A049711 -id:A049711 - cp's OEIS frontend

A049847 a(n) = (2n + 1 - prevprime(2n+1))/2.

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 5, 6, 7, 1, 2, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 4

Offset: 2

N. J. A. Sloane

nonn

Cf. A049613, A049653, A049711, A049716.

Table[(2n+1-NextPrime[2n+1,-1])/2,{n,2,100}] (* Harvey P. Dale, Jul 25 2015 *)

A336497 Numbers that cannot be written as a product of superfactorials A000178.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A336426 in having 360.

			The sequence of terms together with their prime indices begins:
     3: {2}        22: {1,5}        39: {2,6}
     5: {3}        23: {9}          40: {1,1,1,3}
     6: {1,2}      25: {3,3}        41: {13}
     7: {4}        26: {1,6}        42: {1,2,4}
     9: {2,2}      27: {2,2,2}      43: {14}
    10: {1,3}      28: {1,1,4}      44: {1,1,5}
    11: {5}        29: {10}         45: {2,2,3}
    13: {6}        30: {1,2,3}      46: {1,9}
    14: {1,4}      31: {11}         47: {15}
    15: {2,3}      33: {2,5}        49: {4,4}
    17: {7}        34: {1,7}        50: {1,3,3}
    18: {1,2,2}    35: {3,4}        51: {2,7}
    19: {8}        36: {1,1,2,2}    52: {1,1,6}
    20: {1,1,3}    37: {12}         53: {16}
    21: {2,4}      38: {1,8}        54: {1,2,2,2}

A093373 is the version for factorials, with complement A001013.
A336426 is the version for superprimorials, with complement A181818.
A336496 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178(n).
A174605 is the maximum prime multiplicity in A000178(n).
A303279 counts prime factors (with multiplicity) of superprimorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[100],facsusing[Rest[Array[supfac,30]],#]=={}&]

A060846 Smallest prime > the n-th nontrivial power of a prime.

Original entry on oeis.org

5, 11, 11, 17, 29, 29, 37, 53, 67, 83, 127, 127, 131, 173, 251, 257, 293, 347, 367, 521, 541, 631, 733, 853, 967, 1031, 1361, 1373, 1693, 1861, 2053, 2203, 2203, 2213, 2411, 2819, 3137, 3491, 3727, 4099, 4493, 4919, 5051, 5333, 6247, 6563, 6863, 6899, 7927

Offset: 1

Labos Elemer, May 03 2001

nonn

			78125=5^7 is followed by 78137.

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

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060845, A068435, A246547.

NextPrime[Select[Range[10^4], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Oct 04 2024 *)

ispp(x) = !isprime(x) && isprimepower(x);
lista(nn) = apply(x->nextprime(x), select(x->ispp(x), [1..nn])); \\ Michel Marcus, Aug 24 2019

from sympy import primepi, integer_nthroot, nextprime
def A060846(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 nextprime(bisection(f,n,n)) # Chai Wah Wu, Sep 15 2024

a(n) = nextprime(A025475(n+1)) = A007918(A025475(n+1)) = Min{p| p>A025475(n+1)}. [corrected by Michel Marcus, Aug 24 2019]

A175078 Number of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = n.

Original entry on oeis.org

0, 0, 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, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

a(123) = 3 (first occurrence of value 3), a(1357324) = 4 (first occurrence of value 4). I offer a prize of 100 liters of Pilsner Urquell to the discoverer of value of first occurrence of value 5. See A175071 (natural numbers m with result 1) and A175072 (natural numbers m with result 2). See A175077 = results 1 or 2 under iterations of {r mod (max prime p < r)} starting at r = n.
Essentially the same as A121561. [R. J. Mathar, Jan 28 2010]
The function r mod (max prime p < r), which appears in the definition, equals r - (max prime p < r) = A049711(r), because p < r < 2*p by Bertrand's postulate, where p is the largest prime less than r. - Pontus von Brömssen, Jul 31 2022

			a(123) = 3; iteration procedure for n = 123: 123 mod 113 = 10, 10 mod 7 = 3, 3 mod 2 = 1.

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

Cf. A049711, A064722, A121559, A121561, A175071, A175072, A175077.

Array[-1 + Length@ NestWhileList[Mod[#, NextPrime[#, -1]] &, #, Not[1 <= # <= 2] &, 1, 120] &, 105] (* Michael De Vlieger, Oct 30 2017 *)

A175078(n) = if(n<=2,0,1+A175078(n%precprime(n-1))); \\ Antti Karttunen, Oct 30 2017

a(n) = A121561(n-1) for n >= 2, because the functions that are iterated (A049711 here, A064722 in A121561) satisfies A049711(r) = A064722(r-1) + 1. - Pontus von Brömssen, Jul 31 2022

Name shortened by Antti Karttunen, Oct 30 2017

A202111 a(n) = sigma(n) - p, where p is the largest prime < sigma(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 7, 1, 1, 5, 1, 1, 11, 4, 4, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 7, 1, 14, 1, 5, 1, 13, 7, 5, 1, 2, 1, 1, 11, 1, 7, 1, 1, 5, 8, 13, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 4, 5, 6, 1, 5, 1, 11, 1, 5

Offset: 2

Michel Lagneau, Dec 11 2011

nonn

			a(12) = sigma(12) - 23 = 28 - 23 = 5.

