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 1-6 of 6 results.

A227006 Numbers k such that k-1 is not squarefree or a prime divisor of k-1 is in the sequence.

Original entry on oeis.org

5, 6, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
Offset: 1

Views

Author

José María Grau Ribas, Jun 27 2013

Keywords

Comments

The sequence is defined recursively. The complement of A227007.

Links

Crossrefs

Cf. A227007.

Programs

  • Mathematica
    Needs["NumberTheory`NumberTheoryFunctions`"];Property[2] = False; Property[{}] = False; Property[n_] := Property[n] = If[ListQ[n],Property[n[[1, 1]]] || Property[Rest[n]], SquareFreeQ[n - 1] == False || Property[fa[n - 1]]]; Select[1 + Range[100], Property]
  • PARI
    is(n)=if(n<7,return(n>4)); if(n>1807 || !issquarefree(n-1), return(1)); fordiv(n-1,d,if(isprime(d) && is(d), return(1))); 0 \\ Charles R Greathouse IV, Nov 13 2013

Formula

a(n) = n + 16 for n > 1791. - Charles R Greathouse IV, Jun 27 2013

Extensions

Definition corrected by Charles R Greathouse IV, Nov 13 2013

A229289 Primes p of the form p = 2^k * m + 1, where (i) m is squarefree and odd, (ii) all primes that divide m are in the sequence, and (iii) k is 0, 1, or 2.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 107, 131, 139, 157, 173, 211, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 421, 431, 461, 463, 547, 557, 599, 643, 659, 661, 683, 691, 709, 733, 743, 787, 827, 853, 859, 863, 911, 941
Offset: 1

Views

Author

José María Grau Ribas, Oct 05 2013

Keywords

Comments

Taking m=1 in the definition we get the primes 2, 3, 5.
If n is in A226960, then n is a product of terms of this sequence.
If k is only allowed to be 0 or 1, we get 2, 3, 7, 43 and no more. - Jianing Song, Feb 21 2021
Also prime factors of terms in A341858. It is conjectured that this sequence is infinite. - Jianing Song, Feb 22 2021

Links

Crossrefs

Cf. A227007, A226960, A007814, A229290, A229291, A289355, A341858.
For the complement, see A289355.
Proper subsequence of A066651.

Programs

  • Mathematica
    fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i, Length[fa[n]]}] ; Os[b_, 1] = True; Os[b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n - 1)/b^IntegerExponent[n - 1, b]] &&IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i, 1]]], {i, Length[fa[n - 1]]}] == {True};G[b_] := Select[Prime[Range[1000]], Os[b, #] &];G[2]
  • PARI
    is(n)=if(!isprime(n),return(0)); if(n<13,return(1)); my(k=valuation(n-1,2), m=n>>k, f); if(k>2,return(0)); f=factor(m); if(lcm(f[,2])>1, return(0)); for(i=1,#f~, if(!is(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2013

Extensions

Revised definition from Charles R Greathouse IV, Nov 13 2013
Terms corrected by José María Grau Ribas, Nov 14 2013

A267503 Primes p such that p-1 is squarefree and all prime divisors of p-1 other than 5 are also in the sequence.

Original entry on oeis.org

2, 3, 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, 6299, 6911, 7283, 7591, 8563, 8863, 9227, 9871, 9931, 10343, 10627, 11887, 11923, 12911
Offset: 1

Views

Author

José María Grau Ribas, Jan 16 2016

Keywords

Comments

Is this sequence infinite?

Links

Crossrefs

Cf. A267504, A267505, A267506, A267507, A005117, A227455, A227007, A227006.

Programs

  • Maple
    N:= 20000: # to get all terms <= N
Res:= 2:
Agenda:= {3,11}:
P:= {2,10}:
g:= proc(t) local s; s:=  p*t; if s < N then s else NULL fi end proc:
while Agenda <> {} do
  p:= min(Agenda);
  Res:= Res, p;
  newP:= map(g , P);
  P:= P union newP;
  Agenda:= Agenda minus {p} union select(isprime, map(`+`,newP,1));
od:
Res; # Robert Israel, Mar 15 2019
  • Mathematica
    fa = FactorInteger; is[2, p_] = True; is[2, p_];
is[n_, p_] := PrimeQ[n] &&  MoebiusMu[n - 1] ≠ 0 && Union@Table[is[fa[n - 1][[i, 1]], p] || fa[n - 1][[ i, 1]] == p , {i, Length[fa[n - 1]]}] == {True}; Select[Prime[Range[10000]], is[#, 5] &]

A267505 Primes p such that p-1 is squarefree and all prime divisors of p-1 other than 13 are also in the sequence.

