A007918 -id:A007918 - cp's OEIS frontend

A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Jan 30 2025

First differs from A047592, A187320, A207481 and A255805 at n = 48: A047592(48) = A187320(48) = A207481(48) = A255805(48) = 54 is not a term of this sequence.
Numbers k such that A020639(k) >= A051903(k).
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = nextprime(k) = A007918(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.84999238500582943243..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p < i} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p < i} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

			6 = 2^1 * 3^1 is a term since 2 >= 1.
8 = 2^3 is not a term since 2 < 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Wikipedia, Rough number.

Cf. A020639, A051903, A007918.
Cf. A047592, A187320, A207481, A255805.
Subsequences: A380692, A380694, A380695.

q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Max[f[[;; , 2]]]]; Select[Range[100], q]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] >= vecmax(e));

A062772 Smallest prime larger than square of n-th prime.

Original entry on oeis.org

5, 11, 29, 53, 127, 173, 293, 367, 541, 853, 967, 1373, 1693, 1861, 2213, 2819, 3491, 3727, 4493, 5051, 5333, 6247, 6899, 7927, 9413, 10211, 10613, 11467, 11887, 12781, 16139, 17167, 18773, 19333, 22229, 22807, 24659, 26573, 27893, 29947, 32051

Offset: 1

Labos Elemer, Jul 18 2001

Subsequence of A007491. - Zak Seidov, Apr 30 2015

			100th prime, 541 immediately follows 529, square of 9th prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A054270, A054271, A000879.

Cf. A007491. - Zak Seidov, Apr 30 2015

with(numtheory): [seq(nextprime(ithprime(w)^2),w=1..100)];

Array[NextPrime[Prime[#]^2] &, 41] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = { nextprime(prime(n)^2) } \\ Harry J. Smith, Aug 10 2009

a(n) = A007918(A001248(n)) = A151800(A001248(n)). - Michel Marcus, Jun 24 2014

a(n) = A007491(A000040(n)). - Zak Seidov, Apr 30 2015

A072680 Difference between (least prime >= n) and (largest prime <= n).

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 8, 8, 8, 8, 8, 8, 8, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4

Offset: 2

Reinhard Zumkeller, Jul 01 2002

nonn

a(n) = 0 iff n is prime.

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A007917, A007918, A013633, A072681, A097106.

f[n_]:=If[PrimeQ[n],0,NextPrime[n]-NextPrime[n,-1]];Array[f,110,2] (* Harvey P. Dale, Sep 22 2011 *)

numlib::prevprime(i)*(-1)-nextprime(i)*(-1)$ i = 2..106 // Zerinvary Lajos, Feb 26 2007

A072680(n) = (nextprime(n) - precprime(n)); \\ Antti Karttunen, Sep 23 2018

a(n) = A007918(n) - A007917(n).

a(n) = A057427(n - A007917(n)) * A001223(A049084(A007917(n))).

A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

Original entry on oeis.org

5, 11, 29, 367, 149, 631, 127, 1949, 541, 907, 3251, 1693, 2503, 10427, 5779, 10831, 10007, 22229, 30631, 25301, 121123, 76207, 93047, 157627, 212557, 35729, 119027, 1121509, 190979, 672439, 693943, 1004027, 259099, 1646101, 675713, 1207841

Offset: 1

Labos Elemer, Jan 09 2004

nonn

			a(7) = 127 because 127-113 = 14 = 2*7 and 121 = 11^2 is between {127,113} closest primes to 121 a suitable square number. Also 127 is the smallest prime with this property.

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

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; de[1] = 0; t=Table[de[w^2], {w, 1, 50000}]; mt=Table[Min[Flatten[Position[t, 2*j]]], {j, 1, 100}]; Table[nex[Part[mt, j]^2], {j, 1, Length[mt]}]

a(n) = nextprime(A090117(n)) = nextprime(A090116(n)^2).

a(n) = A007918(A090117(n)) = prime(1+pi(A090117(n))).

Name corrected by Jason Yuen, Jun 23 2025

A104081 Smallest prime >= 3^n.

Original entry on oeis.org

2, 3, 11, 29, 83, 251, 733, 2203, 6563, 19687, 59051, 177167, 531457, 1594331, 4782971, 14348909, 43046747, 129140197, 387420499, 1162261523, 3486784409, 10460353259, 31381059613, 94143178859, 282429536483, 847288609457

Offset: 0

Cino Hilliard, Mar 03 2005

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A104080 (for 2^n).
Cf. A104088 (largest prime <= 3^n).
Cf. A000244, A007918, A014211.

[NextPrime(3^n-1): n in [0..50]]; // G. C. Greubel, Nov 08 2018

NextPrime[3^Range[0,50]-1] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

g(n,b=3) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = A014211(n), n > 1. - R. J. Mathar, Dec 13 2008

a(n) = A007918(A000244(n)). - Michel Marcus, Nov 08 2018

A273283 Least prime not less than the geometric mean of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 5, 5, 2, 17, 3, 19, 3, 5, 5, 23, 3, 5, 7, 3, 5, 29, 5, 31, 2, 7, 7, 7, 3, 37, 7, 7, 3, 41, 5, 43, 5, 5, 7, 47, 3, 7, 5, 11, 5, 53, 3, 11, 3, 11, 11, 59, 3, 61, 11, 5, 2, 11, 5, 67, 5, 11, 5, 71, 3, 73, 11, 5, 5, 11, 5, 79, 3, 3, 11

Offset: 2

Giuseppe Coppoletta, May 19 2016

nonn

A079870(n) <= a(n) <= A006530(n) <= n and a(n) = n iff n is prime, while a(n)= A079870(n) iff A079870(n) is prime.

			a(46)=7 because 7 is the least prime not less than sqrt(2*23).
a(84)=5 and A273282(84)=3 because A001222(84)=4 and 3 < 84^(1/4) < 5.

