A005235 -id:A005235 - cp's OEIS frontend

A216418 Fortunate numbers with increasing multiplicity.

3, 23, 61, 331, 2837, 18169

Offset: 1

Charles R Greathouse IV, Sep 06 2012

a(n) appears in A005235 more often than any preceding term.

Is this sequence infinite?

			3 is the first term of A005235, so a(1) = 3. A005235(5) = A005235(8) = 23, while smaller primes appear only once, so a(2) = 23.

Cf. A005235, A046066.

ct(p)=my(pr=1,s);forprime(q=2,p-1,if(!ispseudoprime((pr*=q)+p), next);forprime(r=q+1,p-1,if(ispseudoprime(pr+r),next(2)));s++);s
r=0;forprime(p=2,2e3,t=ct(p);if(t>r,r=t;print1(p", ")))

a(6) from Charles R Greathouse IV, Sep 17 2012

A226525 Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

Original entry on oeis.org

2, 2, 2, 5, 2, 3, 2, 7, 3, 19, 11, 3, 23, 5, 61, 29, 31, 3, 29, 31, 13, 19, 5, 7, 23, 47, 3, 53, 47, 19, 13, 7, 41, 53, 2, 43, 7, 103, 2, 61, 59, 71, 17, 59, 79, 43, 167, 71, 97, 7, 151, 37, 103, 83, 127, 103, 11, 53, 29, 7, 67, 83, 151, 107, 37

Offset: 1

Sean A. Irvine, Jonathan Vos Post and Robert G. Wilson v, Jun 09 2013

This is the semiprime analogous to A005235.
Just like the Fortunate primes (A005235) we conjecture that all terms are prime!
If instead of semiprimorials (A112141), we use primorials the sequence would be {2, 3, 3, 3, 5, 1, 5, 1, 1, 3, 4, 4, 11, 1, 4, 7, 4, 1, 1, 23, 1, 29, 1, 9, 32, 1, 71, 31, 4, 32, 23, 5, 125, 1, 97, 1, 11, 7, 27, 1, 29, 61, 11}; not very interesting.
If instead of m > 1, we start with m > 0, we identify the semiprimorials +1 which are semiprimes by index: 2, 3, 8, 9, 15, 16, 19, 21, 23, 27, 29, ....
A112141(66) + 197 has been completed through the 2030 curve.

			a(3) = 3 since sp(3) = 4*6*9*10 = 2160 and the least number greater than the fourth semiprimorial which is the semiprime is 2165 = 5*433. Therefore the difference is a(3) which equals 3.
A112141(47) + a(47) = 24011725937636436154291480954413133199 * 68433092684820794078956407785220072996675433.
A112141(55) + a(55) = 795251036594717254131632161591406578993 * 650971642564884068706166933685477027845256102005635827825839.
A112141(63) + a(63) = 168586841653003537 * 40363128914158968243564625304355041082304983807201816858670871447070744 83558441664261096307889392423.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Cyril Banderier, Fortune's conjecture (Fortunate and unfortunate primes: Nearest primes from a prime factorial)

Cf. A005235.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{k = 1, m = Times @@ NestList[ NextSemiPrime, 2^2, n-1]}, While[ PrimeOmega[m + k] != 2, k++]; k]; Do[ Print@ f[n], {n, 50}]

The difference between the n-th semiprimorial and the next semiprime greater than that semiprimorial plus 1.

A248714 a(n) = p - prime(n)#^2, where prime(n)# is the product of the first n primes and p is the smallest prime > prime(n)#^2 + 1.

Original entry on oeis.org

3, 5, 7, 11, 17, 29, 23, 41, 29, 37, 89, 79, 89, 71, 439, 389, 163, 79, 151, 73, 89, 211, 113, 113, 419, 167, 139, 199, 173, 137, 487, 197, 401, 167, 739, 641, 461, 199, 223, 331, 379, 401, 293, 223, 251, 647, 593, 613, 317, 271, 257, 947, 331, 347, 593, 433

Offset: 1

Werner D. Sand, Oct 12 2014

nonn

Conjecture: Analogous to Fortune's Conjecture (A005235) all a(n) are prime, so are all members of a(n)=p-k*prime(n)#, k=natural number.

Besides, many powers p-prime(n)#^m, m=natural number, behave as well, e.g. p-prime(n)#^29 does, p-prime(n)#^30 does not.

Cf. A002110, A005235, A037153.

q:=1;p:=1;for i from 1 to 100 do q:=nextprime(q+1);p:=p*q;N:=nextprime(p^2+2)-p^2;print(i,N);end_for: \\ Werner D. Sand, Oct 13 2014

a(n) = {hp = prod(ip=1, n, prime(ip)); nextprime(hp^2+2) - hp^2;} \\ Michel Marcus, Oct 12 2014

A249241 a(n) = p - prime(n)!/prime(n)#, where p is the smallest prime number > prime(n)!/prime(n)#+1.

Original entry on oeis.org

2, 2, 3, 5, 11, 7, 29, 17, 17, 397, 47, 67, 23, 41, 31, 157, 409, 31, 151, 109, 199, 191, 131, 61, 103, 547, 179, 269, 389, 317, 181, 331, 307, 173, 1259, 1289, 619, 131, 223, 683, 139, 241, 191, 101, 1039, 1367, 1153, 241, 1187, 479, 149, 181, 487, 1093, 571, 1151, 809, 199, 823, 491, 191, 151, 1321, 197, 163, 337, 467, 659, 673, 877, 487, 743, 313, 673, 857, 677, 1021

Offset: 1

Werner D. Sand, Oct 23 2014

nonn

Conjecture: All terms are prime.

