A143293 -id:A143293 - cp's OEIS frontend

A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).

14, 45, 74, 198, 5114, 10295, 65174, 1086194, 20485574, 40354813, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253

Offset: 1

Antti Karttunen, Oct 10 2019

nonn

From David A. Corneth, Oct 12 2019: (Start)
Let k' be the arithmetic derivative of k. Then to find terms of the form k = p * q where p, q are prime, we could see that k' = p + q. Then as one of them needs to be two, say p, needs to be 2, we have q = A143293(m) - 2 a prime. This would give terms 2 * q.
If terms are of the form k = p * q * r where p, q, r are distinct primes then k' = p*q + p*r + q*r. For m we like, we could solve p*q + p*r + q*r = A143293(m). checking p * q below some bound, we can solve for r and get r = (A143293(m) - p*q) / (p + q). With some extra constraints and searching different prime signatures, one might confirm terms found are all below some chosen upper bound. (End)
See sequences A369239 and A369240 for more observations and insights about the terms of this sequence. - Antti Karttunen, Jan 22 2024

Giovanni Resta, Table of n, a(n) for n = 1..39 (terms < 10^13)

Sequence A369240 sorted into ascending order.

Cf. A003415, A143293, A327969, A327978, A328313, A369239.

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
k=0; for(n=1,A002620(A143293(6)),if(isA328243(n), k++; print1(n,", ")));

A327969(a(n)) <= 5 for all n.

a(12)-a(25) from David A. Corneth and Giovanni Resta, Oct 12 2019

A328313 For all such terms k in A143293 (partial sums of primorials) for which A129251(k) = 0, the term A276085(k) is included here.

Original entry on oeis.org

2, 4, 2312, 3217644767340672907899084554132

Offset: 1

Antti Karttunen, Oct 12 2019

nonn

Construction: apply A276085 to the terms in the intersection of A143293 and A048103.
The next terms are quite big and can be found in the b-file. Note the nonmonotonic order, a(8) < a(5), a(6) and a(7).
A276086(a(n)) is in A143293, A276086(A276086(a(n))) is one of the primorials, A002110, and A276086(A276086(A276086(a(n)))) is a prime.
A327969(a(n)) <= 5 for all n.

Antti Karttunen, Table of n, a(n) for n = 1..8

Cf. A002110, A048103, A129251, A143293, A276085, A276086, A327969, A328243.

A002110(n) = prod(i=1,n,prime(i));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
k=0; for(n=1,12,t = A143293(n); u = A276085(t); if(A276086(u) == t, k++; print1(u,", ")));

A369240 Irregular triangle read by rows, where row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th partial sum of primorials, A143293(n). Rows of length zero are simply omitted, i.e., when A369239(n) = 0.

Original entry on oeis.org

14, 45, 74, 198, 5114, 10295, 65174, 1086194, 40354813, 20485574, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253, 307379277253, 321317084917, 342666536237, 348440115979

Offset: 1

Antti Karttunen, Jan 19 2024

Only two nonsquarefree terms are currently known: 45, 198.

See comments in A369239 for an explanation why rows with an odd n generally have more terms than those with an even n.

