25469, 26249, 26833, 28649, 29599
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2009

Keywords

Comments

There are primes like p = 20393, 3905513, 5177033, 28398833, or 10877895569 which have more than one representation p=x^4-y^3, with x,y>=1.
My guess is that the number of duplicates is infinite.

Programs

  • PARI
    difffourthcube(n) =
{
local(a,c=0,c2=0,j,k,y);
a=vector(floor(n^2/log(n^2)));
for(j=1,n,
for(k=1,n,
y=j^4-k^3;
if(ispseudoprime(y),
c++;
\\ print(j","k","y);
a[c]=y;
);
);
);
a=vecsort(a);
for(j=2,c,
if(a[j]!=a[j-1]&&a[j]!=0,
c2++;
print1(a[j]",");
if(c2>100,break);
);
);
}

Formula

If x^4-y^3 is prime for integers x >=1, y>=1, list it.

A161747 Primes of the form x^5-y^4, where x,y >= 1.

Original entry on oeis.org

31, 227, 1051, 3109, 7151, 15511, 18127, 30367, 32143, 32687, 144719, 151051, 165311, 186343, 234191, 302399, 369997, 371281, 374239, 407503, 454303, 509263, 531263, 537743, 759359, 1053007, 1088081, 1182287, 1185601, 1354321, 1381441
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2009

Keywords

Comments

If a prime has multiple representations of the format, it is entered only once.

Examples

			2^5 - 1^4 = 31.

Programs

  • PARI
    diffpowers(n,m) =
{
local(a,c=0,c2=0,j,k,y);
a=vector(floor(n^2/log(n^2)));
for(j=1,n,
for(k=1,n,
y=j^m-k^(m-1);
if(ispseudoprime(y),
c++;
\\ print(j","k","y);
a[c]=y;
);
);
);
a=vecsort(a);
for(j=2,length(a),
if(a[j]!=a[j-1]&&a[j]!=0,
c2++;
print1(a[j]",");
if(c2>100,break);
);
);
}

Formula

If x^5-y^4 is prime for integers x,y list without duplicates.

A161748 Smallest prime in the set of primes of the form x^n - y^(n-1), 1<=x, 1<=y.

Original entry on oeis.org

2, 2, 2, 17, 31, 971, 127, 856073, 19427, 58537, 176123, 529393, 8191, 128467258961, 977123207545039, 43013953, 131071, 3814697134553, 524287, 79792266297087713
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2009

Keywords

Comments

The function x^n -y^(n-1) has some prime values if x and y are covering the first quadrant. The smallest of these primes defines a(n).

Examples

			3^1 - 1^0 = 2, 2^2 - 2 = 2, 3^3 - 5^2 = 2, so 2,2,2 are the first 3 entries.

Programs

  • PARI
    diffpowers(n,m) =
{
local(a,c=0,c2=0,j,k,y);
a=vector(floor(n^2/log(n^2)));
for(j=1,n,
for(k=1,n,
y=j^m-k^(m-1);
if(ispseudoprime(y), c++; a[c]=y;);
);
);
a=vecsort(a);
for(j=2,length(a),
if(a[j]!=a[j-1]&&a[j]!=0, c2++; print1(a[j]","); if(c2>100,break););
);
}

Extensions

Definition reworded - R. J. Mathar, Aug 30 2010

A161749 Smallest prime of the form x^(2n+1) - y^(2n-1), x,y >= 1.

Original entry on oeis.org

3, 5, 127, 3299, 1967249047, 8191, 30450469261, 131071, 524287, 476562280296181, 70358283824461
Offset: 1

Views

Author

Cino Hilliard, Jun 17 2009

Keywords

Comments

For even n > 4 = 2m, x^(2m) - y^(2m-2) = (x^m)^2 - y^((m-1))^2 is divisible by x^m - y^(m-1) which is not prime. This accounts for the omission of even powers in the definition.

