A070844 -id:A070844 - cp's OEIS frontend

A035095 Smallest prime congruent to 1 (mod prime(n)).

3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467

Offset: 1

Labos Elemer

nonn

This is a version of the "least prime in special arithmetic progressions" problem.
Smallest numbers m such that largest prime factor of Phi(m) = prime(n), the n-th prime, also seems to be prime and identical to n-th term of A035095. See A068211, A068212, A065966: Min[x : A068211(x)=prime(n)] = A035095(n); e.g., Phi(a(7)) = Phi(103) = 2*3*17, of which 17 = p(7) is the largest prime factor, arising first here.
It appears that A035095, A066674, A125878 are probably all the same, but see the comments in A066674. - N. J. A. Sloane, Jan 05 2013
Minimum of the smallest prime factors of F(n,i) = (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)]. Every prime factor of F(n,i) is congruent to 1 modulo prime(n). - Vladimir Shevelev, Nov 26 2014
Conjecture: a(n) is the smallest prime p such that gpf(p-1) = prime(n). See A023503. - Thomas Ordowski, Aug 06 2017
For n>1, a(n) is the smallest prime congruent to 1 mod (2*prime(n)). - Chai Wah Wu, Apr 28 2025

			a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence, 191 is the smallest prime.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Bd 1 (reprinted Chelsea 1953).
E. C. Titchmarsh, A divisor problem, Renc. Circ. Math. Palermo, 54 (1930) pp. 414-429.
P. Turan, Über Primzahlen der arithmetischen Progression, Acta Sci. Math. (Szeged), 8 (1936/37) pp. 226-235.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On some application of Brun's method, Acta Sci. Math (Szeged), v. 13, 1949, pp. 57-63.
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions J. Lond Math Soc s2-41 (2) (1990), pp. 193-200.
D. R. Heath-Brown, almost-primes in arithmetic progressions in short intervals, Math Proc Cambr. Phil Soc v 83 (1978), pp. 357-375.
D. R. Heath-Brown, Siegel zeros and the least prime in arithmetic progression, Quart. J. of Math 41 (49) (1990), pp. 405-418.
H.-J. Kanold, Uber Primzahlen in arithmetischen Folgen, Math. Ann. v 156 (1964) pp. 393-395.
U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math (N.S.) v 15 (57) (1944), pp 139-178. MR0012111
C. Pomerance, A note on the least prime in an arithmetic progression J. Number Theory 12 (2) (1980), pp. 218-223.
K. Prachar, Uber die kleinste Primzahl in einer arithmetischen Reihe, J Reine Angew Math. 206 (1961) pp. 3-4.
A. Schinzel, Remark on the paper of K. Prachar Uber die kleinste.., J. Reine Angew Math. v 210 (1962) pp. 122-122.
S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp., 32 (1978) pp. 583-591.
S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp., 33 (147) (1979) pp. 1073-1080.
Index entries for sequences related to primes in arithmetic progressions

Cf. A034694, A032448, A006530, A006093, A035096, A000040, A019434, A058383.
Cf. A068211, A068212, A065966, A000010, A070844-A070858, A061092.
Cf. A000040.

a[n_] := Block[{p = Prime[n]}, r = 1 + p; While[ !PrimeQ[r], r += p]; r]; Array[a, 51] (* Jean-François Alcover, Sep 20 2011, after PARI *)
a[n_]:=If[n<2,3,Block[{p=Prime[n]},r=1+2*p;While[!PrimeQ[r],r+=2*p]];r];Array[a,51] (* Zak Seidov, Dec 14 2013 *)

a(n)=local(p,r);p=prime(n);r=1;while(!isprime(r),r+=p);r

{my(N=66); forprime(p=2, , forprime(q=p+1,10^10, if((q-1)%p==0, print1(q,", "); N-=1; break)); if(N==0,break)); } \\ Joerg Arndt, May 27 2016

from itertools import count
from sympy import prime, isprime
def A035095(n): return 3 if n==1 else next(filter(isprime,count((p:=prime(n)<<1)+1,p))) # Chai Wah Wu, Apr 28 2025

According to a long-standing conjecture (see the 1979 Wagstaff reference), a(n) <= prime(n)^2 + 1. This would be sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1) = A000040(n). This in turn would be sufficient to imply that no value occurs twice in this sequence. - Franklin T. Adams-Watters, Jun 18 2010

a(n) = 1 + A035096(n)*A000040(n). - Zak Seidov, Dec 27 2013

Edited by Franklin T. Adams-Watters, Jun 18 2010
Minor edits by N. J. A. Sloane, Jun 27 2010
Edited by N. J. A. Sloane, Jan 05 2013

A070858 Smallest prime == 1 mod L, where L = LCM of 1 to n.

Original entry on oeis.org

2, 3, 7, 13, 61, 61, 421, 2521, 2521, 2521, 55441, 55441, 4324321, 4324321, 4324321, 4324321, 85765681, 85765681, 232792561, 232792561, 232792561, 232792561, 10708457761, 10708457761, 26771144401, 26771144401, 401567166001, 401567166001, 18632716502401, 18632716502401

Offset: 1

Amarnath Murthy, May 16 2002

nonn

Beginning with 3, smallest prime p = a(n) such that p + k is divisible by k + 1 for each k = 1, 2, ..., n. For example: 61 --> 62, 63, 64, 65 and 66 are divisible respectively by 2, 3, 4, 5 and 6. - Robin Garcia, Jul 23 2012

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from R. J. Mathar)

Cf. A060357, A075059, A003418, A070844 to A070856, A035091.

