A049711 -id:A049711 - cp's OEIS frontend

A060360 a(n) = k - (largest prime < k), where k = lcm(1..n).

1, 1, 1, 1, 1, 1, 17, 17, 19, 19, 23, 23, 23, 17, 43, 43, 1, 1, 1, 1, 1, 1, 29, 29, 41, 41, 1, 1, 43, 1, 1, 1, 1, 1, 43, 43, 43, 43, 47, 47, 83, 83, 83, 83, 1, 1, 83, 83, 83, 83, 61, 61, 61, 61, 61, 61, 149, 149, 1, 1, 1, 97, 97, 97, 89, 89, 89, 89, 109, 109

Offset: 3

N. J. A. Sloane, Apr 01 2001

nonn

Cf. A003418, A049711.

lcmp[n_]:=Module[{c=LCM@@Range[n]},c-NextPrime[c,-1]]; Array[lcmp,80,3] (* Harvey P. Dale, Dec 22 2012 *)

a(n) = my(lc = lcm([1..n])); lc - precprime(lc-1); \\ Michel Marcus, Mar 20 2018

a(n) = A049711(A003418(n)). - Michel Marcus, Mar 20 2018

A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 1, 2, 3, 2, 8, 12, 1, 2, 2, 5, 6, 6, 2, 3, 6, 6, 2, 2, 8, 3, 4, 2, 12, 2, 9, 8, 18, 2, 2, 6, 4, 12, 2, 3, 6, 4, 2, 6, 12, 8, 2, 6, 2, 1, 6, 8, 2, 2, 14, 4, 6, 2, 6, 2, 3, 20, 2, 12, 2, 2, 8, 14, 10, 18, 8, 6, 2, 2, 2, 12, 12, 19, 2, 6, 6, 20, 2, 2, 2, 8, 8, 2, 2, 8, 20, 12, 15, 2, 4

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2.

			78125=5^7 follows 78121, the difference is 4.

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

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A246547.

N:= 10^5: # to consider prime powers <= N
P:= select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]):
PP:= sort([seq(seq(p^k,k=2..ilog[p](N)),p=P)]):
map(t -> t - prevprime(t), PP); # Robert Israel, Nov 13 2024

from sympy import primepi, integer_nthroot, prevprime
def A060847(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return (a:=bisection(f,n,n))-prevprime(a) # Chai Wah Wu, Sep 13 2024

a(n) = A246547(n)-prevprime(A246547(n)) = A246547(n)-A049711(A246547(n)).

A060849 Difference between 2 consecutive primes between which a nontrivial power of prime is found.

Original entry on oeis.org

2, 4, 4, 4, 6, 6, 6, 6, 6, 4, 14, 14, 4, 6, 10, 6, 10, 10, 8, 12, 18, 12, 6, 14, 14, 10, 34, 6, 24, 14, 14, 24, 24, 6, 12, 16, 16, 22, 8, 6, 10, 10, 12, 10, 18, 10, 6, 16, 8, 18, 10, 18, 6, 20, 20, 34, 18, 14, 10, 12, 30, 24, 8, 16, 14, 6, 36, 20, 12, 28, 12, 10, 12, 14, 20, 22, 22

Offset: 1

Labos Elemer, May 03 2001

nonn

			59049=3^10 is between 59029 and 59051, so the corresponding term is 59051-59029=22.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A246547, A007917, A007918, A013632, A013633, A049711.

ispp(n) = isprimepower(n) >= 2; \\ A246547
lista(nn) = {for (n=1, nn, if (ispp(n), print1(nextprime(n) - precprime(n), ", ")););} \\ Michel Marcus, Mar 23 2020

a(n) = nextprime(A246547(n)) - prevprime(A246547(n)) = A013633(A246547(n)). [corrected by Michel Marcus, Mar 23 2020]

A078772 a(n) = phi(n-p) where p is largest prime < n, a(1) = a(2) = 1 by convention.

