A227006 -id:A227006 - cp's OEIS frontend

A227007 Numbers k such that k-1 is squarefree and every prime divisor of k-1 is in the sequence.

Original entry on oeis.org

2, 3, 4, 7, 8, 15, 22, 43, 44, 87, 130, 259, 302, 603, 904, 1807

Offset: 1

José María Grau Ribas, Jun 27 2013

The complement of A227006.

1807 is the last term in this sequence. - Charles R Greathouse IV, Jun 27 2013

J. M. Grau, A. M. Oller-Marcén and J, Sondow, On the congruence 1^n+2^n+...+n^n == d (mod n), where d divides n, arXiv preprint arXiv:1309.7941, 2013

Cf. A227006.

Needs["NumberTheory`NumberTheoryFunctions`"];Is[2] = True; Is[{}] = True; Is[n_] := Is[n] = If[ListQ[n], Is[n[[1, 1]]] && Is[Rest[n]], SquareFreeQ[n - 1] && Is[fa[n - 1]]]; Select[1 + Range@10000, Is]

A227455 Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p-1 is not in the sequence.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 23, 24, 25, 27, 29, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 50, 51, 53, 54, 55, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 83, 84, 85, 87, 89, 90, 92, 93, 95, 96, 99, 100, 102, 105, 106, 108, 110

Offset: 1

José María Grau Ribas, Jul 12 2013

nonn

Consider a two-player game in which players take turns and a player given the position k = p_1^s_1 * ... * p_j^s_j must choose one of the j possible moves p_1 - 1, ..., p_j - 1, and the player's chosen move becomes the position given to the other player. The first player whose only possible move is 1 loses. Terms in this sequence are the winning positions for the player whose turn it is.

			Numbers of the form 2^k are not in the sequence because their unique prime divisor is p = 2 and p-1 = 1 is in the sequence.
Numbers of the form 3^k are in the sequence because 3-1 = 2 is not in the sequence.
Numbers of the form 5^k are in the sequence because 5-1 = 4 = 2^2, and 2 is not in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A227006, A227691, A027748.

a227455 n = a227455_list !! (n-1)
a227455_list = 1 : f [2..] [1] where
   f (v:vs) ws = if any (`notElem` ws) $ map (subtract 1) $ a027748_row v
                    then v : f vs (v : ws) else f vs ws
-- Reinhard Zumkeller, Dec 08 2014

fa=FactorInteger;win[1] = True; win[n_] := win[n] = ! Union@Table[win[fa[n][[i, 1]] - 1], {i, 1, Length@fa@n}] == {True}; Select[Range[300], win]

Edited by Jon E. Schoenfield, Jan 23 2021

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

José María Grau Ribas, Jan 16 2016

nonn

Is this sequence infinite?

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

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

José María Grau Ribas, Jan 16 2016

nonn

Is this sequence infinite?

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

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] &]

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

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

José María Grau Ribas, Jan 16 2016

nonn

Is this sequence infinite?

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

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

José María Grau Ribas, Jan 16 2016

nonn

Is this sequence infinite?

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

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] &]

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A227007 Numbers k such that k-1 is squarefree and every prime divisor of k-1 is in the sequence.

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A227455 Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p-1 is not in the sequence.

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

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

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

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

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