A007918 -id:A007918 - cp's OEIS frontend

A067792 a(n) is the least prime >= sigma(n).

2, 3, 5, 7, 7, 13, 11, 17, 13, 19, 13, 29, 17, 29, 29, 31, 19, 41, 23, 43, 37, 37, 29, 61, 31, 43, 41, 59, 31, 73, 37, 67, 53, 59, 53, 97, 41, 61, 59, 97, 43, 97, 47, 89, 79, 73, 53, 127, 59, 97, 73, 101, 59, 127, 73, 127, 83, 97, 61, 173, 67, 97, 107, 127, 89, 149, 71, 127

Offset: 1

Benoit Cloitre, Feb 07 2002

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A007918, A070801, A074495.

Array[If[PrimeQ@ #, #, NextPrime@ #] &@ DivisorSigma[1, #] &, 68] (* Michael De Vlieger, Nov 16 2017 *)

A067792(n) = nextprime(sigma(n)); \\ Antti Karttunen, Nov 16 2017

From Antti Karttunen, Nov 17 2017: (Start)
a(n) = A007918(A000203(n)).
a(n) <= A074495(n).
(End)

Definition clarified by Antti Karttunen, Nov 16 2017

A069549 Smallest composite k such that phi(k) > k*(1-1/n).

Original entry on oeis.org

4, 4, 9, 25, 25, 49, 49, 121, 121, 121, 121, 169, 169, 289, 289, 289, 289, 361, 361, 529, 529, 529, 529, 841, 841, 841, 841, 841, 841, 961, 961, 1369, 1369, 1369, 1369, 1369, 1369, 1681, 1681, 1681, 1681, 1849, 1849, 2209, 2209, 2209, 2209, 2809, 2809, 2809

Offset: 1

Benoit Cloitre, Apr 21 2002

Or, least composite k such that k is coprime to the n-1 numbers k+1 ... k+n-1. E.g., a(4) = 25 because 25 is coprime to 26, 27 and 28. - Amarnath Murthy, Apr 20 2004

Cf. A000010 (phi), A007918, A052349.

a[n_] := NextPrime[n-1]^2; Array[a, 50] (* Amiram Eldar, May 08 2025 *)

a(n) = nextprime(n)^2; \\ Amiram Eldar, May 08 2025

a(n) = nextprime(n)^2 = A007918(n)^2.

Edited by David Wasserman, Apr 23 2007

A071328 Smallest prime q such that q - prime(n) >= n.

Original entry on oeis.org

3, 5, 11, 11, 17, 19, 29, 29, 37, 41, 43, 53, 59, 59, 67, 71, 79, 79, 89, 97, 97, 101, 107, 113, 127, 127, 131, 137, 139, 149, 163, 163, 173, 173, 191, 191, 197, 211, 211, 223, 223, 223, 239, 239, 251, 251, 263, 271, 277, 281, 293, 293, 307, 307

Offset: 1

Reinhard Zumkeller, May 19 2002

nonn

a(n) = A007918(n + A000040(n));

a(n) = A071329(n) iff a(n) = A061068(k) for some k.

			a(10) = A007918(10 + A000040(10)) = A007918(10 + 29)= A007918(39) = 41;
a(6) = A007918(6 + A000040(6)) = A007918(6 + 13)= A007918(19) = 19 = A071329(6) = A061068(4).

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

seq(nextprime(n + ithprime(n)-1),n=1..100); # Robert Israel, Jul 12 2019

A076873 Smallest prime not less than sum of first n primes.

Original entry on oeis.org

2, 5, 11, 17, 29, 41, 59, 79, 101, 131, 163, 197, 239, 281, 331, 383, 443, 503, 569, 641, 719, 797, 877, 967, 1061, 1163, 1277, 1373, 1481, 1597, 1721, 1861, 1993, 2129, 2281, 2437, 2591, 2749, 2917, 3089, 3271, 3449, 3643, 3833, 4049, 4229, 4441, 4663

Offset: 1

Reinhard Zumkeller, Nov 27 2002

nonn

a(n) = A007918(A007504(n)).

			n=5: 2+3+5+7+11 <= 29 = a(5).

