A006862 -id:A006862 - cp's OEIS frontend

A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

1, 5, 29, 11, 2309, 30029, 61, 53, 37, 79, 228737, 229, 304250263527209, 141269, 191, 87337, 27600124633, 1193, 163, 260681003321, 313, 163, 139, 23768741896345550770650537601358309, 66683, 2990092035859, 15649, 17515703, 719, 295201, 15098753, 10172884549, 20962699238647, 4871, 673, 311, 1409, 1291, 331, 1450184819, 23497, 711427, 521, 673, 519577, 1372062943, 56543, 811, 182309, 53077, 641, 349, 389

Offset: 1

Labos Elemer, Oct 25 2000

nonn

			6th term in the sequence corresponds to 7th primorial = 510510 and 510509 = 61 * 8369, so a(7) = 61.

Amiram Eldar, Table of n, a(n) for n = 1..99
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A002585, A006862, A020639, A057588.

Map[If[PrimeQ@ #, #, FactorInteger[#][[1, 1]]] &, FoldList[#1 #2 &, Prime@ Range@ 36] - 1] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = A020639(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from Klaus Brockhaus, Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Apr 02 2001

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 22 2024

Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum of the form 3*(p+q) + p*q, where p and q are odd primes.
Question: Will there be an eventual growth spurt for this sequence? Even though all solutions must be multiples of 3 (but not of 9), because A006862(n) == 1 (mod 3), for n > 1, and the solutions belong to a set listed by A369461.
Similar sequence A369242 grows more vigorously because A033312(n) == -1 (mod 3) for n >= 3, thus allowing non-multiples of 3 as solutions. See comments in A369252.

			a(4) = 1 as there exists a natural number 399 = 3 * 7 * 19, whose arithmetic derivative (indicated with 399', see A003415) is computed as ((3*7) + (3*19) + (7*19)) = 211 = 1 + prime(4)# = A006862(4), and because 399 is the unique term in A046316 that satisfies the condition.
a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).
For other cases, see examples in A369246.

Cf. A002110, A003415, A006862, A046316, A369054, A369246 (the solutions), A369252, A369461.

Cf. also A116979, A369000, A369239 for similar counts, also A369241, A369242 and A369247.

\\ Needs also program from A369054.
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = A369054(A002110(n)+1);

\\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.
search_for_3k1_cases(n) = if(3!=(n%4),0, my(p = 5, q, c=0); while(1, q = (n-(3*p)) / (3+p); if(q < p, return(c), if(1==denominator(q) && isprime(q),c++; write("b369246_by_search_order_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = search_for_3k1_cases(A002110(n)+1);

a(n) = A369054(A006862(n)).

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

Original entry on oeis.org

2, 3, 4, 7, 8, 12, 31, 32, 36, 60, 211, 212, 216, 240, 420, 2311, 2312, 2316, 2340, 2520, 4620, 30031, 30032, 30036, 30060, 30240, 32340, 60060, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092871, 223092872

Offset: 0

Antti Karttunen, Feb 29 2024

After the initial 2, numbers with either one 2 or two 1's in their primorial base representation (A049345), with all the other digits zeros.

			Triangle begins as:
        2;
        3,       4;
        7,       8,      12;
       31,      32,      36,      60;
      211,     212,     216,     240,     420;
     2311,    2312,    2316,    2340,    2520,    4620;
    30031,   30032,   30036,   30060,   30240,   32340,   60060;
   510511,  510512,  510516,  510540,  510720,  512820,  540540,  1021020;
  9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Cf. A002110, A049345, A087112, A276086, A276150, A370129 (arithmetic derivative applied to this triangle).
Cf. A006862 (left edge), A088860 (right edge).
Cf. A177689 (same triangle without the right edge), A370134 (without the leftmost column).
Subsequence of A370132.
Cf. also A173786.

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };

For n >= 1, A276150(a(n)) = 2.

For n >= 1, A276086(a(n)) = A087112(1+n).

A377871 Numbers k such that neither k nor A276085(k) has divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 42, 43, 45, 46, 47, 50, 53, 58, 59, 61, 62, 63, 66, 67, 70, 71, 73, 74, 75, 78, 79, 82, 83, 86, 89, 90, 94, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 122, 125, 126, 127, 130, 131, 134, 137, 138, 139, 142

