A037153 -id:A037153 - cp's OEIS frontend

A105049 a(n) is the smallest prime p such that p+n! is prime and p differs from all a(i), i

2, 3, 5, 7, 11, 13, 19, 23, 17, 41, 29, 43, 67, 71, 149, 73, 31, 37, 89, 53, 47, 127, 97, 131, 107, 59, 137, 101, 223, 163, 241, 79, 139, 103, 61, 577, 311, 151, 269, 173, 211, 167, 109, 191, 239, 521, 233, 83, 383, 337, 271, 827, 449, 443, 229, 157, 179, 283, 293, 277

Offset: 1

R. J. Mathar, Aug 22 2007

nonn

A variant of A130807.

If the requirement is dropped that a(n) be distinct from earlier primes in the list, the sequence becomes 2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 67, 19, 43, 23, 31, 37, 89,.. which presumably duplicates A037153.

A105049 := proc(nmin) local i,a,iused; a := [] ; iused := {} ; for n from 1 to nmin do i := 1; while not isprime(ithprime(i)+n!) or i in iused do i := i+1 ; od; iused := iused union {i} ; a := [op(a),ithprime(i)] ; od ; RETURN(a) ; end: A105049(80);

A082432 a(n) = p - A072181(n), where p is the least prime > A072181(n) + 1.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 13, 13, 13, 13, 13, 17, 17, 17, 23, 59, 47, 41, 23, 23, 23, 83, 293, 383, 383, 103, 563, 107, 107, 71, 1399, 1399, 1399, 1399, 2803, 983, 983, 983, 10589, 5693, 5693, 19553, 827, 31699, 31699, 33001, 12193

Offset: 1

Naohiro Nomoto, Apr 25 2003

Is a(n) always prime?

			a(4) = 17 - A072181(4) = 17 - 12 = 5.

Cf. A005235, A055211, A037153, A037155, A072181.

a(36)-a(47) from Iain Fox, Nov 23 2017

a(48) from Iain Fox, Nov 29 2017

A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

Original entry on oeis.org

2, 2, 3, 7, 8, 15, 8, 18, 16, 19, 12, 20, 11, 8, 11, 6, 12, 23, 24, 15, 31, 21, 27, 15, 16, 26, 25, 17, 17, 29, 20, 27, 27, 30, 23, 16, 28, 23, 25, 29, 15, 24, 19, 36, 36, 39, 15, 36, 24, 44, 35, 29, 27, 25, 36, 22, 37, 31, 32, 41, 29, 55, 27, 45, 29, 59, 34, 37, 24, 49, 25, 40

Offset: 0

Frederick Magata (frederick.magata(AT)web.de), May 13 2009

nonn

The conjectures from A037153 and A087202 can be rephrased using a(n):
Is a(n)>=2 for all n>=0 and a(n)>=3 for all n>=2?
Also compare this with the conjecture on the fortunate numbers A005235.
Is the following true: for every m there is an N such that for all n>N a(n)>m?
There even seems to be the estimate a(n)>log(n+1)*sqrt(n+1)/2.

			a(3)=7: The seven primes following 3!+1=7 are 11,13,17,19,23,29 and 31.
Subtracting 3!=6 from each of them gives 5,7,11,13,17,23 and 25.
The first six values are prime, while the seventh 25=5^2 is not.

Cf. A005235, A037153, A087202.

a:=proc(n) option remember; local k:
for k from 1 while isprime((nextprime@@k)(n!+1)-n!) do od:
k; end;

A385430 Least number k such that k and k + n! have the same number of divisors.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 46, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 62, 41, 59, 47, 67, 159, 107, 127, 79, 37, 97, 61, 131, 86, 43, 97, 53, 61, 97, 71, 47, 94, 101, 233, 53, 83, 61, 249, 53, 71, 158, 71, 149, 107, 134, 254, 206, 166, 131, 271

Offset: 1

Robert G. Wilson v, Jul 31 2025

Inspired by A284783.

First differs from A037153 at n=13 (and when they differ a(n) is a composite < A037153(n)).

			a(1) = 2 since d(2) = d(3) = 2;
a(5) = 7 since d(7) = d(7+5!) = 2;
a(13) = 46 since d(46) = d(46+13!) = 4; etc.

Sean A. Irvine, Table of n, a(n) for n = 1..82

Cf. A000005, A000142, A006881, A037153, A284783.

a[n_] := Block[{k = 2}, While[ DivisorSigma[0, k] != DivisorSigma[0, k + n!], k++]; k]; Array[ a, 51]

a(n) = my(k=1); while (numdiv(k) != numdiv(k+n!), k++); k; \\ Michel Marcus, Aug 02 2025

More terms from Sean A. Irvine, Aug 08 2025

A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 11, 13, 31, 23, 19, 37, 41, 29, 31, 43, 53, 41, 53, 79, 59, 97, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 101, 229, 109, 139, 113, 227, 131, 191, 163, 139, 199, 151, 139, 181, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233

Offset: 1

Labos Elemer, Nov 14 2000

nonn

Analogous to Fortunate numbers and like them so far proved to be primes. This holds for x<=421: if Q is the first follower prime, then Q(421)-lcm(1,...421) = 557. For first some cases when 1+LCM is also a prime, the 2nd primes give 3,5,5,7,11,11,.. deviations, i.e. give primes.

