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.

Showing 1-7 of 7 results.

A126790 Prime numbers that are not Higgs' primes (numbers in A000040 not in A007459).

Original entry on oeis.org

17, 41, 73, 83, 89, 97, 103, 109, 113, 137, 163, 167, 179, 193, 227, 233, 239, 241, 251, 257, 271, 281, 293, 307, 313, 337, 353, 359, 379, 389, 401, 409, 433, 439, 443, 449, 457, 467, 479, 487, 499, 503, 521, 541, 563, 569, 577, 587, 593, 601, 613, 617, 619
Offset: 1

Views

Author

Alonso del Arte, Feb 19 2007

Keywords

Examples

			a(1) = 17 because 4 * 9 * 25 * 49 * 121 * 169 = 901800900 and 901800900 is congruent to 4 mod (17 - 1).

Links

Crossrefs

Cf. A007459.

Programs

  • Mathematica
    (* After running the program given at A007459 *) Complement[Prime[Range[144]], A007459]

A057447 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 107, 109, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 229, 233, 251, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 347, 349, 359
Offset: 1

Views

Author

Robert G. Wilson v, Sep 25 2000

Keywords

Comments

No prime of the form a*b^k + 1, with a > 0, b > 1 and k > 3 (including those in A037896) belongs to the sequence. - Mauro Fiorentini, Aug 09 2023

Links

Crossrefs

Cf. A007459, A037896, A057448.

Programs

  • Mathematica
    NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; f[ n_List ] := (a = n; b = Apply[ Times, a^3 ]; d = NextPrime[ a[[ -1 ]] ]; While[ ! IntegerQ[ b/(d - 1) ] && d < b+2, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]

A057459 a(n+1) = smallest prime p in the range a(n) < p < a(1)*a(2)*...*a(n) such that p-1 divides a(1)*a(2)*...*a(n); or if no such prime p exists, then a(n+1) = smallest prime > a(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 31, 43, 47, 67, 71, 139, 211, 283, 311, 331, 431, 463, 659, 683, 691, 863, 947, 967, 1291, 1303, 1319, 1367, 1427, 1699, 1867, 1979, 1987, 2011, 2111, 2131, 2311, 2531, 3011, 3083, 4099, 4423, 4643, 4691, 4831, 5171, 5179, 5683, 5839
Offset: 1

Views

Author

Robert G. Wilson v, Sep 26 2000

Keywords

Examples

			a(3) = 5. Since the product of a(1)*a(2) is 6, there is no prime p < 6 such that p-1 | 6 so the next prime greater than a(2) is 5.
a(9) = 47 since 46 (2*23) | 2*3*5*7*11*23*31*43.

Links

Crossrefs

Cf. A007459. See A282027 for another version.

Programs

  • Maple
    with(numtheory): a:=[2]; P:=1; j:=1;
for n from 2 to 36 do
sw:=-1; P:=P*a[n-1];
  for i from j+1 to 1000 do
  if (ithprime(i)N. J. A. Sloane, Feb 13 2017
  • Mathematica
    f[s_List] := Block[{b = Times @@ s, p = NextPrime@ Sort[s][[-1]]}, While[ Mod[b, p -1] > 0 && p < b, p = NextPrime@ p]; If[p > b, p = 2; While[ MemberQ[s, p], p = NextPrime@ p]]; Append[s, p]];; Nest[ f, {2}, 50] (* and modified by Robert G. Wilson v, Feb 13 2017 *)

A282027 a(n+1) = smallest prime p > a(n) such that p-1 divides a(1)*a(2)*...*a(n); or if no such prime p exists, then a(n+1) = smallest prime > a(n).

Original entry on oeis.org

2, 3, 7, 43, 47, 283, 659, 1319, 1699, 9227, 11887, 55399, 71359, 159707, 396719, 558643, 793439, 794039, 1117379, 1117943, 1143887, 2235887, 5554067, 6707747, 6863323, 13734803, 15667447, 16663963, 18214099, 20123239, 45196799, 46954223, 55937239, 93908447
Offset: 1

Views

Author

N. J. A. Sloane, Feb 13 2017

Keywords

Links

Crossrefs

Inspired by A007459 and A057459.

