A006093 -id:A006093 - cp's OEIS frontend

A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe, Apr 08 2004
Note that n+1 must be prime for all n in this sequence. - T. D. Noe, Apr 28 2004
Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.
M. Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0A006093.
Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.
Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]

Corrected by R. H. Hardin

A373674 Last element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373673.
Consists of all powers of primes k such that k+1 is not a power of primes.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A001359, min A006512, length A027833.
For composite runs we have A006093, min A008864, length A176246.
For prime runs we have A067774, min A025584, length A251092 or A175632.
For squarefree runs we have A373415, min A072284, length A120992.
For nonsquarefree runs we have min A053806, length A053797.
For runs of prime-powers:
- length A174965
- min A373673
- max A373674 (this sequence)
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A068780, A068781, A073051, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Max/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 522547200, 522547200, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

a(n) = a(n-1) iff n is prime. Thus a(1)=a(2)=a(3)=1 is the only triple in this sequence. - Franz Vrabec, Sep 10 2005
a(k) = a(k+1) for k in A006093. - Lekraj Beedassy, Aug 03 2006
Partial products of A048671. - Peter Luschny, Sep 09 2009

			a(5) = 2 as 5!/lcm(1..5) = 120/60 = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Liam Solus, Simplices for Numeral Systems, arXiv:1706.00480 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to lcm's

Cf. A000142, A002541, A006093, A003418, A048671, A025529, A205957.

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

Original entry on oeis.org

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

A060800 a(n) = p^2 + p + 1 where p runs through the primes.

Original entry on oeis.org

7, 13, 31, 57, 133, 183, 307, 381, 553, 871, 993, 1407, 1723, 1893, 2257, 2863, 3541, 3783, 4557, 5113, 5403, 6321, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 16257, 17293, 18907, 19461, 22351, 22953, 24807, 26733, 28057, 30103, 32221

Offset: 1

Jason Earls, Apr 27 2001

Terms are divisible by 3 iff p is of the form 6*m+1 (A002476). - Michel Marcus, Jan 15 2017

			a(3) = 31 because 5^2 + 5 + 1 = 31.

Harry J. Smith, Table of n, a(n) for n = 1..1000
R. J. Mathar, No common terms in the sequences sigma(p^i) and sigma(p^(i+1)) as p runs through the primes.

Cf. A001248, A131991, A131992, A131993, A253905.

Cf. A008864, A000203. - Zak Seidov, Feb 13 2016

[p^2+p+1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 20 2014

A060800:= n -> map (p -> p^(2)+p+1, ithprime(n)):
seq (A060800(n), n=1..41); # Jani Melik, Jan 25 2011

#^2 + # + 1&/@Prime[Range[200]] (* Vincenzo Librandi, Mar 20 2014 *)

{ n=0; forprime (p=2, prime(1000), write("b060800.txt", n++, " ", p^2 + p + 1); ) } \\ Harry J. Smith, Jul 13 2009

a(n) = A036690(n) + 1.
a(n) = 1 + A008864(n)*A000040(n) = (A030078(n) - 1)/A006093(n). - Reinhard Zumkeller, Aug 06 2007
a(n) = sigma(prime(n)^2) = A000203(A000040(n)^2). - Zak Seidov, Feb 13 2016
a(n) = A000203(A001248(n)). - Michel Marcus, Feb 15 2016
Product_{n>=1} (1 - 1/a(n)) = zeta(3)/zeta(2) (A253905). - Amiram Eldar, Nov 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001

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A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

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A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

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A023506 Exponent of 2 in prime factorization of prime(n) - 1.

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A373673 First element of each maximal run of powers of primes (including 1).

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A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

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A373674 Last element of each maximal run of powers of primes (including 1).

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A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

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A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

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