Original entry on oeis.org

2, 3, 7, 43, 79, 547, 3319, 6163, 36979, 42667, 258847, 1553119, 1573207, 1834639, 1854763, 11131927, 20224159, 20451679, 124027567, 141569107, 141588763, 467477683, 1840398379, 3278780359, 5276533183, 6089163523, 6155955079, 11168428363, 11185512199, 31655671459
Offset: 1

Views

Author

José María Grau Ribas, Jan 16 2016

Keywords

Comments

Is this sequence infinite?

Links

Crossrefs

Cf. A267503, A267504, A267506, A267507, A005117, A227455, A227007, A227006.

Programs

  • Mathematica
    fa = FactorInteger; is[2, p_] = True; is[2, p_];
is[n_, p_] := PrimeQ[n] &&  MoebiusMu[n - 1] ≠ 0 && Union@Table[is[fa[n - 1][[i, 1]], p] || fa[n - 1][[ i, 1]] == p , {i, Length[fa[n - 1]]}] == {True}; Select[Prime[Range[100000]], is[#, 13] &]
  • PARI
    leastdiv(v, pred, inf)={ \\ finds least divisor d satisfying pred(d) && d>=inf
  my(recurse(k,d,lim)= if(d >= lim, lim, if(d>=inf && pred(d), d, k++; if(k<=#v, lim=self()(k, d*v[k], lim); self()(k, d, lim), lim))));
  my(stop=vecprod(v), lim=inf, m=4);
  while(lim<=stop, lim*=m; my(d=recurse(0,1,lim)); if(disprime(d+1), S[#S]); if(t==oo, break); t++; print1(t, ", "))} \\ Andrew Howroyd, Nov 13 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Nov 13 2018

A267504 Primes p such that p-1 is squarefree and all prime divisors of p-1 other than 11 are also in the sequence.

Original entry on oeis.org

2, 3, 7, 23, 43, 47, 67, 139, 283, 463, 659, 947, 967, 1319, 1699, 1979, 3083, 4423, 5683, 5839, 6299, 9227, 10627, 11887, 13259, 18679, 19183, 19447, 19867, 21407, 26539, 30559, 35863, 37379, 39199, 41539, 44087, 44483, 45403, 55399, 63823, 71347, 71359, 74759, 91127, 91463, 115099, 130687, 132527
Offset: 1

Views

Author

José María Grau Ribas, Jan 16 2016

Keywords

Comments

Is this sequence infinite?

Crossrefs

Cf. A267503, A267505, A267506, A267507, A005117, A227455, A227007, A227006.

Programs

  • Mathematica
    fa = FactorInteger; is[2, p_] = True; is[2, p_];
is[n_, p_] := PrimeQ[n] &&  MoebiusMu[n - 1] ≠ 0 && Union@Table[is[fa[n - 1][[i, 1]], p] || fa[n - 1][[ i, 1]] == p , {i, Length[fa[n - 1]]}] == {True}; Select[Prime[Range[10000]], is[#, 11] &]

A267506 Primes p such that p-1 is squarefree and all prime divisors of p-1 other than 17 are also in the sequence.

Original entry on oeis.org

2, 3, 7, 43, 103, 239, 479, 619, 3347, 4327, 10039, 24379, 25999, 30703, 48859, 123583, 143879, 147703, 150587, 170647, 186019, 288359, 344639, 421639, 593003, 689279, 690719, 1029827, 1381439, 1779007, 2651899, 3089479, 3558019, 4242983
Offset: 1

Views

Author

José María Grau Ribas, Jan 16 2016

Keywords

Comments

Is this sequence infinite?

Crossrefs

Cf. A267503, A267504, A267505, A267507, A005117, A227455, A227007, A227006.

Programs

  • Mathematica
    fa = FactorInteger; is[2, p_] = True; is[2, p_];
is[n_, p_] := PrimeQ[n] &&  MoebiusMu[n - 1] ≠ 0 && Union@Table[is[fa[n - 1][[i, 1]], p] || fa[n - 1][[ i, 1]] == p , {i, Length[fa[n - 1]]}] == {True}; Select[Prime[Range[10000]], is[#, 17] &]
Showing 1-6 of 6 results.

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