			Row 1 has no terms because there are no numbers whose arithmetic derivative is equal to 3 = A143293(1).
Row 2 has just one term: 14 (= 2 * 7), with A003415(14) = 2+7 = 9 = A143293(2).
Row 3 has two terms: 45 (= 3^2 * 5) and 74 (= 2 * 37), with A003415(3*3*5) = (3*3) + (3*5) + (3*5) = 39, and A003415(2*37) = 2+37 = 39 = A143293(3).
Row 4 has one term: 198 (= 2 * 3^2 * 11).
Row 5 has two terms: 5114 (= 2 * 2557) and 10295 (= 5 * 29 * 71).
Row 6 has one term: 65174 (= 2 * 32587).
Row 7 has two terms: 1086194 (= 2 * 543097) and 40354813 (= 97 * 541 * 769).
Row 8 has one term: 20485574 (= 2 * 10242787).
Row 9 has 27 terms:
  465779078 (= 2 * 1049 * 222011),
  12101385979 (= 79 * 151 * 1014451),
  15237604243 (= 67 * 2659 * 85531),
  18046312939 (= 79 * 3931 * 58111),
  29501083259 (= 179 * 431 * 382391),
  52467636437 (= 233 * 8501 * 26489),
  65794608773 (= 449 * 761 * 192557),
  86725630997 (= 449 * 2213 * 87281),
  87741700037 (= 449 * 2381 * 82073),
  131833085077 (= 613 * 12241 * 17569),
  etc., up to the last one of them:
  680909375411 (= 8171 * 8219 * 10139).
Row 10 has no terms.
Row 11 has 319 terms, beginning as:
  293420849770 (= 2 * 5 * 157 * 186892261),
  414527038034 (= 2 * 207263519017),
  12092143168139 (= 59 * 5231 * 39180191),
  16359091676491 (= 79 * 91291 * 2268319),
  20784361649963 (= 167 * 251 * 495845639),
  etc., up to the last one of them:
  17866904665985941 (= 224869 * 248041 * 320329).
Row 12 has just one term: 318745032938881 (= 71 * 173 * 307 * 1259 * 67139).
Row 13 probably has thousands of terms. Interestingly, many of them appear in clusters that share a smallest prime factor. For example the following five:
  390120053091860677 (= 1321 * 23563 * 12533283799),
  407566547631686353 (= 1321 * 121687 * 2535429439),
  410999481465461617 (= 1321 * 547999 * 567752023),
  411668623600396429 (= 1321 * 1701571 * 183144919),
  411913933485848977 (= 1321 * 8787799 * 35483263),
  and also these:
  3846842704473466739 (= 20231 * 31601 * 6017086469),
  4300947161911032233 (= 20231 * 43319 * 4907590697),
  4437898843097002379 (= 20231 * 47969 * 4572980861),
  6130224093530040341 (= 20231 * 692459 * 437587529),
  6210584908378844243 (= 20231 * 1275569 * 240664037).

Antti Karttunen, Table of n, a(n) for n = 1..357; all terms up to the row 12 of the table.
Antti Karttunen, PARI program for computing terms of this and related sequences.

Cf. A328243 (same sequence sorted into ascending order).
Cf. A369239 (number of terms on row n), A369243 (the first element of each row), A369244 (the last element of each row).
Cf. A003415, A143293.
Cf. also A366890.

A283985 Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 39, 40, 42, 43, 48, 49, 51, 52, 249, 250, 252, 253, 258, 259, 261, 262, 288, 289, 291, 292, 297, 298, 300, 301, 2559, 2560, 2562, 2563, 2568, 2569, 2571, 2572, 2598, 2599, 2601, 2602, 2607, 2608, 2610, 2611, 2808, 2809, 2811, 2812, 2817, 2818, 2820, 2821, 2847, 2848, 2850, 2851, 2856, 2857, 2859, 2860, 32589

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A002110, A030308, A143293, A276085, A276156, A283477, A283984.

A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from Charles R Greathouse IV, Feb 05 2014
A030308(n,k) = bittest(n,k);
A283985(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A143293(i));

from sympy import primorial, primepi, prime, primerange, factorint
from operator import mul
from functools import reduce
def a002110(n): return 1 if n<1 else primorial(n)
def a276085(n):
    f=factorint(n)
    return sum([f[i]*a002110(primepi(i) - 1) for i in f])
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a276085(a108951(a019565(n)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

(define (A283985 n) (A276085 (A283477 n)))

a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).
a(n) = A276085(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A143293(n).

A225727 Numbers k such that sum of first k primorials (A143293) is divisible by k.

Original entry on oeis.org

1, 3, 17, 51, 967, 2901, 16439, 49317, 147951, 1331559

Offset: 1

Alex Ratushnyak, May 13 2013

a(5) = 967 is a prime,
a(6) = a(5) * 3,
a(7) = a(5) * 17,
a(8) = a(5) * 51,
a(9) = a(5) * 51 * 3,
a(10) = a(5) * 51 * 27.
The next term, if it exists, is greater than 15600000. - Alex Ratushnyak, Jun 16 2013

			Sum of first 3 primorials is 1+2+6=9, because 9 is divisible by 3, the latter is in the sequence.
Sum of first 17 primorials is A143293(17) = 1955977793053588026279. Because A143293(17) is divisible by 17, the latter is in the sequence.

