A060270 -id:A060270 - cp's OEIS frontend

A340006 Number of times the n-th prime (=A000040(n)) occurs in A060270.

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 3, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 2, 2, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2

Offset: 1

A.H.M. Smeets, Dec 26 2020

nonn

Each term in A060270 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A060270.

By excluding the terms that equal one from A060270, we observe the smallest value of A060270(n)/log(A002110(n)) in the range n = 2..1000 to be ~1.014. From this it is believed that the primes less than 0.9*log(A002110(1001))*1.014 (~ 7138) will not occur anymore in the sequence A060270 for n > 1000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 7138 will no longer occur in A060270.

			The prime number 7 does not occur in A060270, and A000040(4) = 7, so a(4) = 0.
The prime number 11 occurs 1 time in A060270, and A000040(5) = 11, so a(5) = 1.

A.H.M. Smeets, Table of n, a(n) for n = 1..916
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A002110, A038711, A060270, A340007.

A038711 a(n) is the smallest m such that A002110(n) + m is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 17, 19, 23, 37, 61, 1, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307

Offset: 0

Labos Elemer, May 02 2000

nonn

Any composite a(n) would disprove Fortune's conjecture, see A005235. - Jeppe Stig Nielsen, Oct 31 2003

			For n=11, 1 + A002110(11) = 200560490131 < 200560490197 = 67 + A002110(11); therefore, a(11)=1 but A005235(11)=67.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A002110, A005235, A035345, A018239, A038710, A060270.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= n-> nextprime(p(n))-p(n):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 16 2020

nmax=2^16384; npd=1;n=1;npd=npd*Prime[n]; While[npdLei Zhou, Feb 15 2005 *)

a(n) = my(P=vecprod(primes(n))); nextprime(P+1) - P; \\ Michel Marcus, Dec 12 2023

a(n) = Min(1, A005235(n)); a(n)=1 for n=1, 2, 3, 4, 5, 11, 75, ...
a(n) = 1 for n=0, 1, 2, 3, 4, 5, 11, 75, ... (A014545); a(n) = A005235(n) otherwise. - Jeppe Stig Nielsen, Oct 31 2003
a(n) = A038710(n) - A002110(n). - Alois P. Heinz, Mar 16 2020

a(0)=1 prepended by Alois P. Heinz, Mar 16 2020

A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 11, 13, 1, 11, 1, 17, 1, 1, 13, 13, 1, 1, 17, 1, 17, 1, 1, 17, 17, 17, 1, 1, 19, 37, 37, 1, 17, 23, 1, 29, 1, 1, 19, 1, 19, 23, 1, 19, 31, 1, 19, 1, 1, 1, 1, 23, 1, 29, 23, 23, 1, 23, 71, 37, 1, 1, 31, 1, 23, 53, 1, 31

Offset: 1

Bill McEachen, May 01 2006

a) Conjecture: entries > 1 will always be prime. The entry will be larger than the largest prime factor of the highly composite number.
b) Will 1 always be the most common entry?
c) While a prime may always be located close to each highly composite number, is the converse false?
d) Is there always a prime between successive highly composite numbers?
From Antti Karttunen, Feb 26 2019: (Start)
The second sentence of point (a) follows as both gcd(n, A151799(n)) = 1 and gcd(A151800(n), n) = 1 for all n > 2 and the fact that the highly composite numbers are products of primorials, A002110 (with the least coprime prime > the largest prime factor). See also the conjectures and notes in A129912 and A141345. (End)

			a(5) = abs(12-11) = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..19999
Bill McEachen, Alternate plot, Wikimedia Commons.
Graeme McRae, Highly Composite Numbers.
Wikipedia, Highly Composite Numbers.
Wikipedia, Divisor Function (sigma).

Cf. A002100, A002182, A007917, A129912, A141345, A151800, A324385.
Sequences tied to conjecture a): A228943, A228945.
Cf. also A005235, A060270.