Giuseppe Coppoletta, Table of n, a(n) for n = 2..10000

Cf. A273282, A273285, A273289, A079870, A079871, A006530, A001222.

Table[NextPrime[(Ceiling[n^(1/PrimeOmega[n])] - 1)], {n,2,50} ] (* G. C. Greubel, May 26 2016 *)

[next_prime(ceil(n^(1/sloane.A001222(n)))-1) for n in (2..82)]

For n >= 2, a(n) = A007918(A079870(n)).

A329570 a(n) is the least prime P such that log(P)/log(p) >= valuation(n,p) for all primes p.

Original entry on oeis.org

2, 2, 3, 5, 5, 3, 7, 11, 11, 5, 11, 5, 13, 7, 5, 17, 17, 11, 19, 5, 7, 11, 23, 11, 29, 13, 29, 7, 29, 5, 31, 37, 11, 17, 7, 11, 37, 19, 13, 11, 41, 7, 43, 11, 11, 23, 47, 17, 53, 29, 17, 13, 53, 29, 11, 11, 19, 29, 59, 5, 61, 31, 11, 67, 13, 11, 67, 17, 23, 7, 71, 11, 73, 37, 29, 19, 11

Offset: 1

M. F. Hasler, Jan 03 2020

nonn

Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: This is the largest prime factor of the bound A329571(n)^2 above which all highly composite numbers are divisible by n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasa Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915): 347-409. A variant of better quality with an additional footnote is available at Ramanujan Papers.

Cf. A002182, A007918, A034699, A199337, A329571.

a[n_] := NextPrime[Max[Power @@@ FactorInteger[n]] - 1]; a[1] = 2; Array[a, 100] (* Amiram Eldar, Jan 17 2025 *)

apply( {A329570(n,f=Col(factor(max(n,2))), P=nextprime(vecmax([log(f[1])*f[2] | f<-f])))=[while( logint(P,f[1]) < f[2], P=nextprime(P+1)) | f<-f]; P}, [1..99])

a(n) = A007918(A034699(n)). - Amiram Eldar, Jan 17 2025

A378366 Difference between n and the greatest non prime power <= n (allowing 1).

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we almost have A010051 (A179278).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
For prime power we have A378457 = A276781-1 (A031218).
For nonsquarefree we have (A378033).
For non perfect power we almost have A075802 (A378363).
Subtracting from n gives (A378367).
The opposite is A378371, adding n A378372.
A000015 gives the least prime power >= n (cf. A378370 = A377282 - 1).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A007920, A013632, A065514, A074984, A113646, A345531, A377051, A377054, A377281, A377289, A378357.

Table[n-NestWhile[#-1&,n,PrimePowerQ[#]&],{n,100}]

a(n) = n - A378367(n).

A053786 a(n) = next prime after n^4.

Original entry on oeis.org

2, 17, 83, 257, 631, 1297, 2411, 4099, 6563, 10007, 14653, 20743, 28571, 38431, 50627, 65537, 83537, 104987, 130337, 160001, 194483, 234259, 279847, 331777, 390647, 456979, 531457, 614657, 707293, 810013, 923539, 1048583, 1185929, 1336337, 1500643

Offset: 1

Enoch Haga, Mar 26 2000

Primes associated with A053785.

			a(5)=631 because 631 is the smallest prime larger than 5^4 = 625.

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

Cf.A000583, A007491, A007918.

[NextPrime(n^4): n in [1..35]]; // Bruno Berselli, Apr 26 2012

seq(nextprime(n^4),n=1..100); # Robert Israel, Jan 29 2018

NextPrime[Range[100]^4] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

vector(50, n, nextprime(n^4)) \\ Michel Marcus, Jan 09 2015

a(n) = A007918(A000583(n)). - Robert Israel, Jan 29 2018

Edited by Jon E. Schoenfield, Jan 09 2015

A056139 a(n) = n^2 - primefloor(n)*primeceiling(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, -13, 4, 23, 0, 1, 0, -25, 4, 35, 0, 1, 0, -37, 4, 47, 0, -91, -42, 9, 62, 117, 0, 1, 0, -123, -58, 9, 78, 149, 0, -73, 4, 83, 0, 1, 0, -85, 4, 95, 0, -187, -90, 9, 110, 213, 0, -211, -102, 9, 122, 237, 0, 1, 0, -243, -118, 9, 138, 269, 0, -133, 4, 143, 0, 1, 0, -291, -142, 9, 162, 317, 0, -157, 4, 167, 0, -331, -162, 9

Offset: 2

Henry Bottomley, Jun 15 2000

a(n)= 0 iff n is prime.

			a(3)=3^2-3*3=0, a(4)=4^2-3*5=1

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A007917, A007918, A030664, A056140, A056141.

A030664(n) = if (n < 2, 1, precprime(n)*nextprime(n)); \\ From A030664
A056139(n) = (n^2 - A030664(n)); \\ Antti Karttunen, Mar 20 2018

a(n) = n^2 - A007917(n)*A007918(n) = A000290(n) - A030664(n).

More terms from Antti Karttunen, Mar 20 2018

cp's OEIS Frontend

A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

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A062772 Smallest prime larger than square of n-th prime.

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Maple

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A072680 Difference between (least prime >= n) and (largest prime <= n).

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A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

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A104081 Smallest prime >= 3^n.

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Magma

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A273283 Least prime not less than the geometric mean of all prime divisors of n counted with multiplicity.

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A329570 a(n) is the least prime P such that log(P)/log(p) >= valuation(n,p) for all primes p.

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