Extensions

Incorrect program removed; definition confined to odd powers; added terms a(8)..a(11). - R. J. Mathar, Feb 27 2012

A162250 Values of the form prime(prime(i)) with a prime digital sum.

Original entry on oeis.org

3, 5, 11, 41, 67, 83, 157, 179, 191, 241, 283, 331, 353, 401, 461, 599, 739, 773, 797, 919, 991, 1031, 1217, 1297, 1433, 1471, 1499, 1523, 1723, 1741, 1787, 2027, 2063, 2081, 2221, 2269, 2351, 2609, 2647, 2683, 2719, 2803, 3019, 3109, 3169, 3259, 3299
Offset: 1

Views

Author

Cino Hilliard, Jun 28 2009

Keywords

Examples

			Prime(prime(6)) = 41. 4+1=5, prime. So 41 is in the sequence.

Crossrefs

Intersection of A006450 and A028834.

Programs

  • PARI
    sodip(n) = {
local(s=0,a,x,y,j,p);
for(x=1,n, p=prime(prime(x)); a=eval(Vec(Str(p))); y=sum(j=1,length(a),a[j]); if(isprime(y),print1(p","));)
}

Formula

{A006450(i) : A007953(A006450(i)) in A000040}. [From R. J. Mathar, Aug 03 2009]

Extensions

Definition rephrased by R. J. Mathar, Sep 11 2009

A162421 Numbers whose prime factors all have the same number of digits.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 45, 47, 48, 49, 50, 53, 54, 56, 59, 60, 61, 63, 64, 67, 70, 71, 72, 73, 75, 79, 80, 81, 83, 84, 89, 90, 96, 97, 98, 100, 101, 103, 105, 107
Offset: 1

Views

Author

Cino Hilliard, Jul 03 2009

Keywords

Comments

The prime numbers A000040 are a subset of this sequence.
A number k>1 is in this sequence, if the count of base-10 digits of all entries in the k-th row of A027746 (=its prime factors) is the same.

Examples

			16 = 2*2*2*2 and the digital length = 1 for all factors. So 16 is in the sequence. 22=2*11 is not in the sequence because the digital length of 11 is not the same as the digital length of 2.

Crossrefs

Cf. A162422

Programs

  • PARI
    factorsmooth(n) =
{
local(x,a,j,f,ln);
for(x=2,n, f=0; a = ifactor(x); ln=length(Str(a[1])); for(j=2,length(a), if(length(Str(a[j]))!=ln,f=1;break);); if(!f,print1(x","));)
};

Formula

{k >1: A055642(A020639(k)) = A055642(A006530(k)) }. - R. J. Mathar, Sep 16 2009

Extensions

Offset set to 1 - R. J. Mathar, Sep 16 2009

A162422 Numbers with at least 2 different numbers of digits among their prime factors.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 145, 146, 148, 152
Offset: 1

Views

Author

Cino Hilliard, Jul 03 2009

Keywords

Comments

Complement of A162421. There are no prime numbers in this sequence.
These numbers can also be called factor rough numbers.
Basically, the number of digits of A020639(k) and of A006530(k) must differ to admit k into the sequence.

Examples

			1111 = 11*101 has factors with different digital lengths. Also it is the first occurrence that differs from A084891.

Crossrefs

Cf. A084891, A162457

Programs

  • PARI
    factorrough(m,n) =
{
local(x,a,j,f,ln);
for(x=m,n, f=0; a = ifactor(x); for(j=2,length(a), ln=length(Str(a[j-1])); if(length(Str(a[j]))!=ln,f=1;break);); if(f,print1(x",")););
}

Formula

{k >1: A055642(A020639(k)) <> A055642(A006530(k)) }. - R. J. Mathar, Sep 16 2009

Extensions

Offset set to 1, definition shortened - R. J. Mathar, Sep 16 2009

A162252 Numbers of the form prime(prime(prime(k))) with a digit sum which is prime.