With[{s = DivisorSigma[0, Range[Product[Prime@ i, {i, 8}]]]}, Map[If[PrimeQ@ #, 0, Min[# - NextPrime[#, -1], NextPrime[#] - #]] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Mar 11 2019 *)

A141345(n) = (nextprime(1+A002182(n))-A002182(n));
A324385(n) = (A002182(n)-precprime(A002182(n)));
A117825(n) = if(1==n,1,min(A141345(n), A324385(n))); \\ Antti Karttunen, Feb 26 2019

a(1) = 1; for n > 1, a(n) = min(A141345(n), A324385(n)). - Antti Karttunen, Feb 26 2019

More terms from Don Reble, May 02 2006

A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 17, 1, 1, 1, 13, 11, 11, 19, 17, 13, 1, 23, 1, 1, 13, 17, 17, 13, 17, 1, 17, 1, 1, 23, 17, 17, 17, 1, 19, 83, 37, 23, 17, 23, 1, 43, 19, 1, 19, 43, 19, 31, 23, 19, 31, 19, 19, 1, 1, 1, 1, 47, 1, 31, 47, 23, 53, 23, 83, 37, 31, 1, 31, 1, 23, 61, 1, 41, 47, 61, 41, 29, 41, 29, 43, 73, 29, 47, 31, 31

Offset: 2

Antti Karttunen, Feb 26 2019

nonn

Like in A141345 it appears (or is conjectured) that no composite numbers ever occur here. Taken together, this leads to McEachen's conjecture given in A117825. Here in range 2..10000 term 1 occurs for 313 times.

The arithmetic mean of a(n)/log(A002182(n)) for the terms 3..10000 is 1.513, i.e., a rough approximation is given by a(n) ~ log(A002182(n)^(3/2)). - A.H.M. Smeets, Dec 02 2020

			A002182(2) = 2, the largest prime <= 2 is 2 itself, thus a(2) = 2-2 = 0.
A002182(7) = 36, the largest prime <= 36 is 31, thus a(7) = 36-31 = 5.

Antti Karttunen, Table of n, a(n) for n = 2..10000

Cf. A002182, A007917, A060270, A117825, A141345.

With[{s = Array[DivisorSigma[0, #] &, 10^6]}, {0}~Join~Map[# - NextPrime[#, -1] &@ FirstPosition[s, #][[1]] &, Drop[Union@ FoldList[Max, s], 2]]] (* or *)
{0}~Join~Map[# - NextPrime[#, -1] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 97, -1]] ] (* Michael De Vlieger, Dec 11 2020 *)

A324385(n) = (A002182(n)-precprime(A002182(n)));

a(n) = A002182(n) - A007917(A002182(n)).

A340007 Number of times the n-th prime (=A000040(n)) occurs in A038711.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 1, 2, 0, 2, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 4, 0, 0, 0, 1, 0, 0, 1, 1, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 1

Offset: 1

A.H.M. Smeets, Dec 26 2020

nonn

Each term in A038711 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A038711.

By excluding the terms that equal one from A038711, we observe the smallest value of A038711(n)/log(A002110(n)) in the range n = 2..1000 to be ~1.017. From this it is believed that the primes less than 0.9*log(A002110(1001))*1.017 (~ 7157) will not occur anymore in the sequence A038711 for n > 1000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 7157 will no longer occur in A038711.

			The prime number 17 occurs 1 time in A038711, and A000040(7) = 17, so a(7) = 1.
The prime number 5 does not occur in A038711, and A000040(3) = 5, so a(3) = 0.

A.H.M. Smeets, Table of n, a(n) for n = 1..916
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A002110, A038711, A060270, A340006.

A268480 Integers k such that A002110(k) is the average of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 8, 38, 40, 64, 73, 89, 236, 480, 486

Offset: 1

Altug Alkan, Mar 21 2016

In other words, the primorial numbers that are considered are those of the form (p + q)/2 where p and q are consecutive primes. Note that the initial values of (p - q)/2 are 1, 1, 1, 23, 239, 191, 331, 373, 1021.
A088256 is a subsequence of these primorials, which in turn are a subsequence of A024675.
Numbers k such that A038711(k) = A060270(k). - Amiram Eldar, May 19 2024

			5 is a term because 2*3*5*7*11 = 2310 = (2309 + 2311)/2.
8 is a term because 2*3*5*7*11*13*17*19 = 9699690 = (9699667 + 9699713)/2.

