A051701 -id:A051701 - cp's OEIS frontend

A227878 Primes occurring twice in A051701.

3, 19, 41, 43, 71, 83, 101, 109, 151, 167, 199, 227, 229, 257, 281, 283, 311, 313, 349, 383, 401, 443, 461, 463, 487, 503, 571, 601, 617, 641, 643, 677, 727, 757, 829, 857, 859, 881, 883, 911, 937, 941, 971, 1033, 1063, 1091, 1093, 1123, 1187, 1217, 1231

Offset: 1

Reinhard Zumkeller, Oct 25 2013

nonn

Consider five consecutive primes p1..p5: (p3 is a term of this sequence) iff (p2-p1 > p3-p2 and p5-p4 > p4-p3).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a227878 n = a227878_list !! (n-1)
a227878_list = f a051701_list where
   f (p:ps@(:p':)) = if p == p' then p : f ps else f ps

A053607 Primes p such that a pure prime power occurs between p and the next prime.

Original entry on oeis.org

3, 7, 13, 23, 31, 47, 61, 79, 113, 127, 167, 241, 251, 283, 337, 359, 509, 523, 619, 727, 839, 953, 1021, 1327, 1367, 1669, 1847, 2039, 2179, 2207, 2399, 2803, 3121, 3469, 3719, 4093, 4483, 4909, 5039, 5323, 6229, 6553, 6857, 6883, 7919, 8191, 9403, 10193

Offset: 1

Labos Elemer, Feb 09 2000

nonn

			127 belongs here because 128 = 2^7 occurs between 127 and 131.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A051701, A025475.

lim = 10^4; pwrs = Sort[Flatten[Table[Prime[n]^i, {n, 1, PrimePi[Sqrt[lim]]}, {i, 2, Log[Prime[n], lim]}]]]; Union[NextPrime[pwrs, -1]] (* T. D. Noe, Mar 11 2013 *)

A116946 Closest semiprime to n-th semiprime S that is different from S (break ties by taking the smaller semiprime).

Original entry on oeis.org

6, 4, 10, 9, 15, 14, 22, 21, 26, 25, 34, 33, 34, 39, 38, 49, 51, 49, 57, 58, 57, 65, 62, 65, 77, 74, 85, 86, 85, 86, 93, 94, 93, 94, 111, 115, 118, 119, 118, 122, 121, 122, 133, 134, 133, 142, 141, 142, 146, 145, 158, 159, 158, 159, 169, 166, 178, 177, 185

Offset: 1

Jonathan Vos Post, Mar 25 2006

			Closest semiprimes to 4, 6, 9, 10, 14, 15, 21 are 6, 4, 10, 9, 15, 14, 22.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A051701 (analog for primes).

csp[{a_,b_,c_}]:=If[c-bHarvey P. Dale, May 06 2014 *)

from sympy import factorint
def semiprime(n): return sum(e for e in factorint(n).values()) == 2
def nextsemiprime(n):
  nxt = n + 1
  while not semiprime(nxt): nxt += 1
  return nxt
def aupton(terms):
  prv, cur, nxt, alst = 0, 4, 6, []
  while len(alst) < terms:
    alst.append(prv if 2*cur - prv <= nxt else nxt)
    prv, cur, nxt = cur, nxt, nextsemiprime(nxt)
  return alst
print(aupton(59)) # Michael S. Branicky, Jun 04 2021

Corrected by Harvey P. Dale, May 06 2014

A053708 Nearest prime to n! (but not equal to n!).

Original entry on oeis.org

2, 3, 5, 23, 113, 719, 5039, 40343, 362867, 3628789, 39916801, 479001599, 6227020777, 87178291199, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709439969, 1124000727777607680031, 25852016738884976639911

Offset: 1

Labos Elemer, Feb 10 2000

nonn

If n! is the average of its closest prime neighbors then the smaller prime is to be chosen (as in A051701).

			For 8! = 40320 the closest upper and lower primes are 40289 and 40343 with d = 31 and d = 23, so 40343 is closer to 8! than the lower neighbor.

Amiram Eldar, Table of n, a(n) for n = 1..449

Cf. A000142, A033932, A037151, A033933, A006990.

f[n_]:=Module[{nf=n!,s,l},s=NextPrime[nf,-1];l=NextPrime[nf];If[nf-s>l-nf,l,s]]
Table[f[i],{i,25}] (* Harvey P. Dale, Dec 08 2010 *)

Corrected by Rick L. Shepherd, Jan 11 2006

a(21)-a(23) from Amiram Eldar, Mar 10 2025

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A227878 Primes occurring twice in A051701.

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A053607 Primes p such that a pure prime power occurs between p and the next prime.

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A116946 Closest semiprime to n-th semiprime S that is different from S (break ties by taking the smaller semiprime).

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A053708 Nearest prime to n! (but not equal to n!).

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Mathematica

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