Cf. A143293, A002110, A057245, A128981.

primes = [2,3]
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5,1000000,6):
  addPrime(n)
  addPrime(n+2)
sum_ = 0
primorial = n = 1
for p in primes:
  sum_ += primorial
  primorial *= p
  if sum_ % n == 0:  print(n, end=', ')
  n += 1

from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A225727_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))) if m % (i+1) == 0)
A225727_list = list(islice(A225727_gen(),6)) # Chai Wah Wu, Feb 23 2022

A357270 a(n) = s(n) mod prime(n+1), where s = A143293.

Original entry on oeis.org

1, 0, 4, 4, 7, 11, 0, 3, 15, 6, 11, 9, 4, 41, 4, 26, 28, 56, 4, 54, 23, 37, 78, 48, 11, 17, 32, 68, 85, 34, 78, 12, 120, 28, 68, 24, 76, 116, 17, 55, 40, 3, 91, 111, 132, 133, 195, 75, 179, 44, 211, 108, 3, 63, 21, 28, 85, 22, 208, 237, 9, 166, 81, 183, 205, 208

Offset: 0

Christopher A. Curry, Sep 21 2022

Let i be some positive integer. Let r(n) = i mod prime(n+1). Let m be defined such that s(m-1) <= i < s(m). Then the first m values in the sequence {r(n)} uniquely define the integer (it is the least positive integer requiring those m remainders modulo those primes). If i is a term of {s(n)}, then a(n) = r(n) for n = 0..m-1.

Christopher A. Curry, Table of n, a(n) for n = 0..9999

Cf. A000040, A143293.

q[n_] := Insert[Table[Mod[Total[Table[Product[Prime[i], {i, 1, r}], {r, m}]] + 1, Prime[m + 1]], {m, n}], 1, 1]; q[65] (* Samuel Harkness, Sep 22 2022 *)

a(n) = if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s % prime(n+1); \\ Michel Marcus, Sep 22 2022

a(n,q=prime(n+1))=my(P=Mod(1,q),s=P); forprime(p=2,q-1, s+=P*=p); lift(s) \\ Charles R Greathouse IV, Sep 22 2022

A079096 Erroneous version of A143293.

Original entry on oeis.org

1, 3, 9, 39, 249, 2559, 5559, 516069, 10215759, 233308629, 6703001859, 207263491989, 7628001626799, 311878265154009, 13394639596824039, 628284422185315449, 33217442899375360179, 1955977793053587999249

Offset: 1

dead

Cf. A002110, A003422.

A225728 Primes p such that sum of primorials (A143293) not including p as a factor is divisible by p.

Original entry on oeis.org

3, 17, 967

Offset: 1

Alex Ratushnyak, May 14 2013

As in A002110, primorial(0)=1, and primorial(n) = primorial(n-1)*prime(n).

The next term, if it exists, is bigger than 10^8.

			Sum of primorials not including 3 as a factor is 1 + 2 = 3. Because it's divisible by 3, the latter is in the sequence.
Sum of primorials not including 17 as a factor is 1 + 2 + 6 + 6*5 + 30*7 + 210*11 + 2310*13 = 32589. Because 32589 is divisible by 17, the latter is in the sequence.

Cf. A002110, A143293, A225727.

s=P=1;forprime(p=2,1e6,s+=P*=p;if(s%p==0,print1(p", "))) \\ Charles R Greathouse IV, Mar 19 2014

is(p)=if(!isprime(p),return(0)); my(s=Mod(1,p),P=s); forprime(q=2,p-1,s+=P*=q); s==0 \\ Charles R Greathouse IV, Mar 19 2014

primes = [2,3]
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5,1000000,6):
  addPrime(n)
  addPrime(n+2)
sum_ = 0
primorial = 1
for p in primes:
  sum_ += primorial
  primorial *= p
  if sum_ % p == 0:  print(p, end=', ')

from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A225728_gen(): return (prime(i+1) for i, m in enumerate(accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))) if m % prime(i+1) == 0)
A225728_list = list(islice(A225728_gen(), 3)) # Chai Wah Wu, Feb 23 2022

A369061 Numbers k such that k + k'*2 is equal to a partial sum of primorial numbers (a term of A143293), where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

1, 7, 37, 99, 2557, 32587, 543097, 10242787, 232889539, 146710424885, 207263519017

Offset: 1

Antti Karttunen, Jan 17 2024

nonn

Numbers k such that A068719(k) = A143293(n), for some n >= 0.
Numbers k for which A276087(A068719(k)) is a prime.
All terms are odd.
Notably each of the terms a(2) .. a(9) map (in the same order) to A143293(2..9), but then k for A143293(10) = 6703028889 is missing, and a(10) and a(11) both map to A143293(11) = 207263519019.

