A007918 -id:A007918 - cp's OEIS frontend

A117101 Numbers k such that nextprime(5k) > 5nextprime(k) and k is composite.

18, 28, 40, 60, 72, 78, 82, 96, 102, 105, 106, 155, 156, 166, 178, 180, 192, 222, 226, 228, 250, 262, 266, 267, 268, 270, 280, 282, 292, 312, 328, 329, 330, 334, 335, 336, 352, 358, 378, 387, 388, 396, 408, 418, 420, 436, 437, 438, 460, 485, 486, 490, 496, 497

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 18 2006

Paolo Xausa, Table of n, a(n) for n = 1..10000

Set difference of A117094 and A000040.
Intersection of A002808 and A117094.
Cf. A007918 (nextprime).

Select[Range[500], CompositeQ[#] && NextPrime[5*#] > 5*NextPrime[#] &] (* Paolo Xausa, Jul 28 2025 *)

for(i=1,300,if(nextprime(5*i)>nextprime(5)*nextprime(i) && !isprime(i),print1(i,",")))

Name simplified by Paolo Xausa, Jul 28 2025

A118754 Smallest prime >= 5*n.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 97, 101, 107, 113, 127, 127, 127, 131, 137, 149, 149, 151, 157, 163, 167, 173, 179, 181, 191, 191, 197, 211, 211, 211, 223, 223, 227, 233, 239, 241, 251, 251, 257, 263, 269, 271, 277, 281

Offset: 0

Jonathan Vos Post, Apr 29 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007918, A008587, A060308, A060264, A118747-A118755.

Table[If[PrimeQ[5n],5n,NextPrime[5n]],{n,0,60}] (* Harvey P. Dale, Nov 29 2024 *)

a(n) = nextprime(5*n); \\ Michel Marcus, Feb 13 2021

a(n) = A007918(A008587(n)). - Michel Marcus, Feb 13 2021

A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

Original entry on oeis.org

4, 6, 9, 10, 14, 22, 25, 35, 49, 55, 65, 77, 85, 91, 119, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 361, 377, 391, 407, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 841, 851, 899, 943, 961, 989, 1073, 1081, 1147, 1189

Offset: 1

Andrew S. Plewe, Nov 13 2007

nonn

An integer n is included if, for some value y >= 0: n = A007918(A000196(n) + y) * A007918(A000196(n) - y) Or: n = nextprime(sqrtint(n) + y) * nextprime(sqrtint(n) - y) Where "nextprime(x)" is the smallest prime number >= to x and "sqrtint(z)" is the integer part of the square root of z.

Has many terms in common with A078972. - Bill McEachen, Dec 24 2020

			25 = nextprime(5 + 0) * nextprime(5 - 0) = 5 * 5 = 25
35 = nextprime(5 + 1) * nextprime(5 - 1) = 7 * 5 = 35
119 = nextprime(10 + 4) * nextprime(10 - 4) = 17 * 7 = 119

Michel Marcus, Table of n, a(n) for n = 1..3406

Cf. A007918, A000196, A078972.

bal(x,y) = nextprime(sqrtint(x)+y) * nextprime(sqrtint(x)-y);
findbal(x) = local(z,y); z=sqrtint(x); while( 0<=z, y=bal(x,z); if(y==x, print1(x", ");break;); z--;);
for (n=1,1200, findbal(n));

A145344 a(n) = smallest prime >= the smallest positive integer with exactly n divisors.

Original entry on oeis.org

2, 2, 5, 7, 17, 13, 67, 29, 37, 53, 1031, 61, 4099, 193, 149, 127, 65537, 181, 262147, 241, 577, 3079, 4194319, 367, 1297, 12289, 907, 967, 268435459, 727, 1073741827, 853, 9221, 196613, 5189, 1277, 68719476767, 786433, 36871, 1693, 1099511627791

Offset: 1

Leroy Quet, Oct 08 2008

nonn

a(n) = smallest prime >= A005179(n).

Amiram Eldar, Table of n, a(n) for n = 1..3322

Cf. A005179, A007918, A145343.

a(n) = A007918(A005179(n)). - Ray Chandler, Oct 12 2008

More terms from R. J. Mathar and Ray Chandler, Oct 10 2008

A166597 Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 2, 2, 4, 4

Offset: 0

Daniel Forgues, Oct 17 2009

nonn

Note the large prime gap of 72 between 31397 and 31469. This is the prime gap with the largest merit (cf. A111870), 72/log(31397)=6.95352 for primes less than 100000. Also 72/(log(31397))^2=0.67154 (cf. conjectures of Cramer-Granville, Shanks and Wolf) is largest for primes less than 100000. - Daniel Forgues, Oct 23 2009

			a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).
a(9) = 4 since the least prime greater than 9 is 11 (gap of 4 from 7 to 11).
a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).

Daniel Forgues, Table of n, a(n) for n = 0..100000
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture.

Cf. A151800, A166594.
Cf. A111870. - Daniel Forgues, Oct 23 2009
See A327441 for the classic G(n) version. - N. J. A. Sloane, Sep 11 2019
Cf. A001223, A000720, A007917, A007918, A151799.

