A013632 -id:A013632 - cp's OEIS frontend

A060845 Largest prime < a nontrivial power of a prime.

3, 7, 7, 13, 23, 23, 31, 47, 61, 79, 113, 113, 127, 167, 241, 251, 283, 337, 359, 509, 523, 619, 727, 839, 953, 1021, 1327, 1367, 1669, 1847, 2039, 2179, 2179, 2207, 2399, 2803, 3121, 3469, 3719, 4093, 4483, 4909, 5039, 5323, 6229, 6553, 6857, 6883, 7919

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 follows 78121

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

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060846.

Take[NextPrime[#,-1]&/@Union[Flatten[Table[Prime[p]^n,{n,2,20},{p,25}]]], 50] (* Harvey P. Dale, Mar 26 2012 *)

{ m=1; for (n=1, 1000, m++; while(sigma(m)*eulerphi(m)*(1 - isprime(m)) <= (m - 1)^2, m++); write("b060845.txt", n, " ", precprime(m - 1)); ) } \\ Harry J. Smith, Jul 19 2009

from sympy import primepi, integer_nthroot, prevprime
def A060845(n):
    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
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return prevprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = prevprime[A025475(n)] = A007917[A025475(n)] = Max{p| p < A025475(n)}

A072917 a(n) = p(n) - phi(n), where p(n) is the least prime greater than phi(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 5, 3, 1, 5, 3, 5, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 5, 1, 1, 5, 3, 1, 3, 1, 1, 5, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1

Offset: 1

Joseph L. Pe, Aug 11 2002

			phi(15) = 8 and the least prime > 8 is 11; hence a(15) = 11 - 8 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A013632, A072344.

a[n_] := Module[{r, p}, p = EulerPhi[n]; r = p + 1; While[ ! PrimeQ[r], r = r + 1]; r - p]; Table[a[i], {i, 1, 100}]
lpg[n_]:=Module[{ep=EulerPhi[n]},NextPrime[ep]-ep]; Array[lpg,200] (* Harvey P. Dale, May 29 2017 *)

A072917(n) = (nextprime(1+eulerphi(n)) - eulerphi(n)); \\ Antti Karttunen, Aug 22 2017

a(n) = A013632(A000010(n)). - Antti Karttunen, Aug 22 2017

A093701 a(n) = smallest m>a(n-1) such that 1+m*n is prime, a(1) = 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 10, 16, 17, 18, 19, 30, 31, 34, 35, 36, 37, 38, 41, 58, 59, 62, 64, 72, 73, 76, 77, 80, 81, 84, 85, 88, 95, 96, 97, 102, 103, 106, 111, 114, 118, 122, 123, 124, 125, 130, 132, 134, 135, 138, 140, 142, 144, 150, 152, 156, 158, 164, 166, 174, 175

Offset: 1

Reinhard Zumkeller, Apr 10 2004

nonn

A093702(n) = 1+a(n)*n.

			For n=3: we have a(2)=2, so we want the smallest number m > 2 such that n*m+1 = 3*m+1 is prime. m=3 fails but m=4 works, so a(3) = m = 4. - _N. J. A. Sloane_, Dec 13 2018

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

Cf. A013632, A093702.

Note that A081942 is a related but distinct sequence.

nxt[{n_,a_}]:=Module[{m=a+1},While[!PrimeQ[m(n+1)+1],m++];{n+1,m}]; NestList[ nxt,{1,1},60][[All,2]] (* Harvey P. Dale, Dec 13 2018 *)

A101597 Number of consecutive composite numbers between balanced primes and their lower or upper prime neighbor.

Original entry on oeis.org

1, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 11, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 11, 11, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Cino Hilliard, Jan 26 2005

nonn

These numbers are not always prime with 35 occurring for prime(n) n<1000000.

The first 35 occurs at a(947). - Antti Karttunen, Dec 16 2017

			53 has the 5 consecutive composites 48,49,50,51,52 below it and the 5 consecutive composites 54,55,56,57,58 above it so 5 is in the second position in the table.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A006562, A013632.

Flatten[Differences /@ Select[Partition[Prime@ Range[1900], 3, 1], #2 == Mean@ {#1, #3} & @@ # &][[All, 1 ;; 2]] - 1] (* Michael De Vlieger, Dec 16 2017 *)

betwixtpr(n) = { local(c1,c2,x,y); for(x=2,n, c1=c2=0; for(y=prime(x-1)+1,prime(x)-1, if(!isprime(y),c1++); ); for(y=prime(x)+1,prime(x+1)-1, if(!isprime(y),c2++); ); if(c1==c2,print1(c1",")) ) }

up_to = 10000; n = 0; forprime(p=1, oo, if((d=(p-precprime(p-1)))==(nextprime(p+1)-p), n++; write("b101597.txt", n, " ", d-1); if(n>=up_to,break))); \\ Antti Karttunen, Dec 16 2017

a(n) = A013632(A006562(n))-1. - Antti Karttunen, Dec 16 2017

Offset changed from 2 to 1 by Antti Karttunen, Dec 16 2017

A123317 Smallest prime power m such that n+m is a prime number.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 8, 5, 4, 3, 2, 1, 2, 1, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 32, 5, 4, 3, 2, 1, 8, 5, 4, 3, 2, 1, 2, 1, 256, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 128, 5, 4, 3, 2, 1, 8, 7, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1