			For 99, A068719(99) = 99 + 99'*2 = 99 + 75*2 = 249 = 1 + 2 + 6 + 30 + 210 = A143293(4), therefore 99 is included in this sequence.
For 2557, which is a prime, 2557 + 2557' * 2 = 2557+2 = 2559 = 1 + 2 + 6 + 30 + 210 + 2310 = A143293(5), therefore 2557 is included in this sequence.

After the initial 1, the even terms of A328243 halved.

Cf. A003415, A068719, A143293, A276086, A276087.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068719(n) = (n+2*A003415(n));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369061(n) = (1==A276150(A276086(A068719(n))));

A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090

Offset: 0

N. J. A. Sloane and J. H. Conway

See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.
a(n) is the least number N with n distinct prime factors (i.e., omega(N) = n, cf. A001221). - Lekraj Beedassy, Feb 15 2002
Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v, Jan 10 2004
Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - Lekraj Beedassy, Mar 31 2005
Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/Pi^2 < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie, Jun 11 2005
Let f be a multiplicative function with f(p) > f(p^k) > 1 (p prime, k > 1), f(p) > f(q) > 1 (p, q prime, p < q). Then the record maxima of f occur at n# for n >= 1. Similarly, if 0 < f(p) < f(p^k) < 1 (p prime, k > 1), 0 < f(p) < f(q) < 1 (p, q prime, p < q), then the record minima of f occur at n# for n >= 1. - David W. Wilson, Oct 23 2006
Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
Records in number of distinct prime divisors. - Artur Jasinski, Apr 06 2008
For n >= 2, the digital roots of a(n) are multiples of 3. - Parthasarathy Nambi, Aug 19 2009 [with corrections by Zak Seidov, Aug 30 2015]
Denominators of the sum of the ratios of consecutive primes (see A094661). - Vladimir Joseph Stephan Orlovsky, Oct 24 2009
Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i = 0..T(sqrt(N))} A005867(i)/A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010
The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. prime(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013
The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015
If a number k and a(n) are coprime and k < (prime(n+1))^b < a(n), where b is an integer, then k has fewer than b prime factors, counting multiplicity (i.e., bigomega(k) < b, cf. A001222). - Isaac Saffold, Dec 03 2017
If n > 0, then a(n) has 2^n unitary divisors (A034444), and a(n) is a record; i.e., if k < a(n) then k has fewer unitary divisors than a(n) has. - Clark Kimberling, Jun 26 2018
Unitary superabundant numbers: numbers k with a record value of the unitary abundancy index, A034448(k)/k > A034448(m)/m for all m < k. - Amiram Eldar, Apr 20 2019
Psi(n)/n is a new maximum for each primorial (psi = A001615) [proof in link: Patrick Sole and Michel Planat, proposition 1 page 2]; compare with comment 2004: Phi(n)/n is a new minimum for each primorial. - Bernard Schott, May 21 2020
The term "primorial" was coined by Harvey Dubner (1987). - Amiram Eldar, Apr 16 2021
a(n)^(1/n) is approximately (n log n)/e. - Charles R Greathouse IV, Jan 03 2023
Subsequence of A267124. - Frank M Jackson, Apr 14 2023

			a(9) = 23# = 2*3*5*7*11*13*17*19*23 = 223092870 divides the difference 5283234035979900 in the arithmetic progression of 26 primes A204189. - _Jonathan Sondow_, Jan 15 2012

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 49.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.
D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.

Alex Ermolaev, Table of n, a(n) for n = 0..350 (terms up to a(100) from T. D. Noe)
Iskander Aliev, Jesús De Loera, Fritz Eisenbrand, Timm Oertel, and Robert Weismantel, The Support of Integer Optimal Solutions, arXiv:1712.08923 [math.OC], 2017.
C. K. Caldwell, The Prime Glossary, Primorial.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
Harvey Dubner, Factorial and primorial primes, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197-203. (Annotated scanned copy)
F. Ellermann, Illustration for A002110, A005867, A038110, A060753.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, pp. 7, 81.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 7.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, preprint, CONCUR 2014 - Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp. 327-341.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, CONCUR 2014 - Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp. 327-341.
F. E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012.
Thomas Morrill, Further Development of "Non-Pythagorean" Musical Scales Based on Logarithms, arXiv:1804.08067 [math.HO], 2018.
J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.
Patrick Sole and Michel Planat, The Robin inequality for 7-free integers, INTEGERS, 2011, #A65.
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
G. Villemin's Almanach of Numbers, Primorielle.
Eric Weisstein's World of Mathematics, Primorial.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