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

Range of A276087, where A276087(n) = A276086(A276086(n)) [the twofold application of the primorial base exp-function].
A276087(0) = 2, and for n >= 0, A276087(A143293(n)) = A000040(n+2), therefore all primes are included.
From Antti Karttunen, Nov 17 2024: (Start)
Even semiprimes > 4 form a subsequence, because A006862 (Euclid numbers) is a subsequence of A048103. Note that A276087(A376416(n)) = A276086(A006862(n)) = A100484(1+n). On the other hand, none of the odd semiprimes, A046315, occur here, because they are all included in A369002, and thus in A377873. Similarly, A276092 after its initial 1 is a subsequence, because A057588 (Kummer numbers) is also a subsequence of A048103.
For k=1..6, there are 6, 52, 486, 4775, 46982, 467372 terms <= 10^k. Question: Does this sequence have an asymptotic density?
(End)

			A276087(A002110(10)) = A276086(A276086(A002110(10))) = A276086(A000040(10+1)) = A276086(31) = 14, therefore 14 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Intersection of A048103 and A377869.
Sequence A276087 sorted into ascending order.
Cf. A377870 (characteristic function).
Subsequences: A000040, A100484 (after its initial 4), A276092 (after its initial 1).
Cf. A006862, A143293, A276086, A376416.

PARI
```
\\ See A377870.
```

A035345 Smallest prime > prime(1)prime(2)...*prime(n)+1.

Original entry on oeis.org

3, 5, 11, 37, 223, 2333, 30047, 510529, 9699713, 223092907, 6469693291, 200560490197, 7420738134871, 304250263527281, 13082761331670077, 614889782588491517, 32589158477190044789, 1922760350154212639131

Offset: 0

N. J. A. Sloane

nonn

			Next prime after 2*3*5 + 1 = 31 is 37, so a(3)=37.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
Eric Weisstein's World of Mathematics, Fortunate Prime

Cf. A002110, A005235, A006862, A038710.

Table[NextPrime[Product[Prime@ k, {k, n}] + 1], {n, 0, 17}] (* Michael De Vlieger, Dec 02 2015 *)

a(n) = nextprime(2+factorback(primes(n))); \\ Michel Marcus, Dec 24 2022

from sympy import nextprime, primorial
def a(n): return nextprime(1 + (primorial(n) if n else 1))
print([a(n) for n in range(18)]) # Michael S. Branicky, Dec 24 2022

a(n) = A002110(n) + A005235(n) for n > 0. - Jonathan Sondow, Dec 02 2015

A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 14, 16, 17, 21, 73, 801, 1971, 3332, 3469, 3509, 4318, 7986, 41292

Offset: 1

N. J. A. Sloane

Positive n such that A002110(n) + A000040(n+1) is prime. - Robert Israel, Dec 02 2015

Subsequence of A265109. - Altug Alkan, Dec 02 2015

			a(10) = 21 because A002110(21) + prime(22) = 40729680599249024150621323549 = 2*3*5*...*67*71*73 + 79 is prime.

Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

Cf. A000040, A002110, A005235, A006862, A035345, A265109.

p:= 3:
A[1]:= 1:
count:= 1:
Primorial:= 2:
for n from 2 to 1000 do
  Primorial:= Primorial*p;
  p:= nextprime(p);
  if isprime(Primorial + p) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Dec 02 2015

Select[Range@ 801, PrimeQ[Product[Prime@ k, {k, #}] + Prime[# + 1]] &] (* Michael De Vlieger, Dec 02 2015 *)

lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(k+1)), print1(k, ", ")); ); } \\ Altug Alkan, Dec 02 2015

a(10)-a(11) were found by Labos Elemer, May 02 2000
a(12) from Ralf Stephan, Oct 20 2002
Offset changed by Altug Alkan, Dec 02 2015
a(13) from Michael De Vlieger, Dec 02 2015
a(14)-a(18) from Altug Alkan, Dec 02 2015
a(19) from Henri Lifchitz, Nov 08 2024

A051265 Maximal value of prime divisors of numbers in reduced residue system for n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer

The smallest number for which a(n)=k is the n-th Euclid number (A006862=A002110 + 1).

Largest value of A001221(k) for 1 <= k <= n such that gcd(k, n) = 1. - Michael De Vlieger, Aug 10 2017

			For n=60 a(n)=1 since in RRS[ 60 ] only 1 and prime powers occur (see A051250).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes, CCCG 2011, Toronto ON, August 10-12, 2011.

Records are A006862.

Cf. A001221, A002110, A051250, A051266, A051267, A051268.

Table[Max@ Map[PrimeNu, Cases[Range[n - 1], k_ /; CoprimeQ[n, k]]] /. k_ /; ! IntegerQ@ k -> 0, {n, 105}] (* Michael De Vlieger, Aug 10 2017 *)

a(n)=my(k=1,s); forprime(p=2,, if(n%p==0, next); k*=p; if(k>n, return(s)); s++) \\ Charles R Greathouse IV, Aug 10 2017

a(n) << log n/log log n. - Charles R Greathouse IV, Aug 10 2017

A051268 Numbers n such that maximal value of prime divisors of reduced residue system for n is 4.