Original entry on oeis.org

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

Offset: 1

Jon Perry, Jan 09 2003

nonn

This sequence is a block of concatenations of vectors of lengths of prime gaps with elements phi(i) for i = 1 to that prime gap. Those vectors are (1), (1, 1), (1, 1, 2, 2), (1, 1, 2, 2, 4, 2), ... - David A. Corneth, Oct 20 2017

			a(10) = phi(10-7) = phi(3) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A001223, A049711.

for (n=1,100, print1(eulerphi(n-precprime(n-1))","))

first(n) = {n = nextprime(n); my(res = vector(n), phimap = Map(), q = 2, v); res[1] = res[2] = 1; forprime(p=3, n, if(!mapisdefined(phimap, p - q), mapput(phimap, p - q, vector(p - q, i, eulerphi(i)))); v = mapget(phimap, p-q); for(i = q + 1, p, res[i] = v[i - q]); q = p); res} \\ David A. Corneth, Oct 20 2017

For n >= 3, a(n) = A000010(A049711(n)). - Antti Karttunen, Oct 20 2017

Description clarified by Antti Karttunen, Oct 20 2017

A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 4, 11, 2, 1, 6

Offset: 1

R. J. Cano, Jan 17 2014

The primes in A013918 would have associated a(n)=0 if not for the qualifier "positive" in the definition.
The sum of the first n primes appears to be close to a prime. For illustration, the maximum for a(n) among the first 5 million terms is a(808500) = 218.
See A013916 for the above mentioned indices, numbers n such that the sum of the first n primes is prime. - M. F. Hasler, Jan 20 2014

			The sum of the first 9 primes is 100, and by adding 1 we get 101. Since 101 is a prime, a(9) = 1.
The sum of the first 10 primes is 129, since 129 - 2 = prime(31) = 127 or 129 + 2 = prime(32) = 131, a(10) = 2.
The sum of the first 129 primes minus 1 is a prime, this is 42468 - 1 = prime(4443), so a(129) = 1.

R. J. Cano, Table of n, a(n) for n = 1..10000

Cf. A007504, A101301.

a(n)=my(u=sum(j=1,n,prime(j)),k=1);while(!(isprime(u+k)||isprime(u-k)),k++);k

Algorithm:
Let S be the sum of the first n primes;
initially, let k=1;
increment k while neither S-k nor S+k is prime;
return a(n)=k.
a(n) = min(A013632(A007504(n)), A049711(A007504(n))). - M. F. Hasler, Jan 20 2014

A289357 Least number k such that n^2 + n - k is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 11, 1, 1, 13, 5, 1, 1, 1, 3, 5, 1, 3, 1, 5, 1, 7, 1, 1, 5, 5, 3, 1, 5, 7, 1, 1, 3, 1, 5, 3, 1, 1, 1, 5, 1, 3, 1, 5, 3, 1, 1, 1, 1, 5, 9, 1, 1, 3, 17, 5, 1, 1, 1, 7, 1, 13, 1, 5, 19, 3, 19, 7, 1, 19, 11, 3, 7, 5, 3, 1, 11, 1

Offset: 1

Gionata Neri, Jul 03 2017

nonn

a(A045546(n)) = 1.

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

Cf. A002378, A045546, A049711, A161550 (resulting primes).

0, seq(n^2 + n - prevprime(n^2+n), n=2..100); # Robert Israel, Jul 03 2017

Table[k = 0; While[! PrimeQ[n^2 + n - k], k++]; k, {n, 85}] (* Michael De Vlieger, Jul 04 2017 *)

for(n=1,85,k={my(k=0);while(!isprime(n^2+n-k),k++);k;};print1(k", "))

a(n) = A049711(A002378(n)). - Robert Israel, Jul 03 2017

A323741 a(n) = m-p where m = (2n+1)^2 and p is the largest prime < m.

Original entry on oeis.org

2, 2, 2, 2, 8, 2, 2, 6, 2, 2, 6, 6, 2, 2, 8, 2, 2, 2, 10, 12, 2, 8, 2, 2, 8, 6, 2, 20, 12, 2, 2, 6, 6, 2, 2, 6, 2, 2, 12, 8, 6, 6, 8, 2, 8, 2, 12, 6, 10, 8, 2, 22, 2, 14, 20, 6, 6, 2, 2, 2, 8, 6, 2, 8, 2, 6, 2, 12, 2, 14, 6, 2, 8, 8, 14, 10, 2, 18, 20, 2, 8, 14, 6, 2, 10, 2, 32, 2, 12, 12, 2, 8, 6, 44, 2, 6, 14, 6, 20, 14

