A067836 -id:A067836 - cp's OEIS frontend

A062894 The prime indices of sequence A067836 (that sequence is conjectured to contain only primes).

1, 2, 3, 4, 6, 5, 7, 8, 9, 12, 21, 10, 11, 14, 22, 16, 23, 19, 13, 15, 41, 35, 42, 27, 20, 17, 45, 39, 29, 33, 28, 54, 26, 37, 46, 61, 47, 59, 40, 31, 57, 18, 24, 58, 36, 43, 49, 64, 80, 106, 67, 60, 65, 63, 48, 94, 85, 51, 73, 122, 105, 100, 113, 138, 104, 115, 75, 32, 82

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Feb 13 2002

nonn

			a(12)=10 because a(12) in sequence A067836 is 29, which is the 10th prime number.

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A067836.

Join[{1}, a = 2; f = 1; Table[f = f*a; a = NextPrime[f + 1] - f; k = 1; While[Prime[k] != a, k++]; k, {n, 2, 69}]] (* Jayanta Basu, Aug 10 2013 *)

from sympy import primepi, nextprime
def A062894_gen(): # generator of terms
    a, f = 2, 1
    yield 1
    while True:
        yield primepi(a:=nextprime((f:=f*a)+1)-f)
A062894_list = list(islice(A062894_gen(),30)) # Chai Wah Wu, Sep 09 2023

Edited by Dean Hickerson, Jun 10 2002

A215463 Number of primes less than B(n) that are missing in the first n terms of Buss's function, A067836.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 10, 9, 8, 7, 7, 6, 6, 5, 4, 3, 20, 19, 19, 18, 17, 16, 18, 17, 16, 15, 14, 22, 21, 20, 19, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 16, 31, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 62, 61, 60, 59, 74, 73, 72

Offset: 1

Robert Price, Mar 09 2013

nonn

			a(10)=2 since B(10)=37, and the two primes 29 and 31 have not yet appeared in the Buss's function sequence, A067836.

Robert Price, Table of n, a(n) for n = 1..1000

A067949 Smallest primes not generated by Buss's function up to n=603 (see A067836).

Original entry on oeis.org

503, 677, 811, 911, 997, 1163, 1193, 1303, 1367, 1459, 1493, 1567, 1613, 2027, 2111, 2113, 2207, 2213, 2287, 2357, 2371, 2381, 2383, 2393, 2467, 2473, 2477, 2617, 2659, 2663, 2693, 2713, 2801, 2851, 2857, 2861, 2917, 2969, 2999, 3019, 3037, 3089, 3109, 3163

Offset: 0

Felice Russo, Mar 05 2002

503, 811, 911, 1193, 1367, 1567 remain after n = 1000. - Robert Price, Mar 09 2013

Cf. A067836.

More terms from Sean A. Irvine, Jan 13 2024

A071290 The sequence f(1), f(2), ... as defined in A067836.

Original entry on oeis.org

1, 2, 6, 30, 210, 2730, 30030, 510510, 9699690, 223092870, 8254436190, 602573841870, 17474641414230, 541713883841130, 23293697005168590, 1840202063408318610, 97530709360640886330, 8095048876933193565390, 542368274754523968881130, 22237099264935482724126330

Offset: 1

Dean Hickerson, Jun 10 2002

nonn

Cf. A067836.

Join[{f = 1}, a = 2; Table[f = f*a; a = NextPrime[f + 1] - f; f, {n, 2, 20}]] (* Jayanta Basu, Aug 10 2013 *)

# Generate f(n) from B(n) sequence using the b067836.txt file:
perl -nE '/(\d+)\s+(\d+)/; $f[$n] = $f[$n-1]*$2; say "$n $f[$n]"; $n++; BEGIN { use Math::GMP qw/:constant/; $f[1]=1; $n=2; say "1 1"; }' b067836.txt
#  Dana Jacobsen, Feb 15 2015

Two more terms from Jayanta Basu, Aug 10 2013

A215464 Smallest prime not generated by Buss's function (A067836) through B(n).

Original entry on oeis.org

3, 5, 7, 11, 11, 17, 19, 23, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 47, 59, 59, 59, 59, 59, 59, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 89, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97

Offset: 1

Robert Price, Mar 09 2013

nonn

			a(5)=11 since B(5)=13, but the prime 11 has not yet appeared in the Buss's function sequence.

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A067836, A062894, A067949, A215463.

A364824 Index of prime(n) in A067836, or -1 if prime(n) does not occur in it.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 12, 13, 10, 19, 14, 20, 16, 26, 42, 18, 25, 11, 15, 17, 43, 118, 33, 24, 31, 29, 212, 40, 68, 30, 98, 22, 45, 34, 109, 28, 39, 21, 23, 46, 143, 27, 35, 37, 55, 47, 123, 58, 90, 132, 32, 139, 91, 41, 44, 38, 52, 36, 77, 54, 48, 53, 83, 51

Offset: 1

Bert Dobbelaere, Aug 09 2023

nonn

All terms in A067836 are distinct, making this sequence defined.
If the conjecture holds that A067836 contains only primes, then a(A062894(n)) = n.
If all primes eventually occur in A067836, then all terms in this sequence are positive and A062894(a(n)) = n.