Original entry on oeis.org

211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 253, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307, 311, 313, 317, 319, 323, 331, 337, 341, 343, 347, 349, 353, 359, 361, 367, 371, 373, 377, 379, 383, 389, 391, 397, 401, 403, 407, 409, 413

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 4 for 1 <= k <= n such that gcd(k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A002110, A006862, A038566, A048597, A051250, A051265, A051266, A051267.

Block[{n = 4, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 175, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

More terms from Michael De Vlieger, Aug 10 2017

A057706 Smaller of twin primes whose average is a primorial number.

Original entry on oeis.org

5, 29, 2309

Offset: 1

Labos Elemer, Oct 24 2000

According to Caldwell, the next term, if it exists, has more than 100000 digits. - T. D. Noe, May 08 2012

			(5+7)/2 = 6 = 2*3, (29+31)/2 = 30 = 2*3*5, (2309+2311)/2 = 2310 = 2*3*5*7*11.

Chris K. Caldwell, Primorial prime.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Index entries for sequences related to primorial numbers.

Cf. A000040 (primes), A002110 (primorials, p#).
Cf. A006862 (Euclid, p#+1), A005234 (prime p#+1), A014545 (index prime p#+1).
Cf. A057588 (Kummer, p#-1), A006794 (prime p#-1), A057704 (index prime p#-1).

Select[FoldList[Times, Prime@ Range@ 40], AllTrue[# + {-1, 1}, PrimeQ] &] - 1 (* Michael De Vlieger, Jul 15 2017 *)

from sympy import isprime, prime, primerange
def auptoprimorial(limit):
  phash, alst = 1, []
  for p in primerange(1, prime(limit)+1):
    phash *= p
    if isprime(phash-1) and isprime(phash+1): alst.append(phash-1)
  return alst
print(auptoprimorial(5)) # Michael S. Branicky, May 29 2021

Offset corrected by Arkadiusz Wesolowski, May 08 2012

A060882 a(n) = n-th primorial (A002110) minus next prime.

Original entry on oeis.org

-1, -1, 1, 23, 199, 2297, 30013, 510491, 9699667, 223092841, 6469693199, 200560490093, 7420738134769, 304250263527167, 13082761331669983, 614889782588491357, 32589158477190044671, 1922760350154212639009

Offset: 0

N. J. A. Sloane, May 05 2001

sign

It is well-known and easy to prove (see Honsbeger) that a(n) > 0 for n > 1. - N. J. A. Sloane, Jul 05 2009

Terms are pairwise coprime with very high probability. I didn't find terms which are pairwise noncoprime, although it may be a case of the "strong law of small numbers." - Daniel Forgues, Apr 23 2012

R. Honsberger, Mathematical Diamonds, MAA, 2003, see p. 79. [Added by N. J. A. Sloane, Jul 05 2009]

Harry J. Smith, Table of n, a(n) for n = 0..100
Hisanori Mishima, P1 * Pn + NextPrime (n = 1 to 100)
Hisanori Mishima, P1 * Pn - NextPrime (n = 1 to 100)
Hisanori Mishima, P1 * Pn + 1 (n = 1 to 100)
Hisanori Mishima, P1 * Pn - 1 (n = 1 to 100)
Hisanori Mishima, WIFC (World Integer Factorization Center)

Cf. A002110, A060881, A064819, A006862, A057588.

pp:=n->mul(ithprime(i),i=1..n);
[seq(pp(n)-ithprime(n+1),n=1..20)];

Join[{-1},With[{nn=20},#[[1]]-#[[2]]&/@Thread[{FoldList[Times,1, Prime[ Range[nn]]],Prime[Range[nn+1]]}]]] (* Harvey P. Dale, May 10 2013 *)

{ n=-1; m=1; forprime (p=2, prime(101), write("b060882.txt", n++, " ", m - p); m*=p; ) } \\ Harry J. Smith, Jul 13 2009

from sympy import prime, primorial
def A060882(n): return primorial(n)-prime(n+1) if n else -1 # Chai Wah Wu, Feb 25 2023

cp's OEIS Frontend

A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

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A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

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A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

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A377871 Numbers k such that neither k nor A276085(k) has divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

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A035345 Smallest prime > prime(1)*prime(2)*...*prime(n)+1.

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A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

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A051265 Maximal value of prime divisors of numbers in reduced residue system for n.

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A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

A035345 Smallest prime > prime(1)prime(2)...*prime(n)+1.