A154598 -id:A154598 - cp's OEIS frontend

A154704 a(n) = smallest number k such that k-1 and k+1 both have exactly n prime divisors (counted with multiplicity).

4, 5, 19, 55, 271, 1889, 10529, 59777, 101249, 406783, 6581249, 12164095, 65071999, 652963841, 6548416001, 13858918399, 145046192129, 75389157377, 943344975871, 23114453401601, 108772434771967, 101249475018751, 551785225781249, 9740041658826751, 136182187711004671, 4560483868737535

Offset: 1

Klaus Brockhaus, Jan 14 2009, Jan 15 2009

nonn

Similar to A154598, where k is restricted to primes.

m=2*a(n) is the least number m such that m-2 and m+2 have exactly n+1 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024

			For k = 4, k-1 = 3 and k+1 = 5 (twin primes) both have one factor and 4 is the smallest such number.
For k = 55, k-1 = 54 = 2*3*3*3 and k+1 = 56 = 2*2*2*7 both have four factors and 55 is the smallest such number.
For k = 59777, k-1 = 59776 = 2*2*2*2*2*2*2*467 and k+1 = 59778 = 2*3*3*3*3*3*3*41 both have eight factors and 59777 is the smallest such number.

David A. Corneth, Table of n, a(n) for n = 1..36
David A. Corneth, Upper bounds on a(n) for n = 1..89

Cf. A115186, A154598.

Cf. A001222 (number of prime divisors of n).

S:=[]; for n:=1 to 10 do k:=3; while not &+[ f[2]: f in Factorization(k-1) ] eq n or not &+[ f[2]: f in Factorization(k+1) ] eq n do k+:=1; end while; Append(~S, k); end for; S;

a[n_]:=Module[{k=2}, While[PrimeOmega[k-1]!=n || PrimeOmega[k+1]!=n, k++]; k]; Array[a,26] (* Stefano Spezia, Apr 02 2024 *)
Flatten[Table[Position[Partition[PrimeOmega[Range[410000]],3,1],?(#[[1]]==#[[3]]==n&),1,1],{n,10}]]+1//Quiet (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Jul 21 2024 *)
Table[SequencePosition[PrimeOmega[Range[410000]],{n,,n},1],{n,10}][[;;,1,1]]+1 (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Aug 29 2024 *)

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q+2) == k, listput(list, m*q+1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Aug 12 2023

a(n) = 2*A115186(n-1) + 1 for n > 1. - Hugo Pfoertner, Apr 02 2024

a(15)-a(23) from Donovan Johnson, Jan 21 2009

a(24)-a(26) from Daniel Suteu, Aug 12 2023

A155800 a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.

Original entry on oeis.org

2, 0, 3, 5, 7, 19, 17, 31, 97, 191, 127, 449, 769, 3329, 6143, 7937, 12799, 51199, 8191, 165887, 65537, 131071, 1179649, 2654209, 7995391, 524287, 10616831, 12910591, 167772161, 113246209, 169869311, 155189249, 1887436799, 3221225473

Offset: 1

J. M. Bergot, Jan 27 2009

nonn

Prime divisors are counted with multiplicity.

			For p=2, the adjacent numbers 1 and 3 together have one prime divisor, hence a(1) = 2. For p=3, the adjacent numbers 2 and 4 together have three prime divisors, hence a(3) = 3. For primes greater than 3, the adjacent numbers are composite and therefore together have at least four prime divisors, so no prime exists whose neighbors together have two prime divisors. Hence a(2) = 0.
For p = 19, p-1 = 18 = 2*3*3 and p+1 = 20 = 2*2*5 together have six prime divisors. All smaller primes are surrounded by numbers which together have fewer or more than six (for 17 there are seven) prime divisors, hence a(6) = 19.

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..46 (terms < 10^13, first 40 terms from Donovan Johnson)

Cf. A001222 (number of prime divisors of n), A154598, A155850, A154704.

Cf. comments in A239669.