2,2,seq(nextprime(n)-prevprime(n+1), n=2..100); # Ridouane Oudra, Dec 28 2024

f[n_]:=Module[{a=If[PrimeQ[n],n,NextPrime[n,-1]]}, NextPrime[n]-a]; Join[{2,2},Array[f,120,2]] (* Harvey P. Dale, May 17 2011 *)

a(n) = nextprime(n+1) - precprime(n); \\ Michel Marcus, Mar 02 2023

From Ridouane Oudra, Dec 28 2024: (Start)
a(n) = A001223(A000720(n)), for n>1.
a(n) = A151800(n) - A007917(n), for n>1.
a(n) = A007918(n+1) - A151799(n+1), for n>1. (End)

Definition rephrased by N. J. A. Sloane, Oct 25 2009

A249053 Defined by (i) a(1)=1; (ii) if you move a(n) steps to the right you must reach a prime; (iii) a(n) = smallest unused composite number greater than a(n-1), unless a(n) is required to be prime by (ii), in which case a(n) is the smallest unused prime greater than a(n-1).

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 11, 12, 13, 14, 17, 18, 20, 23, 24, 25, 26, 29, 30, 31, 32, 37, 38, 41, 42, 44, 45, 47, 48, 53, 54, 55, 59, 60, 62, 63, 67, 68, 71, 72, 73, 74, 79, 80, 81, 82, 83, 84, 89, 90, 97, 98, 101, 102, 104, 105, 106, 108, 109, 110, 113, 114, 115

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Eric Angelini, Posting to Sequence Fans Mailing List, Mar 17 2008 (the definition was clarified by Gabriel Cunningham).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A249054 for another version.

Cf. A002808, A007918.

import Data.Map (singleton, findMin, delete, insert)
a249053 n = a249053_list !! (n-1)
a249053_list = 1 : f 1 1 a002808_list (singleton 1 1) where
   f x z cs m
     | k == x    = p : f (x + 1) p cs (insert (x + p) 0 $ delete x m)
     | otherwise = c : f (x + 1) c cs' (insert (x + c) 0 m)
     where p = a007918 z
           (c:cs') = dropWhile (<= z) cs
           (k,_) = findMin m
-- Reinhard Zumkeller, Nov 01 2014

A266620 a(n) = least non-divisor of n!.

Original entry on oeis.org

2, 3, 4, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71

Offset: 1

Jeffrey Shallit, Jan 01 2016

It appears that a(n) = A151800(n) with the exception of n = 3. - Robert Israel, Jan 13 2016

			For n = 4 the least non-divisor of 4! = 24 = 2^3 * 3 is 5.
For n = 5 the least non-divisor of 5! = 120 = 2^3 * 3 * 5 is 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007918, A007978, A066169, A115627, A151800.

N:= 100: # to get a(1)..a(N)
m:= 1 + numtheory:-pi(N):
Primes:= [seq(ithprime(i),i=1..m)]:
for i from 1 to m do pindex[Primes[i]]:= i od:
V:= Vector(m):
k:= 0:
for n from 1 to N do
  for f in ifactors(n)[2] do
    q:= pindex[f[1]];
    V[q]:= V[q] + f[2];
    k:= max(k, q);
  od:
  a[n]:= min(seq(Primes[i]^(1+V[i]),i=1..k),Primes[k+1]);
od:
seq(a[n],n=1..N); # Robert Israel, Jan 13 2016

Table[Complement[Range[2n], Divisors[n!]][[1]], {n, 30}] (* Alonso del Arte, Sep 23 2017 *)
Table[Block[{m = n!, k = n + 1}, While[Divisible[m, k], k++]; k], {n, 67}] (* Michael De Vlieger, Sep 23 2017 *)

from sympy import nextprime
def A266620(n): return 4 if n == 3 else nextprime(n) # Chai Wah Wu, Feb 22 2023

a(n) = min_{k >= 1} prime(k)^(1 + v(n!, prime(k))) where v(m, p) is the p-adic order of m. - Robert Israel, Jan 13 2016

a(n) = prime(pi(n) + 1) except for n = 3, in which case the least non-divisor of 3! is 4, not 5. - Alonso del Arte, Sep 23 2017

A272899 Product of next n prime numbers greater than n.

Original entry on oeis.org

1, 2, 15, 385, 5005, 323323, 7436429, 955049953, 35336848261, 1448810778701, 62298863484143, 14107860812636383, 832363787945546597, 261682369333342226303, 18579448222667298067513, 1356299720254712758928449, 107147677900122307955347471, 46558817449894322874479515781

Offset: 0

Matthew Goers, May 09 2016

a(n) is of comparable size to n^n. - Charles R Greathouse IV, May 09 2016

a(n) is the product of the terms of the n-th row of A084754. - Michel Marcus, May 09 2016

			a(0) = 1 (the empty product).
a(1) = 2 = 2.
a(2) = 3 * 5 = 15.
a(3) = 5 * 7 * 11 = 385.
a(4) = 5 * 7 * 11 * 13 = 5005.