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

Cf. A007504, A007918.

P:= [seq(ithprime(i),i=1..100)]:
S:= ListTools:-PartialSums(P):
map(s -> nextprime(s-1),S); # Robert Israel, Jul 04 2019

If[PrimeQ[#],#,NextPrime[#]]&/@Accumulate[Prime[Range[50]]]  (* Harvey P. Dale, Feb 04 2011 *)

A086757 Smallest prime p such that n is a palindrome in base-p representation.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 13, 5, 3, 13, 2, 3, 2, 5, 23, 3, 2, 23, 3, 5, 29, 3, 2, 3, 31, 29, 2, 7, 2, 37, 37, 5, 41, 37, 41, 3, 5, 13, 47, 43, 2, 5, 53, 7, 53, 7, 2, 3, 59, 17, 59, 3, 5, 59, 61, 11, 67, 5, 2, 7, 2, 67, 5, 3, 71, 13, 7, 5, 2, 73, 79, 37, 79, 5, 83, 3, 83, 3, 5, 11, 2

Offset: 1

Reinhard Zumkeller, Aug 01 2003

A016026(n) <= a(n) <= A007918(n).

Cf. A007918, A016026.

Cf. A006995 (a(n)=2).

isok(p,n) = my(d=digits(n,p)); d == Vecrev(d);
a(n) = my(p=2); while (!isok(p,n), p=nextprime(p+1)); p; \\ Michel Marcus, Jan 30 2024

from sympy import sieve
from sympy.ntheory import is_palindromic
def a086757(n): return next(p for p in sieve if is_palindromic(n, p)) # Dumitru Damian, Jan 29 2024

A090120 Numbers k such that nextprime(k^2) - prevprime(k^2) = 4.

Original entry on oeis.org

3, 4, 9, 10, 14, 15, 20, 21, 26, 33, 40, 110, 117, 124, 146, 206, 237, 250, 273, 303, 309, 326, 340, 350, 387, 429, 436, 440, 441, 447, 470, 513, 561, 573, 609, 634, 686, 704, 807, 897, 920, 1004, 1035, 1054, 1060, 1071, 1113, 1124, 1143, 1156, 1233, 1239

Offset: 1

Labos Elemer, Jan 09 2004

nonn

Note that the gap = 4 is partitioned either as 2+2 or as 3+1; 1+3 never occurs since n^2-1 is composite if n>2.

			k = 3 is a term since, k^2 = 9 is surrounded by the closest primes: {7,[9],11}.
k = 10 is a term since k^2 = 100 is surrounded by {97,[100],101}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A090116, A090117, A090118, A090119, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

Select[Range[3,1500], NextPrime[#^2] == NextPrime[#^2, -1] + 4 &] (* Giovanni Resta, May 26 2018 *)

isok(n) = nextprime(n^2) - precprime(n^2) == 4; \\ Michel Marcus, May 26 2018

Solutions to {x; A007918(x^2)-A007917(x^2) = 4}.

A091228 Smallest m >= n, such that m is irreducible when interpreted as GF(2)[X]-polynomial.

Original entry on oeis.org

2, 2, 2, 3, 7, 7, 7, 7, 11, 11, 11, 11, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 55, 55, 55, 55, 55, 55, 55, 55, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 73, 73, 73

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Analogous to A007918.

a(n) = n + A091229(n).

A113459 Least number that begins an arithmetic progression of n numbers with the same prime signature.

Original entry on oeis.org

1, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13

Offset: 1

David Wasserman, Jan 08 2006

Initial terms of arithmetic progressions described in A113460. - N. J. A. Sloane, Oct 18 2007
Conjecture: For n > 1, a(n) = A007918(n). - David Wasserman, Jan 08 2006
I disagree with that conjecture! Ignoring the initial terms, this will agree with A007918 up to some point and then (presumably) drop below A007918. The initial term in the arithmetic progression (of length n) must be >= n, but it is likely to be less than A007918(n) if n is large. - N. J. A. Sloane, Oct 18 2007

Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A007918, A087309, A113460.

