Daniel D Gibson - cp's OEIS frontend

A380450 Number of integers k such that prime(n) - primorial(k) is prime.

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

Offset: 1

Daniel D Gibson, Jun 22 2025

nonn

Conjecture A: Each value occurs an infinite number of times in the sequence.

Conjecture B: All natural numbers occur in the sequence.

			For prime(n=6): 13 - 2 = 11, and 13 - 6 = 7, so a(6) = 2.

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A385210, A000040, A002110, A175974 (zeros (primes)), A115785 (record positions (primes)).

a[n_]:=Module[{c=0},Do[d=Prime[n]-Fold[Times, 1, Prime[Range[k-1]]];If[PrimeQ[d]&&d>0,c++],{k,n}];c];Array[a,90] (* James C. McMahon, Jun 27 2025 *)

pri(n) = vecprod(primes(n)); \\ A002110
a(n) = my(nb=0, p=prime(n)); for (k=0, n, if (isprime(p-pri(k)), nb++); ); nb; \\ Michel Marcus, Jun 22 2025

More terms from Michel Marcus, Jun 22 2025

A385210 Number of integers k such that prime(n) + primorial(k) is prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 2, 5, 5, 4, 5, 8, 3, 5, 4, 7, 7, 8, 7, 8, 8, 7, 6, 5, 11, 8, 9, 8, 3, 6, 6, 5, 3, 7, 10, 10, 7, 8, 9, 5, 6, 7, 8, 6, 8, 6, 12, 5, 11, 10, 14, 8, 7, 8, 8, 7, 6, 6, 9, 9, 11, 8, 10, 10, 9, 12, 8, 8, 8, 6, 9, 11, 11, 7, 13, 5, 11, 5, 9, 10, 9, 9, 7, 8

Offset: 1

Daniel D Gibson, Jun 21 2025

nonn

			For prime(n=3): 5 + 2 = 7, 5 + 6 = 11, and 5 + any higher primorial will be composite, so a(3) = 2.

Cf. A000040, A002110.

nn = 120; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn]]]; Table[q = Prime[n]; Total@ Array[Boole@ PrimeQ[q + P[# - 1]] &, n], {n, nn}] (* Michael De Vlieger, Jun 22 2025 *)

pri(n) = vecprod(primes(n)); \\ A002110
a(n) = my(nb=0, p=prime(n)); for (k=0, n-1, if (isprime(p+pri(k)), nb++);); nb; \\ Michel Marcus, Jun 21 2025

More terms from Michel Marcus, Jun 21 2025

A378018 Primes p which can be written as p = (A060735(k) +- next largest prime factor not in A060735(k)) for some k.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 53, 67, 83, 97, 113, 127, 157, 173, 199, 409, 431, 619, 641, 829, 1039, 1061, 1249, 1459, 1481, 1669, 1879, 1901, 2089, 2111, 2297, 6917, 9227, 13873, 16183, 18493, 23087, 25423, 27733, 30013, 30047, 60077, 90073

Offset: 1

Daniel D Gibson, Nov 14 2024

nonn

A060735 can be thought of as multiples of primorials, up to the next prime not found in the given primorial. This sequence adds or subtracts that next prime to produce new prime values.
31 is the first prime this does not produce (other than 2).
143 is the first nonprime value that this pattern produces (other than values < 3).
Conjectured to be infinite.

			23 is a term because 23 = 2*3*5 - 7 and it is prime.
67 is a term because 67 = 2*2*3*5 + 7 and it is prime.

