A062251 -id:A062251 - cp's OEIS frontend

A060324 a(n) is the minimal prime q such that n*(q+1)-1 is prime, that is, the smallest prime q so that n = (p+1)/(q+1) with p prime; or a(n) = -1 if no such q exists.

2, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 3, 7, 2, 3, 2, 3, 2, 5, 2, 3, 5, 5, 2, 5, 3, 3, 2, 5, 2, 13, 3, 3, 2, 3, 2, 11, 2, 5, 5, 3, 3, 5, 2, 3, 2, 5, 3, 5, 2, 19, 5, 3, 7, 7, 2, 3, 2, 5, 2, 7, 11, 3, 2, 5, 2, 5, 3, 11, 5, 3, 5, 13, 5, 5, 2, 3, 2, 7, 2, 7, 5, 3, 2, 5, 2, 3, 2, 17, 2, 7, 3, 5, 2, 3, 3, 11, 2, 5, 5

Offset: 1

Matthew Conroy, Mar 29 2001

A conjecture of Schinzel, if true, would imply that such a q always exists.

			1 = (2+1)/(2+1), so the first term is 2; 3(2+1) - 1 = 8 which is not prime, yet 3(3+1) - 1 = 11 is prime (3 = (11+1)/(3+1)) so the 3rd term is 3.

T. D. Noe, Table of n, a(n) for n = 1..10000
Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.
Peter Luschny, Schinzel-Sierpinski conjecture and Calkin-Wilf tree.

Cf. A060424. Values of p are given in A062251.

Cf. A000040, A010051.

a060324 n = head [q | q <- a000040_list, a010051' (n * (q + 1) - 1) == 1]
-- Reinhard Zumkeller, Aug 28 2014

a:= proc(n) local q;
       q:= 2;
       while not isprime(n*(q+1)-1) do
          q:= nextprime(q);
       od; q
    end:
seq(a(n), n=1..300); # Alois P. Heinz, Feb 11 2011

a[n_] := (q = 2; While[!PrimeQ[n*(q + 1) - 1], q = NextPrime[q]]; q); a /@ Range[100] (* Jean-François Alcover, Jul 20 2011, after Maple prog. *)

a(n) = {my(q=2); while (!isprime(n*(q+1)-1), q = nextprime(q+1)); q;} \\ Michel Marcus, Nov 20 2017

a(n) = (A062251(n)+1) / n - 1. - Reinhard Zumkeller, Aug 28 2014

A249800 a(n) is the smallest prime q such that n(q+1)+1 is prime, that is, the smallest prime q such that n = (p-1)/(q+1) with p prime; or a(n) = -1 if no such q exists.

Original entry on oeis.org

3, 2, 3, 2, 5, 2, 3, 11, 3, 2, 5, 2, 3, 2, 3, 5, 5, 3, 11, 2, 5, 2, 5, 2, 3, 2, 3, 3, 7, 5, 11, 2, 5, 2, 5, 2, 3, 5, 3, 5, 17, 2, 3, 7, 3, 2, 5, 3, 3, 2, 5, 2, 13, 2, 5, 5, 3, 3, 11, 2, 5, 5, 5, 2, 7, 2, 3, 5, 3, 2, 7, 5, 3, 2, 7, 2, 5, 3, 3, 2, 5, 113, 5, 3, 11

Offset: 1

Paolo P. Lava, Nov 06 2014

Variation on Schinzel's Hypothesis.

			For n=1 the minimum primes p and q are 5 and 3: (p-1)/(q+1) = (5-1)/(3+1) = 4/4 = 1. Therefore a(1)=3.
For n=2 the minimum primes p and q are 7 and 2: (p-1)/(q+1) = (7-1)/(2+1) = 6/3 = 2. Therefore a(2)=2.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Cf. A060324, A062251, A064632, A249801, A249802, A249803.

with(numtheory): P:=proc(q) local k,n;
for n from 1 to q do for k from 1 to q do
if isprime(n*(ithprime(k)+1)+1) then print(ithprime(k)); break; fi;
od; od; end: P(10^5);

a249800[n_Integer] := Module[{q}, q = 2; While[CompositeQ[n (q + 1) + 1], q = NextPrime[q]]; q]; a249800/@Range[120] (* Michael De Vlieger, Nov 19 2014 *)

a(n) = my(q=2); while(! isprime(n*(q+1)+1), q = nextprime(q+1)); q; \\ Michel Marcus, Nov 07 2014

A249801 Take smallest prime q such that n*(q+1)+1 is prime (A249800), that is, the smallest prime q so that n = (p-1)/(q+1) with p prime; sequence gives values of p; or -1 if A249800(n) = -1.