Programs

  • Maple
    A[1]:= 2: P:= 1:
for n from 2 to 30 do
  P:= A[n-1]*P;
  p0:= nextprime(A[n-1]);
  p:= p0;
  while p-1 <= P and P mod (p-1) <> 0 do
    p:= nextprime(p)
  od:
  if p-1 > P then A[n]:= p0
  else A[n]:= p
  fi;
od:
seq(A[i],i=1..30); # Robert Israel, Mar 17 2017
  • PARI
    lista(nn) = {my(d, k, m, t, v=List([2])); for(n=2, nn, k=1; m=oo; while((d=prod(i=1, t=k, v[i]))m || t==n-1, t++); forsubset([t, k], w, if(ispseudoprime(d=prod(i=1, k, v[w[i]])+1) && d>v[n-1], m=min(m, d)))); listput(v, if(mJinyuan Wang, Nov 21 2020

Extensions

Corrected and extended by Robert Israel, Mar 17 2017
More terms from Jinyuan Wang, Nov 21 2020

A057448 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^5.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281
Offset: 1

Views

Author

Robert G. Wilson v, Sep 25 2000

Keywords

Crossrefs

Cf. A007459 & A057447.

Programs

  • Mathematica
    NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; f[ n_List ] := (a = n; b = Apply[ Times, a^5 ]; d = NextPrime[ a[[ -1 ]] ]; While[ ! IntegerQ[ b/(d - 1) ] && d < b+2, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]

A131457 a(n+1) is the next semiprime such that a(n+1)-1 divides (a(1)...a(n))^2.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 26, 33, 34, 35, 46, 49, 51, 55, 57, 58, 65, 69, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 161, 166, 169, 177, 178, 183, 185, 187, 201, 202, 203, 205, 206, 209, 213
Offset: 1

Views

Author

Jonathan Vos Post, Oct 21 2007

Keywords

Comments

This is to semiprimes A001358 as A007459 is to primes A000040.

Examples

			a(1) = 4 because 4 = 2^2 is the first semiprime.
a(2) = 9 because 9 = 3^2 is the next semiprime after 4, where 9-1=8 divides 4^2 = 16.
a(3) = 10 because 10 = 2*5 is the next semiprime after 9 where 10-9=9 divides (4*9)^2.
a(4) = 21 because 21 = 3*7 is the next semiprime after 10, where 10-1=9 divides (4*9*10)^2.
a(5) = 22 because 22 = 2*11 is the next semiprime after 21, where 21-1=20 divides (4*9*10*21)^2.

Crossrefs

Cf. A000040, A001358, A007459, A070552, A109373.

Programs

  • Maple
    isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false; fi ; end: A131457 := proc(n) option remember ; local a,prevpr; if n =1 then 4; else prevpr := (mul(A131457(i),i=1..n-1))^2 ; a := A131457(n-1)+1 ; while not isA001358(a) or prevpr mod (a-1) <> 0 do a := a+1 ; od; RETURN(a) ; fi ; end: seq(A131457(n),n=1..80) ; # R. J. Mathar, Oct 30 2007
  • Mathematica
    semiprimeQ[n_] := PrimeOmega[n] == 2;
a[n_] := a[n] = Module[{k, prevpr}, If[n == 1, 4, prevpr = Product[a[i], {i, 1, n-1}]^2; k = a[n-1]+1; While[!semiprimeQ[k] || Mod[prevpr, k-1] != 0, k++]; Return[k]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 28 2024, after R. J. Mathar *)

Extensions

Corrected and extended by R. J. Mathar, Oct 30 2007

A214396 Number of HSI-algebras on n elements, up to isomorphism.

Original entry on oeis.org

1, 5, 44, 657, 13577, 672740
Offset: 1

Views

Author

Charles R Greathouse IV, Aug 21 2012

Keywords

Comments

An HSI-algebra is a structure (1, +, *, ^) over some set such that Tarski's high-school identities hold: addition and multiplication are commutative and associative, multiplication distributes over addition, 1 is the multiplicative identity, x^1 = x, 1^x = 1, x^y * x^z = x^(y+z), (xy)^z = x^z * y^z, and (x^y)^z = x^(y*z).
Burris & Lee (1992) find a(3) = 44.

Examples

			From _Bert Dobbelaere_, Sep 13 2020: (Start)
The following operator definitions over the set of elements {1,A,B} is consistent with the identities. There are 44 such solutions that cannot be transformed into eachother by swapping symbols, hence a(3) = 44.
x + y | y = 1  A  B       x * y | y = 1  A  B       x ^ y | y = 1  A  B
------+--------------    -------+--------------    -------+--------------
x = 1 |     A  A  1       x = 1 |     1  A  B       x = 1 |     1  1  1
    A |     A  A  A           A |     A  A  B           A |     A  A  1
    B |     1  A  B           B |     B  B  B           B |     B  B  B
(End).

Links

Crossrefs

Cf. A007459.

Formula

Trivial upper bound: a(n) <= n^(3n^2+1). - Charles R Greathouse IV, Jun 19 2013

Extensions

a(4) from Bert Dobbelaere, Sep 13 2020
a(5)-a(6) from Choiwah Chow, Oct 21 2024
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.