Cf. A060735, subset of A000040, superset of A367182, superset of A038708.

rad(n) = factorback(factorint(n)[, 1]);
lista(nn) = my(a=1, list = List()); for (n=2, nn, my(f = factor(a), p = if (a==1, 2, nextprime(vecmax(f[,1])+1))); if (isprime(a-p), listput(list, a-p)); if (isprime(a+p), listput(list, a+p)); a = a + rad(a);); vecsort(Set(list)); \\ Michel Marcus, Dec 14 2024

(not in order) primorial(i) * m +- prime(i+1) where 0

A377887 a(n) is the number of ways of writing prime(n) as k-q with q a prime and k a primorial.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 3, 3, 3, 1, 3, 4, 4, 5, 4, 5, 3, 4, 4, 5, 4, 4, 7, 8, 6, 5, 7, 4, 5, 9, 5, 6, 5, 9, 7, 4, 5, 8, 5, 8, 6, 7, 5, 9, 6, 5, 7, 6, 4, 3, 9, 6, 5, 12, 8, 5, 6, 7, 7, 6, 9, 8, 7, 13, 4, 8, 4, 6, 6, 7, 4, 7, 11, 5, 8, 8, 9, 6, 7, 7, 6, 12, 4, 10, 11, 11, 7, 8, 11

Offset: 1

Daniel D Gibson, Nov 10 2024

nonn

Except at n=2, the largest k which must be considered is the product of the first n-1 primes, since if k includes p=prime(n) itself then q = k-p is divisible by p and so not prime.

			For n=4, p = prime(4) = 7 can be written as p = 30 - 23, where 30 is a primorial and 23 is prime, but no other primorials satisfy this condition, so a(3) = 1.

Cf. A002110, A175933.

a[n_] := Count[FoldList[Times, Prime[Range[n - 1]]] - Prime[n], ?(# > 0 && PrimeQ[#] &)]; a[2] = 1; Array[a, 100] (* _Amiram Eldar, Nov 14 2024 *)

a(n)={my(p=prime(n), s=0, t=1); forprime(q=2, p, t*=q; if(t>p && isprime(t-p), s++)); s} \\ Andrew Howroyd, Nov 14 2024

A377884 Composite numbers k without prime factors that are divisors of the greatest primorial less than k.

Original entry on oeis.org

25, 49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781, 793, 799

Offset: 1

Daniel D Gibson, Nov 10 2024

nonn

			The greatest primorial below 25 is 6, the factors of which are 2 and 3, neither of which are factors of 25=5^2.

Intersection of A002808 and A335284.

Cf. A002110, A260188.

q[k_] := Module[{p = 2, r = 2}, If[CompositeQ[k], While[r < k && ! Divisible[k, p], p = NextPrime[p]; r *= p]; r >= k, False]]; Select[Range[800], q] (* Amiram Eldar, Nov 14 2024 *)

P(n)=my(t=1, k); forprime(p=2, , k=t*p; if(k>n, return(t), t=k)); \\ A260188
isok(k) = (k>1) && !isprime(k) && (gcd(k, P(k-1)) == 1); \\ Michel Marcus, Nov 12 2024

A367182 Primes of the form (k-th primorial) - (k+1)st prime.

Original entry on oeis.org

23, 199, 2297, 30013, 9699667, 3217644767340672907899084554047, 267064515689275851355624017992701, 23768741896345550770650537601358213, 1492182350939279320058875736615841068547583863326864530259

Offset: 1

Daniel D Gibson, Nov 08 2023

nonn

Conjecture: sequence is infinite.

			primorial(4) - prime(4+1) = 2*3*5*7 - prime(5) = 210 - 11 = 199, which is prime, so 199 is a term.

Cf. A002110, A093078, A249798 (corresponding k's).
The prime numbers in A060882.
A038708 with subtraction instead of addition.

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User: Daniel D Gibson

A380450 Number of integers k such that prime(n) - primorial(k) is prime.

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A385210 Number of integers k such that prime(n) + primorial(k) is prime.

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A378018 Primes p which can be written as p = (A060735(k) +- next largest prime factor not in A060735(k)) for some k.

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A377887 a(n) is the number of ways of writing prime(n) as k-q with q a prime and k a primorial.

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A377884 Composite numbers k without prime factors that are divisors of the greatest primorial less than k.

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A367182 Primes of the form (k-th primorial) - (k+1)st prime.

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