Cf. A000961, A003418, A051451, A057019, A037153, A035346, A005235, A054272, A055211, A037155, A045493, A038710 etc

N=1; for(n=2,1e3, if(isprimepower(n,&p), N*=p; print1(nextprime(N+2)-N", "))) \\ Charles R Greathouse IV, Nov 18 2015

Name corrected by Charles R Greathouse IV, Nov 18 2015

A082433 a(n) = A072181(n) - p, where p is the largest prime < A072181(n) - 1.

Original entry on oeis.org

3, 5, 7, 7, 11, 11, 11, 11, 13, 23, 17, 17, 17, 41, 191, 47, 31, 53, 53, 53, 31, 179, 61, 61, 337, 131, 523, 523, 419, 223, 223, 223, 223, 79, 3821, 3821, 3821, 23399, 21269, 21269, 3607

Offset: 3

Naohiro Nomoto, Apr 25 2003

Are all terms prime?

All terms are odd. - Michael S. Branicky, Sep 05 2021

			a(4) = A072181(4)-7 = 12-7 = 5.

Cf. A005235, A037153, A037155, A072181, A082432.

from sympy import factorint, isprime
def afindn(terms):
    prev_factors, prevan, prevk, n = dict(), 1, None, 2
    for n in range(2, terms+1):
        n_factors, an = factorint(n), 1
        for pi in set(prev_factors.keys()) | set(n_factors.keys()):
            ei = prev_factors[pi] if pi in prev_factors else 1
            fi = n_factors[pi] if pi in n_factors else 1
            an *= pi**(ei*fi)
        if n >= 3:
            if an != prevan:
                k = 3
                while not isprime(an - k): k += 2
            else:
                k = prevk
            print(k, end=", ")
            prevk = k
        prev_factors, prevan = factorint(an), an
afindn(36) # Michael S. Branicky, Sep 05 2021

a(36)-a(40) from Jinyuan Wang, Sep 05 2020

a(41)-a(43) from Michael S. Branicky, Sep 05 2021

A190801 Least semiprime whose prime factors differ by n!.

Original entry on oeis.org

6, 15, 55, 145, 889, 5089, 55561, 927889, 6169249, 39916921, 678585889, 13891047241, 417210398089, 1656387533161, 56229997825849, 481224167424529, 11026310270976961, 236887827111937369, 10826413936386055921

Offset: 1

Michel Lagneau, May 20 2011

nonn

Appears to be the same as A037152(n) * A037153(n).

			a(5) = 889 because 889 = 7 * 127 , and 127 - 7 = 120 = 5!

f[n_] := Block[{p=2}, While[! PrimeQ[p+n!], p=NextPrime[p]]; p*(p+n!)]; Table[f[n], {n, 60}]

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

A343593 a(n) is the smallest number k > 0 such that n! + n + k is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 15, 8, 1, 6, 17, 54, 5, 28, 7, 14, 19, 70, 9, 10, 9, 74, 107, 16, 33, 20, 39, 194, 77, 96, 47, 4, 63, 26, 95, 274, 5, 58, 13, 20, 55, 28, 3, 194, 55, 186, 5, 34, 11, 220, 1, 18, 169, 16, 93, 50, 225, 234, 211, 708, 69, 208, 3, 128, 217

Offset: 0

Ventsislav D. Tsenov, Apr 21 2021

nonn

			For n = 2: the smallest value of k such that 2! + 2 + k = 4 + k is prime is 1.

Cf. A000040, A037153, A090786.

a:= n-> (t-> nextprime(t)-t)(n!+n):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 21 2021

a[n_] := NextPrime[(m = n! + n)] - m; Array[a, 100, 0] (* Amiram Eldar, Apr 21 2021 *)

a(n) = my(s=n!+n); nextprime(s+1) - s; \\ Michel Marcus, Apr 21 2021

a(n) = A037153(n) - n for n >= 1.

a(n) = A090786(n) + 1.

More terms from Alois P. Heinz, Apr 21 2021

cp's OEIS Frontend

A105049 a(n) is the smallest prime p such that p+n! is prime and p differs from all a(i), i

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A082432 a(n) = p - A072181(n), where p is the least prime > A072181(n) + 1.

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A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

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A385430 Least number k such that k and k + n! have the same number of divisors.

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A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

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A082433 a(n) = A072181(n) - p, where p is the largest prime < A072181(n) - 1.

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Python

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A190801 Least semiprime whose prime factors differ by n!.

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Mathematica

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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A343593 a(n) is the smallest number k > 0 such that n! + n + k is prime.

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