A053177 -id:A053177 - cp's OEIS frontend

A259978 Terms in A053177 that are relatively prime to 3.

35, 95, 119, 143, 203, 215, 275, 299, 335, 395, 455, 515, 527, 539, 623, 635, 695, 707, 767, 779, 803, 899, 923, 935, 959, 1007, 1043, 1115, 1127, 1139, 1175, 1199, 1235, 1295, 1355, 1403, 1547, 1595, 1643, 1655, 1679, 1715, 1727, 1763, 1775, 1859, 1883, 1895

Offset: 1

N. J. A. Sloane, Jul 12 2015, following a suggestion from R. P. Boas, May 19 1974

Lars Blomberg, Table of n, a(n) for n = 1..4332
R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A053177.

Select[2 Prime@ Range@ 162 + 1, CompositeQ@ # && GCD[3, #] == 1 &] (* Michael De Vlieger, Jul 13 2015 *)

isok(n) = (n % 2) && !isprime(n) && isprime((n-1)/2) && (gcd(n, 3) == 1); \\ Michel Marcus, Jul 28 2018

a(n) = 2*A275770(n) + 1. - Hilko Koning, Jul 23 2018

More terms from Lars Blomberg, Jul 13 2015

A087656 Let f be defined on the rationals by f(p/q) =(p+1)/(q+1)=p_{1}/q_{1} where (p_{1},q_{1})=1. Let f^k(p/q)=p_{k}/q_{k} where (p_{k},q_{k})=1. Sequence gives least k such that p_{k}-q_{k} = 1 starting at n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41, 82

Offset: 3

Benoit Cloitre, Oct 04 2003

nonn

Proof that this is the same as A059975 except for offset, from Joseph Myers, Feb 21 2004. Claim: a(n+1) = A059975(n). If p is the least prime factor of n then the rule here gives (n+1)/1 -> (n+2)/2 -> ... -> (n+p)/p = (n/p + 1)/1 so a(n+1) = a(n/p + 1) + (p-1) and clearly A059975(n) = A059975(n/p) + (p-1). The natural start for the induction is A059975(1) = a(2) = 0 (one place before the currently listed sequences start).

			6 -> (6+1)/(1+1) = 7/2 -> (7+1)/(2+1) = 8/3 -> (8+1)/(3+1) = 9/4 -> (9+1)/(4+1) = 2/1 and 2-1 = 1 hence a(6) = 4.

Same as A059975 apart from offset.

a(x)=if(x<0, 0, c=0; while(abs(numerator(x)-denominator(x)-1)>0, x=(numerator(x)+1)/(denominator(x)+1); c++); c)

If p is prime a(p+1)=p-1; it appears that a(n)=(n-1)/2 iff n is in A079148 or in A053177.

A281317 Primes p such that p == i mod d(i) where d(i) are the prime divisors of 2p+1.

Original entry on oeis.org

7, 13, 37, 67, 157, 337, 367, 607, 787, 937, 1093, 3037, 3307, 7717, 9187, 12757, 15187, 19687, 27337, 35437, 42187, 49207, 69457, 75937, 267907, 347287, 683437, 744187, 797161, 882367, 1148437, 1458607, 1736437, 2067187, 2870437, 2929687, 3125587, 4823437

Offset: 1

Michel Lagneau, Jan 20 2017

nonn

Subsequence of A053176.
a(n)== 1 mod 6 or a(n)== 1, 7 mod 12. A majority of members of the sequence are congruent to 7 mod 10.
omega(2*a(n)+1) = 1 for n = 2, 11, 29,... => 2*a(n)+1 = 3^3, 3^7, 3^13,... where omega(n) = A001221(n).

			157 is in the sequence because  2*157 + 1 = 315 = 3 ^ 2 * 5 * 7 => 157 == 1 (mod 3), 157 == 2 (mod 5) and 157 == 3 (mod 7).

Cf. A001221, A053176, A053177.

with(numtheory):
for n from 2 to 10^5 do:
  p:=ithprime(n):q:=2*p+1:x:=factorset(q):n1:=nops(x):j:=0:
   for i from 1 to n1 do:
     if irem(p,x[i])=i
      then j:=j+1:
      else
     fi:
   od:
    if j=n1
     then
     printf(`%d, `,p):
     else
    fi:
  od:

Select[Prime@ Range[10^6], Function[p, Function[i, Times @@ Boole@ MapIndexed[Mod[p, #1] == First@ #2 &, FactorInteger[i][[All, 1]]] > 0][2 p + 1]]] (* Michael De Vlieger, Jan 20 2017 *)

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A259978 Terms in A053177 that are relatively prime to 3.

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A087656 Let f be defined on the rationals by f(p/q) =(p+1)/(q+1)=p_{1}/q_{1} where (p_{1},q_{1})=1. Let f^k(p/q)=p_{k}/q_{k} where (p_{k},q_{k})=1. Sequence gives least k such that p_{k}-q_{k} = 1 starting at n.

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PARI

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A281317 Primes p such that p == i mod d(i) where d(i) are the prime divisors of 2p+1.

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Maple

Mathematica