A060389 -id:A060389 - cp's OEIS frontend

A000918 a(n) = 2^n - 2.

-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 68719476734, 137438953470

Offset: 0

N. J. A. Sloane

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011
For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002
For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004
Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004
For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005
Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005
From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005
The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006
Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007
Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007
It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009
First differences of A005803. - Reinhard Zumkeller, Oct 12 2011
For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014
For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015
Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015
Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020
For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019
The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020
For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020
For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022
For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

			a(4) = 14 because the 14 = 6 + 4 + 4 rationals (in lowest terms) for n = 3 are (see the Jun 14 2017 formula above): 1/2, 1, 3/2, 2, 5/2, 3; 1/4, 3/4, 5/4, 7/4; 1/8, 3/8, 5/8, 7/8. - _Wolfdieter Lang_, Jun 14 2017

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. - Mohammad K. Azarian, Oct 27 2011
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 86, Exercise 3.16.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
S. Bilotta, E. Grazzini, and E. Pergola, Enumeration of Two Particular Sets of Minimal Permutations, J. Int. Seq. 18 (2015) 15.10.2
R. B. Campbell, The effect of inbreeding constraints and offspring distribution on time to the most recent common ancestor, Journal of Theoretical Biology, Volume 382, 7 October 2015, Pages 74-80.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one. - _N. J. A. Sloane_, Sep 27 2010
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 77
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Japanese Mathematical Olympiad 1993, Final Round - Problem 2, Feb 11 1993.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
T. Manneville, V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Kanstantsin Pashkovich, Symmetry in Extended Formulations of the Permutahedron [sic], arXiv:0912.3446 [math.CO], 2009-2013. [_Jonathan Vos Post_, Dec 17 2009]
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Pratik Poddar, Consecutive Heads Puzzle, Oct 2009.
H. P. Robinson, Letter to N. J. A. Sloane, Sep 1975
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Eric Weisstein's World of Mathematics, Sphere Line Picking
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index to sequences related to Olympiads.

Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121, A110146.
Cf. A033484, A000225, A000325, A095151, A125128.
Cf. A058809 (3^n-3, n>0).

a000918 = (subtract 2) . (2 ^)
a000918_list = iterate ((subtract 2) . (* 2) . (+ 2)) (- 1)
-- Reinhard Zumkeller, Apr 23 2013

[2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011

Maple
```
seq(2^n-2,n=0..20) ;
```

Table[2^n - 2, {n, 0, 29}] (* Alonso del Arte, Dec 01 2012 *)

a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011

def A000918(n): return (1<Chai Wah Wu, Jun 10 2025

a(n) = 2*A000225(n-1).
G.f.: 1/(1 - 2*x) - 2/(1 - x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
For n >= 1, a(n) = A008970(n + 1, 2). - Philippe Deléham, Feb 21 2004
G.f.: (3*x - 1)/((2*x - 1)*(x - 1)). - Simon Plouffe in his 1992 dissertation for the sequence without the leading -1
a(n) = 2*a(n - 1) + 2. - Alexandre Wajnberg, Apr 25 2005
a(n) = A000079(n) - 2. - Omar E. Pol, Dec 16 2008
a(n) = A058896(n)/A052548(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = A164874(n - 1, n - 1) for n > 1. - Reinhard Zumkeller, Aug 29 2009
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n > 0. - Reinhard Zumkeller, Feb 28 2010
a(n + 1) = A027383(2*n - 1). - Jason Kimberley, Nov 02 2011
G.f.: U(0) - 1, where U(k) = 1 + x/(2^k + 2^k/(x - 1 - x^2*2^(k + 1)/(x*2^(k + 1) - (k + 1)/U(k + 1) ))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n+1) is the sum of row n in triangle A051601. - Reinhard Zumkeller, Aug 05 2013
a(n+1) = A127330(n,0). - Reinhard Zumkeller, Nov 16 2013
a(n) = Sum_{k=1..n-1} binomial(n, k) for n > 0. - Dan McCandless, Nov 14 2015
From Miquel Cerda, Aug 16 2016: (Start)
a(n) = A000225(n) - 1.
a(n) = A125128(n-1) - A000325(n).
a(n) = A095151(n) - A125128(n) - 1. (End)
a(n+1) = 2*(n + Sum_{j=1..n-1} (n-j)*2^(j-1)), n >= 1. This is the number of the rationals k/2, k = 1..2*n for n >= 1 and (2*k+1)/2^j for j = 2..n, n >= 2, and 2*k+1 < n-(j-1). See the example for n = 3 below. Motivated by the proposal A287012 by Mark Rickert. - Wolfdieter Lang, Jun 14 2017