A070858 := proc(n)
    local l,p;
    l := ilcm(seq(i,i=1..n)) ;
    for p from 1 by l do
        if isprime(p) then
            return p;
        end if;
    end do:
end proc; # R. J. Mathar, Jun 25 2013

a[n_] := Module[{m = 1, lcm = LCM @@ Range[n]}, While[!PrimeQ[m], m += lcm]; m]; Array[a, 30] (* Amiram Eldar, Mar 15 2025 *)

a(n)=my(L=lcm(vector(n,i,i)),k=1);while(!ispseudoprime(k+=L),); k \\ Charles R Greathouse IV, Jun 25 2013

More terms from Sascha Kurz, Feb 02 2003

A035091 Smallest prime == 1 mod (n^2).

Original entry on oeis.org

2, 5, 19, 17, 101, 37, 197, 193, 163, 101, 727, 433, 677, 197, 1801, 257, 3469, 1297, 10831, 401, 883, 1453, 12697, 577, 11251, 677, 1459, 3137, 10093, 1801, 15377, 12289, 2179, 3469, 7351, 1297, 5477, 18773, 9127, 1601, 16811, 3529, 22189, 11617

Offset: 1

Labos Elemer

nonn

Smallest prime of form (n^2)*k+1, i.e., an arithmetic progression with n^2 differences; k is the subscript of the progressions.

			a(5) = 101 because in 5^2k + 1 = 25k + 1 progression k=4 generates the smallest prime (this is 101) and 26, 51, and 76 are composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Robert Price)
Index entries for sequences related to primes in arithmetic progressions.

Analogous case is A034694. Special case is A002496.

Cf. A070844 to A070858, A061092, A035095.

With[{prs=Prime[Range[2500]]},Flatten[Table[Select[prs,Mod[#-1,n^2]==0&,1],{n,50}]]] (* Harvey P. Dale, Sep 22 2021 *)

a(n) = if(n == 1, 2, my(s = n^2); forprime(p = 1, , if(p % s == 1, return(p)))); \\ Amiram Eldar, Mar 16 2025

A070845 Largest n-digit number with exactly n divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 97, 961, 9998, 83521, 999981, 4826809, 99999994, 999887641, 9999999824, 25937424601, 999999999981, 3138428376721, 99999999998144, 999997242389289, 9999999999999997, 45949729863572161, 999999999999999252

Offset: 1

Amarnath Murthy, May 12 2002

Ray Chandler, Table of n, a(n) for n=1..41

Cf. A070844.

If n is a prime >= 29 then a(n) = 0. - Chandler

a(6) from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 07 2003

Extended by Ray Chandler, Feb 10 2009

A073913 Number of staircase polygons on the square lattice with perimeter 2n and one (possibly rotated) staircase polygonal hole.

Original entry on oeis.org

1, 12, 94, 604, 3461, 18412, 93016, 452500, 2139230, 9890404, 44921002, 201099320, 889594210, 3896177956, 16920602244, 72954802376, 312595497011, 1332153819572, 5650155211024, 23864065957572, 100418115489408

Offset: 8

Olivier Gérard, Feb 14 2009, based on data from the web site of Iwan Jensen

nonn

The old entry with this A-number was a duplicate of A070844.

Iwan Jensen, Table of n, a(n) for n = 8..125 [Data from web page]
Iwan Jensen, Polygon enumerations.
Iwan Jensen and Andrew Rechnitzer, The exact perimeter generating function for a model of punctured staircase polygons, J. Phys. A: Math. Theor. 41 (2008) 215002, Table 1.

Cf. A057410, A057414.

G.f.: -(1/4)*(f1(x)-f2(x)+f3(x)-f4(x)) where f1(x) = (1-8*x+16*x^2-4*x^3)/(1-4*x), f2(x) = (1-6*x+6*x^2)/sqrt(1-4*x), f3(x) = (1/sqrt(2))*(sqrt(2+sqrt(3+4*x))*(3-8*x+2*x^2-sqrt(3+4*x)*(1-2*x)))/(1-4*x)^(3/4), f4(x) = (1/sqrt(2))*((3-8*x+2*x^2+sqrt(3+4*x)*(1-2*x)))/(1-4*x)^(1/4)/sqrt(2+sqrt(3+4*x)) [from Jensen and Rechnitzer, 2008]. - Sean A. Irvine, Dec 27 2024

Offset corrected by Sean A. Irvine, Dec 27 2024

cp's OEIS Frontend

A035095 Smallest prime congruent to 1 (mod prime(n)).

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A070858 Smallest prime == 1 mod L, where L = LCM of 1 to n.

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A035091 Smallest prime == 1 mod (n^2).

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A070845 Largest n-digit number with exactly n divisors, or 0 if no such number exists.

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A073913 Number of staircase polygons on the square lattice with perimeter 2n and one (possibly rotated) staircase polygonal hole.

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Extensions