Join[{2,0},With[{m=SortBy[{PrimeOmega[#-1]+PrimeOmega[#+1],#}&/@Prime[ Range[200000]],First]},Transpose[Flatten[Table[Select[m,#[[1]]==n&,1],{n,3,24}],1]][[2]]]] (* Harvey P. Dale, Sep 24 2013 *)

N=10^7; default(primelimit, N); M=26;
/* M must be determined empirically for each N. Program-generated zeros other than a(2) appearing as terms when N and M are increased must be regarded as provisional */
for(m=1,M,t=0;forprime(p=1,N,if(bigomega(p-1)+bigomega(p+1)==m,t=1;print1(p", ");break));if(t==0,print1("0, "))) \\ Chris Boyd, Mar 24 2014

Edited, 1151 replaced by 769, 3457 replaced by 3329, extended beyond a(14) by Klaus Brockhaus, Jan 28 2009
a(29)-a(32) from Klaus Brockhaus, Jan 30 2009
a(33)-a(34) from Donovan Johnson, Aug 03 2009

A155850 a(n) = smallest k > 1 such that k-1 and k+1 together have n prime divisors.

Original entry on oeis.org

2, 4, 3, 5, 7, 15, 17, 31, 65, 129, 127, 449, 511, 2561, 1025, 7937, 12799, 20481, 8191, 28673, 65537, 131071, 458751, 360449, 966655, 524287, 4194303, 2097151, 29360129, 34865153, 67108865, 134217729, 33554431, 608174081, 268435457, 536870911, 4831838207

Offset: 1

Klaus Brockhaus, Jan 28 2009, Jan 31 2009

nonn

Prime divisors are counted with multiplicity.
Similar to A155800, where k is restricted to primes.
Terms of the form 2^m-1 or 2^m+1 seem to occur frequently.

			Adjacent to 2 are the numbers 1 and 3 which together have one prime divisor, hence a(1) = 2. Adjacent to 3 are 2 and 4; together they have three prime divisors, hence a(3) = 3. Adjacent to 4 are the primes 3 and 5, each having one prime divisor; hence a(2) = 4.
For k = 129, k-1 = 128 = 2*2*2*2*2*2*2 and k+1 = 130 = 2*5*13 together have ten prime divisors. For all numbers k < 129 the adjacent numbers k-1 and k+1 together have fewer or more than ten (for 127 there are eleven) prime divisors, hence a(10) = 129.

Cf. A001222 (number of prime divisors of n), A155800, A154704, A154598.

{for(n=2, 150000000, s=bigomega(n-1)+bigomega(n+1); if(v[s]==0, v[s]=n)); v}

a(34)-a(37) from Donovan Johnson, Nov 02 2013

A247114 Primes sandwiched between 4-almost primes (A014613).

Original entry on oeis.org

89, 151, 197, 233, 307, 349, 461, 491, 569, 571, 739, 857, 859, 1013, 1061, 1097, 1277, 1291, 1303, 1483, 1667, 1747, 1831, 1913, 1973, 2003, 2131, 2357, 2503, 2531, 2621, 2683, 3011, 3067, 3163, 3209, 3229, 3259, 3271, 3581, 3797, 3929, 4013, 4027, 4073, 4219, 4327, 4597, 4793, 4877, 4903

Offset: 1

Zak Seidov, Jan 10 2015

nonn

Primes p such that p - 1 and p + 1 are 4-almost primes.

			89 - 1 = 2^3*11, 89 + 1 = 2*3^2*5.

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

Cf. A014612, A014613, A046314, A154598, A063644, A247088.

Select[Prime[Range[1000]], 4 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &]

forprime(p= 1,5000, if(4==bigomega(p-1)&&4==bigomega(p+1), print1(p", ")))

is(n)=bigomega(n-1)==4 && bigomega(n+1)==4 && isprime(n) \\ Charles R Greathouse IV, Apr 27 2015

cp's OEIS Frontend

A154704 a(n) = smallest number k such that k-1 and k+1 both have exactly n prime divisors (counted with multiplicity).

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A155800 a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.

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A155850 a(n) = smallest k > 1 such that k-1 and k+1 together have n prime divisors.

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A247114 Primes sandwiched between 4-almost primes (A014613).

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