Maple programs fixed by Vaclav Kotesovec, Dec 13 2014

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

Original entry on oeis.org

1, 2, 12, 360, 75600, 174636000, 5244319080000, 2677277333530800000, 25968760179275365452000000, 5793445238736255798985527240000000, 37481813439427687898244906452608585200000000, 7517370874372838151564668004911177464757864076000000000, 55784440720968513813368002533861454979548176771615744085560000000000

Offset: 0

N. J. A. Sloane

Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
Superprimorials, from primorials by analogy with superfactorials.
Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
Subsequence of A130091. - Reinhard Zumkeller, May 06 2007
Hankel transform of A171448. - Paul Barry, Dec 09 2009
This might be a good place to explain the name "Chernoff sequence" since his name does not appear in the References or Links as of Mar 22 2014. - Jonathan Sondow, Mar 22 2014
Pickover (1992) named this sequence after Paul Chernoff of California, who contributed this sequence to his book. He was possibly referring to American mathematician Paul Robert Chernoff (1942 - 2017), a professor at the University of California. - Amiram Eldar, Jul 27 2020

			a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.

T. D. Noe, Table of n, a(n) for n=0..25

Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
Cf. A002110, A051357.
A000142 counts divisors of superprimorials.
A000325 counts uniform divisors of superprimorials.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 is a sister-sequence.
A118914 has row a(n) equal to {1..n}.
A124010 has row a(n) equal to {n..1}.
A130091 lists numbers with distinct prime multiplicities.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336426 gives non-products of superprimorials.
Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.

a006939 n = a006939_list !! n
a006939_list = scanl1 (*) a002110_list -- Reinhard Zumkeller, Jul 21 2012

[1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // G. C. Greubel, Oct 14 2018

a := []; printlevel := -1; for k from 0 to 20 do a := [op(a),product(ithprime(i)^(k-i+1),i=1..k)] od; print(a);

Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* Harvey P. Dale, Jul 07 2011 *)
Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* Alonso del Arte, Sep 30 2011 *)

a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from math import prod
from sympy import prime
def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # Chai Wah Wu, Aug 12 2025

a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - N. J. A. Sloane, Feb 20 2005
a(0) = 1, a(n) = a(n - 1)p(n)#, where p(n)# is the n-th primorial A002110(n) (the product of the first n primes). - Alonso del Arte, Sep 30 2011
log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - Charles R Greathouse IV, Mar 14 2011
A181796(a(n)) = A000110(n+1). It would be interesting to have a bijective proof of this theorem, which is stated at A181796 without proof. See also A336420. - Gus Wiseman, Aug 03 2020

Corrected and extended by Labos Elemer, May 30 2001

A070826 One half of product of first n primes A000040.

Original entry on oeis.org

1, 3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 3234846615, 100280245065, 3710369067405, 152125131763605, 6541380665835015, 307444891294245705, 16294579238595022365, 961380175077106319535, 58644190679703485491635, 3929160775540133527939545, 278970415063349480483707695

Offset: 1

Wolfdieter Lang, May 10 2002

Also, with offset 0, product of first n odd primes. - N. J. A. Sloane, Feb 26 2017
Identical to A002110(n)/2, n>=1.
a(n+1) is the least odd number with exactly n distinct prime divisors. - Labos Elemer, Mar 24 2003
Also, odd numbers n for which sigma(n)*phi(n)/n^2 reaches a new record low, monotonically decreasing to the lower bound 8/Pi^2. - M. F. Hasler, Jul 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A003266 (for Fibonacci), A070825 (for Lucas), A003046 (for Catalan).
Cf. also A002110, A024451, A060389, A091852, A276086, A203008 [= A003415(a(1+n))].
Range of A196529.