Offset: 1

Reinhard Zumkeller, Sep 27 2006

nonn

			n=23: 23+1=3*2^3, 23+2=5^2, 23+3=13*2, 23+2^2=3^3, 23+5=7*2^2, 23+7=5*3*2, but 23+8=31=A000040(11), therefore a(23)=8;
n=24: 24+1=5^2, 24+2=13*2, 24+3=3^3, 24+2^2=7*2^2, but 24+5=29=A000040(10), therefore a(24)=5;
the smallest occurring proper odd prime power is 9=3^2:
n=118: 118+1=17*7, 118+2=5*3*2^3, 118+3=11^2, 118+2^2=61*2, 118+5=41*3, 118+7=5^3, 118+2^3=7*2*3^2, but 118+3^2=127=A000040(31), therefore a(118)=9.

Cf. A013632, A000961.

A123317 := proc(n)
    local m ;
    m :=1 ;
    if isprime(n+m) then
        return m ;
    end if;
    for m from 2 do
        if nops(numtheory[factorset](m)) = 1 then
            if isprime(n+m) then
                return m;
            end if;
        end if;
    end do:
end proc:
seq(A123317(n),n=1..102) ; # R. J. Mathar, Aug 09 2019

A123318(n) = n + a(n);

a(A006093(n)) = 1; a(A040976(n)) = 2 for n>2.

A204669 Primes p such that q-p = 62, where q is the next prime after p.

Original entry on oeis.org

34061, 190409, 248909, 295601, 305147, 313409, 473027, 479639, 531731, 633497, 682079, 693881, 724331, 777479, 877469, 896201, 1011827, 1088309, 1137341, 1152527, 1179047, 1181777, 1190081, 1210289, 1216619, 1226117, 1272749, 1281587, 1286711, 1305449, 1343801, 1345361, 1357361, 1464179

Offset: 1

N. J. A. Sloane, Jan 17 2012

nonn

All terms == 5 mod 6. - Zak Seidov, Jan 01 2013
There are no two consecutive primes in the sequence, while there are such primes p=prime(m) that q=prime(m+2) is also a term.
First such p's are at indices 554, 908, 1902, 2588, 3035, 5320, 6213, 6881, 7853, 8262, which correspond to 10237391, 15442121, 27374771, 36040469, 41216027, 66544301, 76313597, 83565611, 93112589, 97515359 (respectively). Note that a(554) = 10237391 = A226657(31). - Zak Seidov, Jul 01 2015
Primes p such that A013632(p) = 62. - Robert Israel, Jul 02 2015

M. F. Hasler, Table of n, a(n) for n = 1..8496 (all terms up to 10^8).
Index entries for primes, gaps between

Cf. A013632, A226657.

[n: n in [2..2*10^6 ] | (NextPrime(n)-NextPrime(n-1)) eq 62]; // Vincenzo Librandi, Jul 02 2015

p:= 2:
count:= 0:
while count < 40 do
  q:= nextprime(p);
  if q - p = 62 then
    count:= count+1;
    A[count]:= p;
  fi;
  p:= q;
od:
seq(A[i],i=1..count); # Robert Israel, Jul 02 2015

Select[Prime@ Range@ 120000, NextPrime@ # - # == 62 &] (* Michael De Vlieger, Jul 01 2015 *)
Select[Partition[Prime[Range[120000]],2,1],#[[2]]-#[[1]]==62&][[All,1]] (* Harvey P. Dale, Apr 01 2017 *)

g=62;c=o=0;forprime(p=1,default(primelimit),(-o+o=p)==g&write("c:/temp/b204669.txt",c++" "p-g))  \\ M. F. Hasler, Jan 18 2012

A226381 Numbers n such that the distance from n to the next prime is the same as the distance from n^2 to the next prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 15, 16, 21, 31, 36, 37, 38, 39, 40, 45, 48, 50, 57, 61, 64, 66, 67, 76, 81, 85, 91, 97, 99, 103, 105, 111, 126, 130, 131, 141, 147, 150, 151, 154, 156, 163, 168, 171, 180, 181, 185, 193, 202, 207, 210, 216, 225, 235, 237, 240, 246, 248, 249, 250, 253

Offset: 1

Gerasimov Sergey, Jun 05 2013

nonn

Numbers n such that (smallest prime > n)- n = (smallest prime > n^2)- n^2.

Primes in the sequence are: 2, 3, 7, 13, 31, 37, 61, 67, 97, 103, 131, 151, 163, 181, 193,...