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A000203, A049711, A070801, A072918.

A202111 := proc(n)
    A049711(numtheory[sigma](n) );
end proc:
seq(A202111(n),n=2..80) ; # R. J. Mathar, Dec 13 2011

f[n_]:=Module[{nf=DivisorSigma[1,n]},nf-NextPrime[nf,-1]];f/@Range[3,90]

A049711(n) = (n-precprime(n-1)); \\ This function from M. F. Hasler, Sep 09 2015
A202111(n) = A049711(sigma(n)); \\ Antti Karttunen, Nov 07 2017

a(n) = A049711(A000203(n)). - R. J. Mathar, Dec 13 2011

More terms from Antti Karttunen, Nov 07 2017

A367035 Numbers k such that the greatest prime less than 2*k is less than twice the greatest prime less than k.

Original entry on oeis.org

8, 14, 18, 20, 32, 33, 38, 39, 44, 48, 60, 61, 62, 63, 68, 72, 73, 74, 80, 81, 98, 102, 103, 104, 105, 108, 109, 110, 111, 128, 138, 140, 150, 151, 152, 153, 158, 164, 165, 168, 182, 183, 194, 198, 200, 212, 213, 214, 215, 224, 228, 230, 242, 243, 258, 259, 260, 264, 265, 266, 267, 268, 269, 270

Offset: 1

Robert Israel, Dec 15 2023

nonn

Numbers k such that A049711(2 * k) > 2 * A049711(k).

			a(3) = 18 is a term because the greatest prime < 18 is 17, the greatest prime < 2*18 = 36 is 31, and 31 < 2 * 17.

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

Includes k+1 for k in A053176. Disjoint from A006254.

Cf. A049711, A151799.

select(k -> prevprime(2*k) < 2*prevprime(k), [$3..300]);

isok(k) = precprime(2*k-1) < 2*precprime(k-1); \\ Michel Marcus, Dec 16 2023

A060845 Largest prime < a nontrivial power of a prime.

Original entry on oeis.org

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)}

A202090 a(n) = Fibonacci(n) - p, where p is the largest prime < Fibonacci(n).

Original entry on oeis.org

2, 1, 2, 2, 3, 2, 6, 5, 4, 4, 3, 4, 14, 5, 4, 2, 7, 4, 8, 17, 8, 14, 31, 14, 10, 37, 20, 26, 9, 20, 22, 11, 6, 12, 15, 32, 18, 17, 18, 16, 43, 24, 6, 17, 20, 26, 27, 20, 6, 9, 12, 34, 29, 36, 30, 47, 48, 4, 45, 32, 54, 27, 132, 22, 31, 4, 32, 11, 12, 60, 7, 76

Offset: 5

Michel Lagneau, Dec 11 2011

nonn

			a(7) = Fibonacci(7) - 19 = 21-19 = 2;
a(11) = Fibonacci(11) - 83 = 89 - 83 = 6.

Cf. A000045.

A049711 := proc(n)
    n-prevprime(n) ;
end proc:
A202090 := proc(n)
    A049711(combinat[fibonacci](n) );
end proc:
seq(A202090(n),n=5..80) ; # R. J. Mathar, Dec 13 2011

f[n_]:=Module[{nf=Fibonacci[n]},nf-NextPrime[nf,-1]];f/@Range[5,90]

a(n) = A049711(A000045(n)). - R. J. Mathar, Dec 13 2011

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;

A013635 a(n) = prevprime(n) + n.

Original entry on oeis.org

5, 7, 8, 11, 12, 15, 16, 17, 18, 23, 24, 27, 28, 29, 30, 35, 36, 39, 40, 41, 42, 47, 48, 49, 50, 51, 52, 59, 60, 63, 64, 65, 66, 67, 68, 75, 76, 77, 78, 83, 84, 87, 88, 89, 90, 95, 96, 97, 98, 99, 100, 107, 108, 109, 110, 111, 112, 119, 120, 123, 124, 125, 126

Offset: 3

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..5000

Cf. A007917, A049711.

[n + PreviousPrime(n): n in [3..70]]; // Vincenzo Librandi, Mar 30 2019

Maple
```
[ seq(prevprime(i)+i,i=3..100) ];
```

Table[n+NextPrime[n,-1],{n,3,60}] (* Harvey P. Dale, Mar 29 2019 *)

a(n) = precprime(n-1) + n; \\ Michel Marcus, Mar 30 2019

cp's OEIS Frontend

A049847 a(n) = (2*n + 1 - prevprime(2*n+1))/2.

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A336497 Numbers that cannot be written as a product of superfactorials A000178.

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A060846 Smallest prime > the n-th nontrivial power of a prime.

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A175078 Number of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = n.

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A202111 a(n) = sigma(n) - p, where p is the largest prime < sigma(n).

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A367035 Numbers k such that the greatest prime less than 2*k is less than twice the greatest prime less than k.

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A060845 Largest prime < a nontrivial power of a prime.

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A202090 a(n) = Fibonacci(n) - p, where p is the largest prime < Fibonacci(n).

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A049847 a(n) = (2n + 1 - prevprime(2n+1))/2.