a:=n->mul(ithprime(j), j=2..n):seq(a(n), n=1..17); # Zerinvary Lajos, Aug 24 2008

Rest[ FoldList[ Times, 1, Prime[ Range[ 18]] ]]/2 (* Robert G. Wilson v, Feb 17 2004 *)
FoldList[Times, 1, Prime[Range[2, 18]]] (* Zak Seidov, Jan 26 2009 *)

a(n) = prod(k=2, n, prime(k)) \\ Michel Marcus, Mar 25 2017, simplified by M. F. Hasler, Jul 09 2025

from sympy import primorial
def A070826(n): return primorial(n)>>1 # Chai Wah Wu, Jul 21 2022

a(n) = A002110(n)/2.
From Antti Karttunen, Feb 06 2024: (Start)
a(1) = 1, and for n > 1, a(n) = A276086(A060389(n-1)).
a(n) = A024451(n) - 2*A203008(n-1).
(End)
a(n) = A000040(n)*a(n-1) for n > 1, a(1) = 1. - M. F. Hasler, Jul 09 2025

Formula corrected by Gary Detlefs, Dec 07 2011

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

0, 2, 6, 4, 30, 8, 210, 6, 12, 32, 2310, 10, 30030, 212, 36, 8, 510510, 14, 9699690, 34, 216, 2312, 223092870, 12, 60, 30032, 18, 214, 6469693230, 38, 200560490130, 10, 2316, 510512, 240, 16, 7420738134810, 9699692, 30036, 36, 304250263527210, 218, 13082761331670030, 2314, 42, 223092872, 614889782588491410, 14

Offset: 1

Antti Karttunen, May 27 2024

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)).

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. also A000040, A000720, A002110, A003961, A034386, A060389, A108951, A276085, A276154, A373984 [= A108951(n)-a(n)], A373985 [= gcd(a(n), A108951(n))], A373986, A373987.

A373158(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*prod(i=1,primepi(f[i, 1]),prime(i))); }; \\ corrected Jun 25 2024

From Antti Karttunen, Jun 25 2024, Oct 28 2024: (Start)
a(n) = A276085(A003961(n)).
For n >= 1, a(A000040(n)) = A002110(n), a(A002110(n)) = A060389(n).
(End)

Data [first incorrect term was at a(8)] and the faulty PARI-program corrected by Antti Karttunen, Jun 25 2024

A203008 (n-1)-st elementary symmetric function of the first n odd primes; a(0) = 0.

Original entry on oeis.org

0, 1, 8, 71, 886, 12673, 230456, 4633919, 111429982, 3343015913, 106868339918, 4054408822031, 169941130770676, 7459593754902673, 357142287146260646, 19235986110046059943, 1151217759731312559002, 71185663518687172418657

Offset: 0

Clark Kimberling, Dec 29 2011

nonn

Arithmetic derivative of the product of first n odd primes. - Antti Karttunen, Jan 31 2024

Primes occur at indices: 3, 19, 23, 117, 119, 127, 161, 209, ..., and they are: 71, 346723099672193960193396979, 15360643606799479140185671512081451, ... - Antti Karttunen, Feb 06 2024

Antti Karttunen, Table of n, a(n) for n = 0..349

Cf. A000035, A003415, A024451, A060389, A070826 (n-th. symm. function), A071148 (1st symm. func), A327860.

f[k_] := Prime[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203008 *)

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A203008(n) = if(!n,n,A003415(A002110(1+n)/2)); \\ Antti Karttunen, Jan 31 2024

From Antti Karttunen, Jan 31 2024 and Feb 06 2024: (Start)
a(n) = A003415(A070826(1+n)) = (1/2)*(A024451(1+n)-A070826(1+n)).
For n >= 1, a(n) = A327860(A060389(n)).
A000035(a(n)) = A000035(n).
(End)

Term a(0) = 0 prepended by Antti Karttunen, Jan 31 2024

A217723 a(n) = (sum of first n primorial numbers) minus 1.

Original entry on oeis.org

1, 7, 37, 247, 2557, 32587, 543097, 10242787, 233335657, 6703028887, 207263519017, 7628001653827, 311878265181037, 13394639596851067, 628284422185342477, 33217442899375387207, 1955977793053588026277, 119244359152460559009547

Offset: 1

Paul Larm, Mar 21 2013

			For n = 4, a(4) = 2 + 2*3 + 2*3*5 + 2*3*5*7 - 1 = 247.