A034386 gives the second version of the primorial numbers.
Subsequence of A005117 and of A064807. Apart from the first term, a subsequence of A083207.
Cf. A001615, A002182, A002201, A003418, A005235, A006862, A034444 (unitary divisors), A034448, A034387, A033188, A035345, A035346, A036691 (compositorial numbers), A049345 (primorial base representation), A057588, A060735 (and integer multiples), A061742 (squares), A072938, A079266, A087315, A094348, A106037, A121572, A053589, A064648, A132120, A260188.
Cf. A061720 (first differences), A143293 (partial sums).
Cf. also A276085, A276086.
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
-- Reinhard Zumkeller, Feb 19 2012, May 03 2011

[1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012

[1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015

A002110 := n -> mul(ithprime(i),i=1..n);

FoldList[Times, 1, Prime[Range[20]]]
primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial,20] (* José María Grau Ribas, Feb 15 2010 *)
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)

a(n)=prod(i=1,n, prime(i)) \\ Washington Bomfim, Sep 23 2008

p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) )  \\ Harry J. Smith, Nov 13 2009

a(n) = factorback(primes(n)) \\ David A. Corneth, May 06 2018

from sympy import primorial
def a(n): return 1 if n < 1 else primorial(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Mar 29 2017

[sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014

; with memoization-macro definec
(definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; Antti Karttunen, Aug 30 2016

Asymptotic expression for a(n): exp((1 + o(1)) * n * log(n)) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
a(n) = A054842(A002275(n)).
Binomial transform = A136104: (1, 3, 11, 55, 375, 3731, ...). Equals binomial transform of A121572: (1, 1, 3, 17, 119, 1509, ...). - Gary W. Adamson, Dec 14 2007
a(0) = 1, a(n+1) = prime(n)*a(n). - Juri-Stepan Gerasimov, Oct 15 2010
a(n) = Product_{i=1..n} A000040(i). - Jonathan Vos Post, Jul 17 2008
a(A051838(n)) = A116536(n) * A007504(A051838(n)). - Reinhard Zumkeller, Oct 03 2011
A000005(a(n)) = 2^n. - Carlos Eduardo Olivieri, Jun 16 2015
a(n) = A035345(n) - A005235(n) for n > 0. - Jonathan Sondow, Dec 02 2015
For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)). - Antti Karttunen, Aug 30 2016
A054841(a(n)) = A002275(n). - Michael De Vlieger, Aug 31 2016
a(n) = A270592(2*n+2) - A270592(2*n+1) if 0 <= n <= 4 (conjectured for all n by Alon Kellner). - Jonathan Sondow, Mar 25 2018
Sum_{n>=1} 1/a(n) = A064648. - Amiram Eldar, Oct 16 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = A132120. - Amiram Eldar, Apr 12 2021
Theta being Chebyshev's theta function, a(0) = exp(theta(1)), and for n > 0, a(n) = exp(theta(m)) for A000040(n) <= m < A000040(n+1) where m is an integer. - Miles Englezou, Nov 26 2024

cp's OEIS Frontend

A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).

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A328313 For all such terms k in A143293 (partial sums of primorials) for which A129251(k) = 0, the term A276085(k) is included here.

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A369240 Irregular triangle read by rows, where row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th partial sum of primorials, A143293(n). Rows of length zero are simply omitted, i.e., when A369239(n) = 0.

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A283985 Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

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A225727 Numbers k such that sum of first k primorials (A143293) is divisible by k.

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A357270 a(n) = s(n) mod prime(n+1), where s = A143293.

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A079096 Erroneous version of A143293.

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A225728 Primes p such that sum of primorials (A143293) not including p as a factor is divisible by p.

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A369061 Numbers k such that k + k'*2 is equal to a partial sum of primorial numbers (a term of A143293), where k' stands for the arithmetic derivative of k, A003415.