Original entry on oeis.org

5, 7, 13, 13, 31, 19, 29, 97, 37, 31, 67, 37, 53, 43, 61, 97, 103, 73, 229, 61, 127, 67, 139, 73, 101, 79, 109, 113, 233, 181, 373, 97, 199, 103, 211, 109, 149, 229, 157, 241, 739, 127, 173, 353, 181, 139, 283, 193, 197, 151, 307, 157, 743, 163, 331, 337, 229

Offset: 1

Paolo P. Lava, Nov 06 2014

Variation on Schinzel's Hypothesis.

			For n=1 the minimum primes p and q are 5 and 3: (p-1)/(q+1) = (5-1)/(3+1) = 4/4 = 1. Therefore a(1)=5.
For n=2 the minimum primes p and q are 7 and 2: (p-1)/(q+1) = (7-1)/(2+1) = 6/3 = 2. Therefore a(2)=7. Etc.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Cf. A060324, A062251, A064632, A249800, A249802-A249803.

with(numtheory): P:=proc(q) local k,n;
for n from 1 to q do for k from 1 to q do
if isprime(n*(ithprime(k)+1)+1) then print(n*(ithprime(k)+1)+1);
break; fi; od; od; end: P(10^5);

a(n) = my(q=2); while(! isprime(p=n*(q+1)+1), q = nextprime(q+1)); p; \\ Michel Marcus, Nov 07 2014

A249802 a(n) is the smallest prime q such that n(q-1)-1 is prime, that is, the smallest prime q so that n = (p+1)/(q-1) with p prime; or a(n) = -1 if no such q exists.

Original entry on oeis.org

5, 3, 2, 2, 5, 2, 3, 2, 3, 3, 5, 2, 19, 2, 3, 3, 5, 2, 3, 2, 3, 3, 7, 2, 7, 5, 3, 7, 7, 2, 3, 2, 5, 3, 5, 3, 3, 2, 7, 3, 5, 2, 7, 2, 3, 19, 7, 2, 3, 5, 3, 3, 5, 2, 3, 5, 3, 7, 7, 2, 19, 2, 5, 3, 7, 3, 7, 2, 3, 3, 5, 2, 67, 2, 3, 3, 5, 5, 3, 2, 11, 3, 5, 2, 7, 11

Offset: 1

Paolo P. Lava, Nov 06 2014

Variation on Schinzel's Hypothesis.

			For n=1 the minimum primes p and q are 3 and 5: (p+1)/(q-1) = (3+1)/(5-1) = 4/4 = 1. Therefore a(1)=5.
For n=2 the minimum primes p and q are 3 and 3: (p+1)/(q-1) = (3+1)/(3-1) = 4/2 = 2. Therefore a(2)=3. Etc.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis

Cf. A060324, A062251, A064632, A249800, A249801, A249803.

with(numtheory): P:=proc(q) local k,n;
for n from 1 to q do for k from 1 to q do
if isprime(n*(ithprime(k)-1)-1) then print(ithprime(k)); break; fi;
od; od; end: P(10^5);

a(n) = my(q=2); while(! isprime(n*(q-1)-1), q = nextprime(q+1)); q; \\ Michel Marcus, Nov 07 2014

A249803 Take smallest prime q such that n(q-1)-1 is prime (A249802), that is, the smallest prime q so that n = (p+1)/(q-1) with p prime; sequence gives values of p; or -1 if A249802(n) = -1.