Offset: 1

Ali Sada, Sep 03 2019

nonn

a(n) cannot be a square: suppose a(n) = k^2; then p=m-a(n) could be factored as (2n+k-1)*(2n-k-1); hence it would not be a prime.
Legendre's conjecture implies a(n) <= 4*n.  Oppermann's conjecture implies a(n) <= 2*n. - Robert Israel, Sep 04 2019
All terms are even. - Alois P. Heinz, Sep 04 2019

			When n=4, m=81, p=79, so a(4) = 81-79 = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A016754, A049711, A088572, A089166, A151799.

seq((2*n+1)^2-prevprime((2*n+1)^2),n=1..100); # Robert Israel, Sep 04 2019

mp[n_]:=Module[{m=(2n+1)^2},m-NextPrime[m,-1]]; Array[mp, 100] (* Harvey P. Dale, Feb 03 2022 *)

a(n) = (2*n+1)^2 - precprime((2*n+1)^2 - 1); \\ Michel Marcus, Sep 05 2019

a(n) = A049711(A016754(n)).

A033557 2n - {largest prime < n}.

Original entry on oeis.org

4, 5, 7, 7, 9, 9, 11, 13, 15, 13, 15, 15, 17, 19, 21, 19, 21, 21, 23, 25, 27, 25, 27, 29, 31, 33, 35, 31, 33, 33, 35, 37, 39, 41, 43, 39, 41, 43, 45, 43, 45, 45, 47, 49, 51, 49, 51, 53, 55, 57, 59, 55, 57, 59, 61, 63

Offset: 3

N. J. A. Sloane

nonn

A049711(n) + n.

f := n->2*n-prevprime(n); [ seq(f(i),i=3..60) ];

A309760 Even numbers k such that k-p is composite where p is the largest prime <= k.

Original entry on oeis.org

122, 148, 190, 208, 220, 250, 292, 302, 326, 346, 418, 430, 476, 518, 532, 538, 556, 586, 628, 640, 670, 700, 718, 782, 796, 806, 820, 838, 848, 872, 896, 902, 928, 962, 1006, 1030, 1048, 1060, 1078, 1084, 1138, 1144, 1150, 1162, 1180, 1210, 1246, 1258, 1268, 1274

Offset: 1

Aaron Schäfers, Aug 15 2019

nonn

Cf. A007917, A018252, A049711, A025583.

Select[Range[4, 1274, 2], CompositeQ[# - NextPrime[#, -1]] &] (* Amiram Eldar, Sep 17 2019 *)

is(n) = if(n%2==1 ||n==2, return(0)); my(c = n - precprime(n)); !isprime(c) && c != 1 \\ David A. Corneth, Aug 21 2019

A333839 a(n) = Sum_{k = 4..n} ((prevprime(k) + nextprime(k))/2 - k).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 2, 4, 5, 5, 4, 2, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 3, 6, 8, 9, 9, 8, 6

Offset: 4

Ctibor O. Zizka, Apr 07 2020

nonn

It looks like a(n) >= 0 for all n >= 4.
If (n,n+2) are twin primes, then a(n) = 0 and a(n+1) = 0.
Partial sums of b(k) = prevprime(k) + nextprime(k) - 2*k; b(k) = 0 for A145025.

			a(4) = (3 + 5)/2 - 4 = 0;
a(5) = (3 + 5)/2 - 4 + (3 + 7)/2 - 5 = 0.

Cf. A013632, A033559, A049711, A145025.

Array[Sum[Total[NextPrime[k, {-1, 1}]]/2 - k, {k, 4, #}] &, 92, 4] (* Michael De Vlieger, Apr 10 2020 *)

a(n) = Sum_{k = 4..n} (A013632(k) - A049711(k)) / 2.

cp's OEIS Frontend

A060360 a(n) = k - (largest prime < k), where k = lcm(1..n).

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A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

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A060849 Difference between 2 consecutive primes between which a nontrivial power of prime is found.

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A078772 a(n) = phi(n-p) where p is largest prime < n, a(1) = a(2) = 1 by convention.

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A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

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PARI

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A289357 Least number k such that n^2 + n - k is prime.

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A323741 a(n) = m-p where m = (2n+1)^2 and p is the largest prime < m.

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