Cf. A113461, A127781, A007917, A061558.

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar.

A117093 Numbers k such that nextprime(3k) > 3nextprime(k) (if p is prime then nextprime(p) = p).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 16, 17, 18, 19, 23, 28, 29, 30, 31, 37, 38, 39, 40, 41, 43, 47, 53, 58, 59, 61, 67, 71, 72, 73, 78, 79, 81, 82, 83, 88, 89, 95, 96, 97, 98, 99, 100, 101, 103, 106, 107, 108, 109, 113, 127, 130, 131, 137, 138, 139, 148, 149, 150, 151, 156, 157

Offset: 1

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

Includes all primes.

			13 is a term since nextprime(3*13) = 41 and 3*nextprime(13) = 3*13=39 and 41 > 39.

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

Union of A000040 and A117100.

Cf. A007918 (nextprime).

filter:= n -> nextprime(3*n) > 3*nextprime(n-1):
select(filter, [$1..1000]); # Robert Israel, Jul 25 2025

fp[k_]:=If[PrimeQ[k],k,NextPrime[k]];Select[Range[160],fp[3#]>3fp[#]&] (* James C. McMahon, Aug 23 2024 *)

for(i=1,100,if(nextprime(3*i)>nextprime(3)*nextprime(i),print1(i,", ")))

a(44)-a(59) from James C. McMahon, Aug 23 2024

A117094 Numbers k such that nextprime(5k) > 5nextprime(k) (if p is prime then nextprime(p) = p).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 18, 19, 23, 28, 29, 31, 37, 40, 41, 43, 47, 53, 59, 60, 61, 67, 71, 72, 73, 78, 79, 82, 83, 89, 96, 97, 101, 102, 103, 105, 106, 107, 109, 113, 127, 131, 137, 139, 149, 151, 155, 156, 157, 163, 166, 167, 173, 178, 179, 180, 181, 191, 192, 193, 197, 199, 211, 222, 223, 226

Offset: 1

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

Includes all primes. - Robert Israel, Jul 25 2025

			13 is a term since nextprime(5*13) = 67 and nextprime(5)*nextprime(13) = 5*13=65 and 67 > 65.

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

Union of A000040 and A117101.

Cf. A007918 (nextprime).

select(n -> nextprime(5*n) > 5*nextprime(n-1), [$1..1000]); # Robert Israel, Jul 25 2025

Select[Range[250], NextPrime[5*#] > 5*NextPrime[# - 1] &] (* Paolo Xausa, Jul 28 2025 *)

for(i=1,150,if(nextprime(5*i)>nextprime(5)*nextprime(i),print1(i,",")))

More terms from Robert Israel, Jul 25 2025

cp's OEIS Frontend

A067792 a(n) is the least prime >= sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A069549 Smallest composite k such that phi(k) > k*(1-1/n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A071328 Smallest prime q such that q - prime(n) >= n.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

A076873 Smallest prime not less than sum of first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A086757 Smallest prime p such that n is a palindrome in base-p representation.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Python

A090120 Numbers k such that nextprime(k^2) - prevprime(k^2) = 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A091228 Smallest m >= n, such that m is irreducible when interpreted as GF(2)[X]-polynomial.

Views

Author

Keywords

Comments

Links

Formula

A113459 Least number that begins an arithmetic progression of n numbers with the same prime signature.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A117093 Numbers k such that nextprime(3*k) > 3*nextprime(k) (if p is prime then nextprime(p) = p).

A117093 Numbers k such that nextprime(3k) > 3nextprime(k) (if p is prime then nextprime(p) = p).

A117094 Numbers k such that nextprime(5k) > 5nextprime(k) (if p is prime then nextprime(p) = p).