Original entry on oeis.org

5, 11, 179, 331, 599, 919, 1297, 1523, 1787, 2221, 3259, 3637, 3943, 4397, 5381, 6113, 6661, 6823, 8221, 9859, 10631, 11953, 12097, 12301, 12547, 12763, 13469, 14723, 15641, 15823, 17627, 18149, 19577, 20063, 20773, 21529, 23431, 26371, 26489
Offset: 1

Views

Author

Cino Hilliard, Jun 28 2009

Keywords

Comments

Members of A038580 with a digit sum which is prime.

Examples

			For k=6, prime(prime(prime(6))) = A038580(6)=179. The digit sum 1+7+9 = 17 is prime, so 179 is in the sequence.

Programs

  • Maple
    read("transforms") ; A038580 := proc(n) ithprime(ithprime(ithprime(n))) ; end:
for n from 1 to 80 do if isprime(digsum(A038580(n))) then printf("%d,", A038580(n)) ; fi; od: # R. J. Mathar, Aug 14 2009
  • Mathematica
    Select[Table[Nest[Prime, x, 3], {x, 1, 100}],
PrimeQ[Total[IntegerDigits[#, 10]]] &]
  • PARI
    sodip2(n,m) = /* m multiple nesting of prime(prime(prime..(n) */
{ local(s=0,a,x,y,j,p);
for(x=1,n, p=prime(x);
for(i=1,m,p=prime(p));
a=eval(Vec(Str(p))); y=sum(j=1,length(a),a[j]); if(isprime(y),print1(p","));)
}

Formula

{A038508(k): A007953(A038508(k)) in A000040, any k}.

Extensions

Edited by R. J. Mathar, Aug 14 2009

A159878 The digits of Pi whose spellings in English contain no i's.

Original entry on oeis.org

3, 1, 4, 1, 2, 3, 7, 3, 2, 3, 4, 2, 4, 3, 3, 3, 2, 7, 0, 2, 4, 1, 7, 1, 3, 3, 7, 1, 0, 2, 0, 7, 4, 4, 4, 2, 3, 0, 7, 1, 4, 0, 2, 2, 0, 2, 0, 3, 4, 2, 3, 4, 2, 1, 1, 7, 0, 7, 2, 1, 4, 0, 1, 3, 2, 2, 3, 0, 4, 7, 0, 3, 4, 4, 0, 0, 2, 2, 3, 1, 7, 2, 3, 4, 0, 1, 2, 4, 1, 1, 1, 7, 4, 0, 2, 4, 1, 0, 2, 7, 0, 1, 3, 2, 1
Offset: 1

Views

Author

Cino Hilliard, Apr 25 2009

Keywords

Comments

Blind Pi: The series of digits of Pi A000796 after removal of any 5, 6, 8 or 9 (see A095763, A089589).
The difference of the value of this constant to Pi is 0.000355..., compared to a difference of 0.0012... = A003077 for 22/7.
The only other alpha language that has no numbers 0 to 9 with an i is Albanian.
It is natural to ask "is the constant defined in this way irrational, transcendental?"

Examples

			Pi = 3.1415... . The digit 5 or five contains an i in the spelling. So 5 is not in the sequence.

Programs

  • Mathematica
    Flatten[ RealDigits[Pi, 10, 174][[1]] /. {5 -> {}, 6 -> {}, 8 -> {}, 9 -> {}}] (* Robert G. Wilson v, May 27 2009 *)
  • PARI
    blindpi(n) =
{
default(realprecision,1000);
local(pi,x);
pi=Vec(Str(Pi*10^99));
default(realprecision,28);
for(x=1,n,
if(pi[x]=="0"||pi[x]=="1"||pi[x]=="2"||pi[x]=="3"||pi[x]=="4"||pi[x]=="7",
print1(pi[x]",");
);
);
}

Extensions

Edited by R. J. Mathar, Apr 28 2009

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