Original entry on oeis.org

3, 3, 2, 3, 19, 5, 13, 7, 17, 19, 43, 11, 233, 13, 29, 31, 67, 17, 37, 19, 41, 43, 137, 23, 149, 103, 53, 167, 173, 29, 61, 31, 131, 67, 139, 71, 73, 37, 233, 79, 163, 41, 257, 43, 89, 827, 281, 47, 97, 199, 101, 103, 211, 53, 109, 223, 113, 347, 353, 59, 1097

Offset: 1

Paolo P. Lava, Nov 06 2014

Variation on Schinzel's Hypothesis.

			For n=1 the minimum primes p and q are 3 and 5: (p+1)/(q-1) = (3+1)/(5-1) = 4/4 = 1. Therefore a(1)=3.
For n=2 the minimum primes p and q are 3 and 3: (p+1)/(q-1) = (3+1)/(3-1) = 4/2 = 2. Therefore a(2)=3.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Cf. A060324, A062251, A064632, A249800, A249801, A249802.

with(numtheory): P:=proc(q) local k,n;
for n from 1 to q do for k from 1 to q do
if isprime(n*(ithprime(k)-1)-1) then print(n*(ithprime(k)-1)-1);
break; fi; od; od; end: P(10^5);

a(n) = my(q=2); while(! isprime(p=n*(q-1)-1), q = nextprime(q+1)); p; \\ Michel Marcus, Nov 07 2014

A062256 Record-setting values of q(n), the minimal prime q such that n(q+1)-1 is a prime p (i.e., q(n) > q(j) for all 0 < j < n).

Original entry on oeis.org

2, 11, 41, 103, 433, 1019, 2423, 6131, 22391, 146519, 398339, 1461359, 2803139, 3943883, 11329061, 37133051, 72486287, 89857919, 152222051, 247964153, 316352087, 927830951, 2030767073, 5359478723, 8908239161, 11980112897, 17219108579, 20740431791, 27651446429

Offset: 1

N. J. A. Sloane, Jul 01 2001

			31 = (433+1)/(13+1).

Amiram Eldar, Table of n, a(n) for n = 1..39
Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.

Cf. A060324, A062251. Values of n are in A060424, values of q in A062252.

a(n) = A060424(n)*(A062252(n)+1) - 1.

More terms from Vladeta Jovovic, Jul 02 2001

A062252 Record-setting values of q(n), the minimal prime q such that n(q+1)-1 is a prime p (i.e., q(n) > q(j) for all 0 < j < n).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 41, 71, 109, 179, 239, 269, 347, 353, 443, 503, 509, 617, 641, 701, 773, 881, 971, 977, 1013, 1019, 1103, 1109, 1223, 1559, 1607, 1709, 1889, 2063, 2297, 2663, 2963, 3137

Offset: 1

N. J. A. Sloane, Jul 01 2001

			31 = (433+1)/(13+1).

Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.

Cf. A060324, A062251. Values of n are in A060424, values of p in A062256.

a(36)-a(39) from Amiram Eldar, Jan 26 2019

cp's OEIS Frontend

A060324 a(n) is the minimal prime q such that n*(q+1)-1 is prime, that is, the smallest prime q so that n = (p+1)/(q+1) with p prime; or a(n) = -1 if no such q exists.

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A249800 a(n) is the smallest prime q such that n(q+1)+1 is prime, that is, the smallest prime q such that n = (p-1)/(q+1) with p prime; or a(n) = -1 if no such q exists.

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A249801 Take smallest prime q such that n*(q+1)+1 is prime (A249800), that is, the smallest prime q so that n = (p-1)/(q+1) with p prime; sequence gives values of p; or -1 if A249800(n) = -1.

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A249802 a(n) is the smallest prime q such that n(q-1)-1 is prime, that is, the smallest prime q so that n = (p+1)/(q-1) with p prime; or a(n) = -1 if no such q exists.

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A249803 Take smallest prime q such that n(q-1)-1 is prime (A249802), that is, the smallest prime q so that n = (p+1)/(q-1) with p prime; sequence gives values of p; or -1 if A249802(n) = -1.

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Maple

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A062256 Record-setting values of q(n), the minimal prime q such that n(q+1)-1 is a prime p (i.e., q(n) > q(j) for all 0 < j < n).

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A062252 Record-setting values of q(n), the minimal prime q such that n(q+1)-1 is a prime p (i.e., q(n) > q(j) for all 0 < j < n).

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