Index entries for sequences related to primorial numbers.

Cf. A002110, A024675, A038711, A060270, A088256.

P:= 2: count:= 0:
for n from 2 to 500 do
  P:= P*ithprime(n);
  # first try d=1
  if isprime(P+1) then
    good:= isprime(P-1);
  elif isprime(P-1) then good:= false
  else
    for d from ithprime(n+1) by 2 do
      if igcd(d,P) > 1 then next fi;
      if isprime(P+d) then
        good:= isprime(P-d); break
      elif isprime(P-d) then
        good:= false; break
      fi
    od;
  fi;
  if good then
     count:= count+1;
     A[count]:= n;
  fi
od:
seq(A[i],i=1..count);  # Robert Israel, Aug 29 2016

prim[n_] := Times @@ Prime[Range[n]]; Select[Range[2, 100], Total[NextPrime[(p = prim[#]), {-1, 1}]] == 2*p &] (* Amiram Eldar, May 19 2024 *)

a002110(n) = prod(k=1, n, prime(k));
for(n=2, 1e3, if((nextprime(a002110(n)) - a002110(n)) == (a002110(n) - precprime(a002110(n))), print1(n, ", ")))

A340041 The prime gap, divided by two, which surrounds p#.

Original entry on oeis.org

1, 1, 6, 1, 9, 24, 23, 40, 51, 37, 60, 36, 68, 87, 66, 84, 99, 95, 115, 88, 117, 143, 51, 177, 182, 168, 139, 243, 221, 193, 204, 516, 260, 154, 182, 306, 239, 216, 191, 211, 303, 263, 672, 303, 615, 417, 312, 378, 275, 375, 322, 445, 312, 294, 354, 492, 399, 348, 461

Offset: 2

Robert G. Wilson v, Jan 22 2021

nonn

If p and q are consecutive primes, we say here that there is a gap of q-p. (Other sequences use different definitions of "gap".) - N. J. A. Sloane, Mar 07 2021

Records: 1, 6, 9, 24, 40, 51, 60, 68, 87, 99, 115, 117, 143, 177, 182, 243, 516, 672, 855, 915, 925, 1100, 1139, 1620, 1863, 2272, 2842, 4177, 4190, 5025, 5692, 6254, 6413, 6879, 7914, 8026, 9928, 10604, ..., .

			For a(1), there are two contiguous primes {2, 3} with 2 being 2#. The prime gap is 1. However, the two primes do not surround 2#, so a(1) like A340013(2) is undefined.
For a(2), the prime gap contains {5, 6, 7}, with 3# = 6 in the middle. The prime gap is 2, therefore a(2) = 1;
For a(3), the prime gap contains {29, 30, 31}, with 5# = 30  in the middle. The prime gap is 2, therefore a(3) = 1.
For a(4), the prime gap contains {199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211}, with 7# =  205 in the middle. The prime gap is 12, therefore a(4) = 6. etc.

Cf. A006862, A007014, A038711, A060270, A340013 (analog for n!).

a[n_] := Block[{p = Times @@ Prime@ Range@ n}, (NextPrime[p, 1] - NextPrime[p, -1])/2]; a[1] = 0; Array[a, 60]

a(n) = (A006862(n) - A007014(n))/2 = (A038711(n) + A060270(n))/2.

a(n) = A058044(n)/2. - Hugo Pfoertner, Jan 22 2021

cp's OEIS Frontend

A340006 Number of times the n-th prime (=A000040(n)) occurs in A060270.

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A038711 a(n) is the smallest m such that A002110(n) + m is prime.

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A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.

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A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).

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A340007 Number of times the n-th prime (=A000040(n)) occurs in A038711.

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A268480 Integers k such that A002110(k) is the average of two consecutive primes.

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A340041 The prime gap, divided by two, which surrounds p#.

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