While Fortune's conjecture (A005235) uses products of primes, this sequence uses products of composite numbers (more exactly: of nonprimes, because 1 belongs to them). It looks like all multiples of prime(n)# (except some powers) lead to a sequence which contains only prime numbers.

			n = 1; prime(1)!/prime(1)# = 2/2 = 1; p = nextprime(1+1) = 3; a(1) = 3-1 = 2.

Cf. A092435.

q:=1; p:=1; for i from 1 to 100 do q:=nextprime(q+1); p:=p*q; N:=nextprime((fact(q)/p)+2)-fact(q)/p; print(i,N); end_for:

A092435(n)=prime(n)!/prod(i=1,n,prime(i))
a(n)=my(t=A092435(n)); nextprime(t+2)-t \\ Charles R Greathouse IV, Oct 23 2014

A305099 Least prime m such that either prime(n)# - m is prime or prime(n)# + m is prime, where p# denotes the product of all primes <= p.

Original entry on oeis.org

3, 3, 7, 11, 13, 17, 19, 23, 37, 41, 67, 59, 47, 47, 67, 59, 61, 89, 89, 103, 79, 83, 89, 97, 103, 131, 113, 127, 223, 191, 163, 179, 389, 239, 151, 167, 173, 239, 199, 191, 199, 223, 233, 593, 293, 457, 227, 311, 373, 257, 307, 313, 283, 277, 271, 307, 307

Offset: 1

David Nicolas Lopez, May 22 2018

nonn

Since it is known that the first 2000 terms of A005235 are primes, and the first 1000 terms of A055211 are primes, then the first 1000 terms of this sequence are also the least m > 1 such that prime(n)# - m is prime or prime(n)# + m is prime. - Amiram Eldar, Nov 02 2018

			For n = 6, the sixth primorial is 30030. The nearest prime such that p(6)# plus or minus prime equals its 30030's closest prime is equal to 17 because 30030+17=30047 which is prime or 30030 - 17 = 30013 which is also prime. Given that we only care about the smallest prime distance to the closest prime to the primorial, then we return 17.
For n = 7, the seventh primorial is 510510. The closest prime to the primorial is 510529 which is 510510 + 19; therefore 19 is in the sequence.

Martin Gardner, The Last Recreations (1997), pp. 194-195.
R. K. Guy, Unsolved Problems in Number Theory, Section A2
Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.

S. W. Golomb, Evidence of Fortunes conjecture, Mathematics Magazine, Vol. 54, No. 4 (Sep., 1981), pp. 209-210.
The Prime Glossary, Fortunate and lesser fortunate numbers

Cf. A005235, A055211.

primorial[n_] := Product[Prime[i], {i, 1, n}]; a[n_] := Module[{k = 2, pr = primorial[n]}, While[! PrimeQ[pr - k] && ! PrimeQ[pr + k], k = NextPrime[k]]; k]; Array[a, 57] (* Amiram Eldar, Oct 31 2018 *)

a(n) = { my(pr = prod(k=1, n, prime(k)), m=2); while (!isprime(pr-m) && !isprime(pr+m), m = nextprime(m+1)); m;} \\ Michel Marcus, Nov 02 2018

# returns quasi-fortunate-numbers up to n
def generateQFN(n):
    quasi_fortunate_numbers = []
    primorialArray = []
    prime = Primes()
    num_length = n+1
    primorial = 1
    for i in range(num_length):
        primorial *= prime[i]
        primorialArray.append(primorial)
    for primorials in primorialArray:
        num = 0
        while num < num_length:
            if is_prime(primorials+prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            elif is_prime(primorials-prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            num += 1
    return quasi_fortunate_numbers
generateQFN(7)

a(n) = min(A005235(n), A055211(n)), for n > 1.

More terms from Amiram Eldar, Oct 31 2018

A326228 Primes p such that m - p#/6 or p#/6 - n is not in A002822 (twin ranks), where m (resp. n) is the next larger (resp. smaller) twin rank w.r.t. p#/6, and # = A034386 (primorial).

Original entry on oeis.org

41, 227, 307, 311, 349, 457, 613

Offset: 1

M. F. Hasler, Jun 24 2019

Related to the "Twin Fortune Conjecture" (A. Dinculescu) which states that the distance between p#/6 and the next larger or smaller n in A002822 (twin rank, such that 6n +- 1 are twin primes) is again a twin rank; very similar to Fortune's Conjecture, cf. A005235.

For a(1) = 41, the non twin rank is p#/6 - n, for all other terms listed here, it is m - p#/6. However, in these cases, the other distance is a twin rank. For all other primes, both distances are twin ranks.

Cf. A002822 (twin ranks), A034386 (primorial), A005235 (Fortunate numbers).

is(p)={ my(m=A034386(p)/6,n=m); until(is_A002822(n-=1),); (is_A002822(m-n) || ((n=m) && !until(is_A002822(m+=1),) && is_A002822(m-n))) && isprime(p)}

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

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A216418 Fortunate numbers with increasing multiplicity.

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A226525 Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

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A248714 a(n) = p - prime(n)#^2, where prime(n)# is the product of the first n primes and p is the smallest prime > prime(n)#^2 + 1.

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A249241 a(n) = p - prime(n)!/prime(n)#, where p is the smallest prime number > prime(n)!/prime(n)#+1.

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A305099 Least prime m such that either prime(n)# - m is prime or prime(n)# + m is prime, where p# denotes the product of all primes <= p.

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A326228 Primes p such that m - p#/6 or p#/6 - n is not in A002822 (twin ranks), where m (resp. n) is the next larger (resp. smaller) twin rank w.r.t. p#/6, and # = A034386 (primorial).

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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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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.