			1 is in the sequence because the distance from 1 to 2 is the same as the distance from 1^2 to 2.
2 is in the sequence because the distance from 2 to 3 is the same as the distance from 2^2 to 5.
3 is in the sequence because the distance from 3 to 5 is the same as the distance from 3^2 to 11.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[Range[235], NextPrime[#] - # == NextPrime[#^2] - #^2 &] (* Giovanni Resta, Jun 09 2013 *)

is(n)=nextprime(n+1)-n==nextprime(n^2)-n^2 \\ Charles R Greathouse IV, Jun 14 2013

{n: A013632(n) = A013632(n^2)}. - R. J. Mathar, Jun 09 2013

Corrected by Giovanni Resta, Jun 09 2013

A338570 Primes p such that q*r mod p is prime, where q is the prime preceding p and r is the prime following p.

Original entry on oeis.org

11, 13, 19, 29, 31, 37, 47, 53, 59, 67, 73, 83, 89, 109, 127, 131, 151, 163, 173, 179, 211, 239, 251, 263, 269, 283, 307, 337, 359, 373, 383, 421, 433, 443, 449, 467, 479, 499, 503, 523, 541, 547, 569, 593, 599, 607, 653, 659, 677, 757, 787, 797, 829, 853, 877, 907, 919, 947, 967, 971, 977, 1033

Offset: 1

J. M. Bergot and Robert Israel, Nov 02 2020

nonn

Primes p such that -A049711(p)*A013632(p) mod p is prime.

Includes primes p such that p-8, p-2 and p+4 are also prime. Dickson's conjecture implies that there are infinitely many of these.

			a(3) = 19 is a member because 19 is prime, the previous and following primes are 17 and 23, and (17*23) mod 19 = 11 is prime.

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

Cf. A013632, A049711, A338566, A338567.

R:= NULL: p:= 0: q:= 2: r:= 3:
count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if isprime(p*r mod q) then count:= count+1; R:= R, q;  fi;
od:
R;

A338578 Primes p such that (r-p)*(r-q) > r, where q and r are the next two primes.

Original entry on oeis.org

2, 3, 5, 11, 17, 19, 29, 43, 47, 83, 109, 199, 283

Offset: 1

Robert Israel, Nov 03 2020

a(14) > 10^8 if it exists.

As soon as the prime gap grows slow enough, for all large enough p we have (r-p)*(r-q) <= r, implying finiteness of this sequence. In particular, finiteness would follow from Cramer's conjecture. - Max Alekseyev, Nov 09 2024

			a(5)=17 is a member because it is prime, the next two primes are 19 and 23, and (23-17)*(23-19)=24 > 23.

Contains A338566.

Cf. A013632, A105161, A338577.

p:= 0: q:=2:r:= 3:  R:= NULL:
while p < 10^4 do
  p:= q: q:= r: r:= nextprime(r);
  if (r-q)*(r-p) > r then R:= R, p; fi
od:
R;

A058018 Difference between LCM(1,...,x) and the smallest prime > LCM(1,...,x), where x is the n-th prime power (A000961).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 13, 1, 13, 31, 23, 19, 1, 41, 1, 31, 43, 1, 41, 53, 79, 59, 1, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 1, 229, 109, 139, 113, 227, 131, 191, 163, 1, 199, 151, 139, 1, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233, 283, 229, 277, 263

Offset: 1

Labos Elemer, Nov 14 2000

nonn

The first value corresponds to x = 1, LCM(1) = 1.
For the first 100 prime powers, the value is either prime or 1.
The values of x are taken to be prime powers so that each distinct LCM occurs exactly once.

			The 6th distinct prime power is A000961(7) = 8, LCM(1,...,8) = 840 and 853 is the first prime that follows, thus a(7) = 853-840 = 13.

Cf. A000961, A003418, A013632, A051451, A058017.

With[{max = 250}, (NextPrime[#] - #)& /@ Exp[Accumulate[Join[{0}, Select[Array[MangoldtLambda, max], # > 0 &]]]]] (* Amiram Eldar, Aug 13 2024 *)

lista(nn) = {for (n=1, nn, if ((n==1) || isprimepower(n), v = lcm(vector(n, x, x)); print1(nextprime(v+1) - v, ", ")););} \\ Michel Marcus, Apr 09 2015

a(n) = A013632(A051451(n)) = A058017(n) - A051451(n). - Amiram Eldar, Aug 13 2024

Edited by Franklin T. Adams-Watters, Aug 15 2006
Offset set to 1 by Michel Marcus, Apr 09 2015
Name corrected by Amiram Eldar, Aug 13 2024

cp's OEIS Frontend

A060845 Largest prime < a nontrivial power of a prime.

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A072917 a(n) = p(n) - phi(n), where p(n) is the least prime greater than phi(n).

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A093701 a(n) = smallest m>a(n-1) such that 1+m*n is prime, a(1) = 1.

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A101597 Number of consecutive composite numbers between balanced primes and their lower or upper prime neighbor.

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Extensions

A123317 Smallest prime power m such that n+m is a prime number.

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Maple

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A204669 Primes p such that q-p = 62, where q is the next prime after p.

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Magma

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A226381 Numbers n such that the distance from n to the next prime is the same as the distance from n^2 to the next prime.

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