A060324 -id:A060324 - cp's OEIS frontend

A062251 Take minimal prime q such that n(q+1)-1 is prime (A060324), that is, the smallest prime q so that n = (p+1)/(q+1) with p prime; sequence gives values of p.

2, 5, 11, 11, 19, 17, 41, 23, 53, 29, 43, 47, 103, 41, 59, 47, 67, 53, 113, 59, 83, 131, 137, 71, 149, 103, 107, 83, 173, 89, 433, 127, 131, 101, 139, 107, 443, 113, 233, 239, 163, 167, 257, 131, 179, 137, 281, 191, 293, 149, 1019, 311, 211, 431, 439, 167, 227

Offset: 1

N. J. A. Sloane, Jul 01 2001

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

			1 = (2+1)/(2+1), 2 = (5+1)/(2+1), 3 = (11+1)/(3+1), 4 = (11+1)/(2+1), ...

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.
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.

Cf. A060424. Values of q are given in A060324.

a062251 n = (a060324 n + 1) * n - 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; n*(q+1)-1
    end:
seq(a(n), n=1..300);

a[n_] := (q = 2; While[ ! PrimeQ[n*(q+1)-1], q = NextPrime[q]]; n*(q+1)-1); Table[a[n], {n, 1, 57}] (* Jean-François Alcover, Feb 17 2012, after Maple *)

a(n) = (A060324(n) + 1) * n - 1. - Reinhard Zumkeller, Aug 28 2014

More terms from Vladeta Jovovic, Jul 02 2001

A078454 Index of the first occurrence of prime(n) in A060324.

Original entry on oeis.org

1, 3, 7, 13, 37, 31, 89, 51, 101, 219, 309, 973, 146, 832, 464, 1031, 2714, 2352, 6403, 311, 6397, 6352, 1487, 1439, 20718, 4252, 11958, 3719, 1332, 4136, 14509, 4601, 4223, 12414, 4043, 38862, 57949, 20257, 4958, 4832, 2213, 96792, 27932, 261337

Offset: 1

Amarnath Murthy, Dec 23 2002

nonn

Smallest k such that k*prime(n) + k - 1 is a prime.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..100

Cf. A060324.

A060324(n)=forprime(q=2,999999999,if(isprime(n*(q+1)-1),return(q)));-1
al(n)=local(v,k,xv);v=vector(n);while(n>0,xv=primepi(A060324(k++));if(xv<=#v&v[xv]==0,v[xv]=k;n--));v (End)

More terms from Franklin T. Adams-Watters, May 25 2010

A060424 Record-setting n's for the function q(n), the minimum prime q such that n(q+1)-1 is prime p (i.e., q(n) > q(j) for all 0 < j < n).

Original entry on oeis.org

1, 3, 7, 13, 31, 51, 101, 146, 311, 1332, 2213, 6089, 10382, 11333, 32003, 83633, 143822, 176192, 246314, 386237, 450644, 1198748, 2302457, 5513867, 9108629, 11814707, 16881479, 18786623, 24911213, 28836722, 34257764, 196457309

Offset: 1

Matthew Conroy, Apr 10 2001

nonn

			a(3)=7, since q(7)=5 and q(j) < 5 for 0 < j < 7.

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. See A062252 and A062256 for the corresponding values of q and p.

q[n_] := Module[{p = 2}, While[! PrimeQ[n*(p+1)-1], p = NextPrime[p]]; p]; record = 0; a[0] = 0; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, If[q[k] > record, record = q[k]; Print[k]; Return[k]]]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Nov 18 2013 *)

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

A255598 a(n) is the minimal number q>1 such that n(q+1)-1 is prime, or -1 if no such q exists.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 3, 7, 2, 3, 2, 3, 2, 5, 2, 3, 4, 5, 2, 5, 3, 3, 2, 5, 2, 13, 3, 3, 2, 3, 2, 11, 2, 5, 4, 3, 3, 5, 2, 3, 2, 5, 3, 5, 2, 9, 5, 3, 4, 7, 2, 3, 2, 5, 2, 7, 6, 3, 2, 5, 2, 5, 3, 11, 4, 3, 4, 13, 5, 5, 2, 3, 2, 7, 2, 7, 4, 3, 2, 5, 2, 3, 2, 15, 2, 7, 3, 5, 2, 3, 3, 11

Offset: 1

Zak Seidov, Feb 27 2015

nonn

a(n) <= A060324(n), and a(n) = A060324(n) iff a(n) is prime.
Among first 10000 terms, in 2453 cases a(n) < A060324(n). First such cases are for n = 22, 40, 51, 54, 62, 70, 72, 82, 89, 100.
Also, among first 10^6 terms, in 324388 cases a(n) < A060324(n).

Cf. A060324.

a(n) = {my(q = 2); while (! isprime(n*(q+1)-1), q ++); q;} \\ Michel Marcus, Feb 27 2015

cp's OEIS Frontend

A062251 Take minimal prime q such that n(q+1)-1 is prime (A060324), that is, the smallest prime q so that n = (p+1)/(q+1) with p prime; sequence gives values of p.

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A078454 Index of the first occurrence of prime(n) in A060324.

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A060424 Record-setting n's for the function q(n), the minimum prime q such that n(q+1)-1 is prime p (i.e., q(n) > q(j) for all 0 < j < n).

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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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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).