Alois P. Heinz, Table of n, a(n) for n = 0..322

Cf. A000720, A002110, A007918, A034386, A084754.

a:= n-> mul((nextprime@@i)(n), i=1..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Jun 24 2024

Table[Times@@Prime[Range[PrimePi[n] + 1, PrimePi[n] + n]], {n, 25}] (* Alonso del Arte, May 09 2016 *)

a(n)=my(v=primes(primepi(n)+n)); prod(i=0,n-1,v[#v-i]) \\ Charles R Greathouse IV, May 09 2016

from math import prod
from sympy import prime, primepi
def a(n): r = primepi(n); return prod(prime(i) for i in range(r+1, r+n+1))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Feb 15 2021

a(n) = A002110(n + A000720(n))/A034386(n), where A002110(n) are the primorials, A000720(n) is the pi(n) prime counting function, and A034386(n) is the primorial of primes less than or equal to n. E.g., a(7) = 955049953 = A002110(11) / A034386(7).

a(0)=1 prepended by Alois P. Heinz, Jun 24 2024

A279105 a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.

Original entry on oeis.org

3, 10, 44, 78, 136, 348, 592, 666, 820, 1272, 1652, 1830, 2144, 2628, 3320, 3738, 4656, 5886, 6328, 7620, 8384, 9042, 10728, 13040, 14532, 15752, 16290, 18528, 21100, 21944, 24084, 25424, 28920, 32382, 32896, 35508, 39340, 42192, 46050, 48828

Offset: 1

Hartmut F. W. Hoft, Dec 06 2016

nonn

A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1.
a(1) = 3 is the only odd number since 1 is the only odd number in A174973.
Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.

			a(3)=44 is the smallest number whose symmetric representation has 2 parts and 7 legs in each part.
a(4)=78 is the smallest number whose symmetric representation has 2 parts and 11 legs in each part.
No number k whose symmetric representation of sigma(k) has 2 parts can have 21 legs in its parts since there is no q in A174973 such that 2*q - 1 = 21.

Right border of A239929.
Supersequence of A246956 and A262259.
Cf. A007918, A174973, A237048, A237270, A237271, A237593, A246956, A249223, A279105.

a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]
a279105[n_] := Map[# * NextPrime[2*#]&, Select[Range[n], a174973Q]]
a279105[150] (* sequence data *)

a(n) = A174973(n) * A007918(2 * A174973(n) + 1).

A286264 a(n) = 2*(ceiling((n^2)/2)+1) - 1.

Original entry on oeis.org

3, 5, 11, 17, 27, 37, 51, 65, 83, 101, 123, 145, 171, 197, 227, 257, 291, 325, 363, 401, 443, 485, 531, 577, 627, 677, 731, 785, 843, 901, 963, 1025, 1091, 1157, 1227, 1297, 1371, 1445, 1523, 1601, 1683, 1765, 1851, 1937, 2027, 2117, 2211, 2305, 2403, 2501

Offset: 1

Ralf Steiner, May 05 2017

			n=2: (1*3*5*7)/(2*4*6*8) = (1*1*5*7)/(2*4*2*8) => a(2) = 5 = A151800(2^2).
n=3: (1*3*5*7*9*11*13*15*17)/(2*4*6*8*10*12*14*16*18) = (1*1*1*1*1*11*13*15*17)/(2*4*2*8*2*12*2*16*2) => a(3) = 11 = A151800(3^2).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A007918 (nextprime), A151800 (version 2).

[3/2 - (-1)^n/2 + n^2 : n in [1..100]]; // Wesley Ivan Hurt, May 05 2017

A286264:=n->3/2 - (-1)^n/2 + n^2: seq(A286264(n), n=1..100); # Wesley Ivan Hurt, May 05 2017

Table[2 (Ceiling[n^2/2] + 1) - 1, {n, 1, 40}]

Vec(x*(3 - x + x^2 + x^3) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ Colin Barker, May 05 2017

a(n) > n^2.
From Colin Barker, May 05 2017: (Start)
G.f.: x*(3 - x + x^2 + x^3) / ((1 - x)^3*(1 + x)).
a(n) = 3/2 - (-1)^n/2 + n^2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4. (End)
Sum_{n>=1} 1/a(n) = Pi*coth(Pi/2)/4 + Pi*tanh(Pi/sqrt(2))/(4*sqrt(2)) - 1/2. - Amiram Eldar, Jul 26 2024

More terms from Colin Barker, May 05 2017

cp's OEIS Frontend

A117101 Numbers k such that nextprime(5*k) > 5*nextprime(k) and k is composite.

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A118754 Smallest prime >= 5*n.

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A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

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A145344 a(n) = smallest prime >= the smallest positive integer with exactly n divisors.

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A166597 Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.

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A249053 Defined by (i) a(1)=1; (ii) if you move a(n) steps to the right you must reach a prime; (iii) a(n) = smallest unused composite number greater than a(n-1), unless a(n) is required to be prime by (ii), in which case a(n) is the smallest unused prime greater than a(n-1).

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A266620 a(n) = least non-divisor of n!.

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A272899 Product of next n prime numbers greater than n.

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A117101 Numbers k such that nextprime(5k) > 5nextprime(k) and k is composite.