Cf. A062894, A067836.

from itertools import count
from sympy import prime, nextprime
def A364824(n):
    a, f, p = 2, 1, prime(n)
    for i in count(1):
        if a == p:
            return i
        a=nextprime((f:=f*a)+1)-f # Chai Wah Wu, Sep 09 2023

A068192 Let a(1)=2, f(n) = 4a(1)a(2)...a(n-1) for n >= 1 and a(n) = f(n)-prevprime(f(n)-1) for n >= 2, where prevprime(x) is the largest prime < x.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 29, 23, 41, 43, 37, 89, 59, 53, 67, 79, 71, 137, 109, 239, 167, 199, 47, 83, 97, 61, 373, 101, 179, 193, 131, 151, 73, 263, 593, 139, 113, 157, 103, 241, 181, 397, 233, 617, 311, 191, 229, 271, 269, 127, 223, 331, 337, 211, 163

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Feb 19 2002

nonn

The terms are easily seen to be distinct. It is conjectured that every element is prime. Do all primes occur in the sequence?

First 1000 terms are primes. - Mauro Fiorentini, Aug 01 2020

Mauro Fiorentini, Table of n, a(n) for n = 1..1000

Cf. A068193 has the indices of the primes in this sequence. A066631 has the sequence of f's. Also see A067836.

Mathematica
```
<
				
```

f := 4:for n from 1 to 50 do a := f-numlib::prevprime(f-2):f := f*a:print(a) end_for

Edited by Dean Hickerson, Jun 10 2002

A217724 Smallest prime producing a gap with the next prime, the size of the gap being a composite number with 2n+1 as a factor.

Original entry on oeis.org

139, 1831, 1129, 2971, 1327, 19333, 81463, 19609, 44293, 89689, 173359, 212701, 265621, 544279, 1100977, 396733, 1098847, 370261, 1349533, 6752623, 5518687, 6371401, 10343761, 2010733, 4652353, 33803689, 83751121, 38394127, 39389989, 79167733, 142414669

Offset: 1

Robert Price, Mar 21 2013

nonn

The "gap" is defined as the number of integers between two primes. If the two primes are p1 and p2, then the size of the gap is p2-p1-1.

For at least n<=100, the gap is minimal and is 3*(2n+1).

			For n=2, the required gap is a composite number with 5 (2n+1) as a factor.  The size of all gaps are odd, so gaps of 15, 25, 35, etc. are required.  The prime 1831 and its next prime of 1847 produces a gap of 15.  1831 is the smallest prime with this property.

Robert Price, Table of n, a(n) for n = 1..100

Cf. A067836.

A259256 With a(1) = 1, a(n) is the smallest number not already in the sequence such that a(n) + Product_{i=1..n-1} a(i) is a square.

Original entry on oeis.org

1, 3, 6, 7, 18, 36, 148, 5625, 351225, 5350321, 151875880681, 247160867363588025, 126888381222131340236953809, 592938336545755964751256254689753896569

Offset: 1

Derek Orr, Jun 22 2015

Giovanni Resta, Table of n, a(n) for n = 1..22 (terms < 10^1000)

Cf. A259255, A094353, A067836.

L = {1}; While[Length[L] < 22, p = Times @@ L; q = Ceiling[Sqrt[p + 1]]; While[ MemberQ[L, q^2 - p], q++]; AppendTo[L, q^2 - p]]; L (* Giovanni Resta, Jun 22 2015 *)

v=[1];n=1;while(n<10^7,s=n+prod(i=1,#v,v[i]);if(issquare(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v

a(11)-a(14) from Giovanni Resta, Jun 22 2015

A372607 Let a(1) = 2, f(n) = a(1)a(2)...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 13, 25, 17

Offset: 1

Davide Rotondo, May 07 2024

Conjecture: every element is a ludic number.

This is the analog of Buss' conjecture (cf. A067836) for ludic numbers instead of primes, and similar to the idea of ludic Fortunate numbers (A376237) in analogy to the usual Fortunate numbers A005235. - M. F. Hasler, Nov 04 2024

Cf. A067836, A003309 (ludic numbers), A376237 (ludic Fortunate numbers).

A372607_upto(n=15, f=1)=vector(n,i,n=if(i>1, next_A003309(1+f*=n)-f,2)) \\ M. F. Hasler, Nov 02 2024

cp's OEIS Frontend

A062894 The prime indices of sequence A067836 (that sequence is conjectured to contain only primes).

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A215463 Number of primes less than B(n) that are missing in the first n terms of Buss's function, A067836.

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A067949 Smallest primes not generated by Buss's function up to n=603 (see A067836).

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A071290 The sequence f(1), f(2), ... as defined in A067836.

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Perl

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A215464 Smallest prime not generated by Buss's function (A067836) through B(n).

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A364824 Index of prime(n) in A067836, or -1 if prime(n) does not occur in it.

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Python

A068192 Let a(1)=2, f(n) = 4*a(1)*a(2)*...*a(n-1) for n >= 1 and a(n) = f(n)-prevprime(f(n)-1) for n >= 2, where prevprime(x) is the largest prime < x.

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A217724 Smallest prime producing a gap with the next prime, the size of the gap being a composite number with 2n+1 as a factor.

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A259256 With a(1) = 1, a(n) is the smallest number not already in the sequence such that a(n) + Product_{i=1..n-1} a(i) is a square.

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PARI

Extensions

A372607 Let a(1) = 2, f(n) = a(1)*a(2)*...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

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A068192 Let a(1)=2, f(n) = 4a(1)a(2)...a(n-1) for n >= 1 and a(n) = f(n)-prevprime(f(n)-1) for n >= 2, where prevprime(x) is the largest prime < x.

A372607 Let a(1) = 2, f(n) = a(1)a(2)...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.