Paul Larm (terms 1..20) and Amiram Eldar (terms 21..100), Table of n, a(n) for n = 1..100
Paul Larm, Table of a(n) for n = 2..25 showing prime factors.

Equals A060389 - 1. Cf. A223546, A223548.

Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] - 1 (* Alonso del Arte, Mar 21 2013 *)
Accumulate[Table[Fold[Times,Prime[Range[n]]],{n,20}]]-1 (* Harvey P. Dale, May 23 2020 *)

a(1) = P(1)# - 1, a(2) = P(1)# + P(2)# -1; where P(n)# is the product of first n prime numbers (primorial#).

A084737 Beginning with 1, numbers such that (a(n+2)-a(n+1))/(a(n+1)-a(n)) = prime(n).

Original entry on oeis.org

1, 2, 4, 10, 40, 250, 2560, 32590, 543100, 10242790, 233335660, 6703028890, 207263519020, 7628001653830, 311878265181040, 13394639596851070, 628284422185342480, 33217442899375387210, 1955977793053588026280

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 14 2003

nonn

Successive differences are primorials.

			a(3) = 4, a(4) = 10 and a(5) = 40 and (40-10)/(10-4) = 5 = prime(3).

Index entries for sequences related to primorial numbers

Cf. A002110, A060389, A088860, A143293, A276085.

Join[{1},Accumulate[FoldList[Times,1,Prime[Range[20]]]]+1] (* Harvey P. Dale, Dec 14 2011 *)

From Antti Karttunen, Feb 06 2024: (Start)
For n >= 1, a(n) = A276085(2*A002110(n-1)).
For n >= 2, a(n) = 1 + A143293(n-2).
For n >= 3, a(n) = 2 + A060389(n-2).
(End)

More terms from Vladeta Jovovic, Jun 17 2003

A225841 Numbers n such that the sum of first n primorial numbers is divisible by n.

Original entry on oeis.org

1, 2, 4, 523, 1046, 2092

Offset: 1

Alex Ratushnyak, May 21 2013

The k-th primorial number is defined as the product of the first k primes.
The next term, if it exists, is greater than 14000000. - Alex Ratushnyak, Jun 13 2013
If a prime p | a(n) for some n, then p = 2, p = 523, or p > 10^8. Any such prime is itself a member of this sequence. From this (and a small amount of additional calculation) it follows that any other terms below 10^10 are of the form 2^k * p for p > 10^8. - Charles R Greathouse IV, Feb 09 2014

			2 + 2*3 + 2*3*5 + 2*3*5*7 = 2 + 6 + 30 + 210 = 248, because 248 is divisible by 4, the latter is in the sequence.

Cf. A060389, A002110, A045345, A143293, A225727.

With[{nn=2100},Select[Thread[{Accumulate[FoldList[Times,Prime[ Range[ nn]]]],Range[nn]}],Divisible[ #[[1]],#[[2]]]&]][[All,2]] (* Harvey P. Dale, Jul 29 2021 *)

list(maxx)={n=prime(1); cnt=1;summ=0;scnt=0;
while(n<=maxx,summ=summ+prodeuler(x=1,prime(cnt),x);
if(summ%cnt==0,scnt++;print(scnt,"  ",cnt) );cnt++;n=nextprime(n+1) ); }
\\note MUST increase precision to 10000+ digits \\Bill McEachen, Feb 04 2014

P=1;S=n=0;forprime(p=2,1e4,S+=P*=p;if(S%n++==0,print1(n", "))) \\ Charles R Greathouse IV, Feb 05 2014

is(n)=my(q=prime(n),P=Mod(1,n),S);forprime(p=2,q,S+=P*=p);!S \\ Charles R Greathouse IV, Feb 05 2014

primes = []
n = 1
sum = 2
primorial = 6
def addPrime(k):
  global n, sum, primorial
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  sum += primorial
  primorial *= k
  n += 1
  if sum % n == 0:  print(n, end=',')
print(1, end=',')
for p in range(5, 100000, 6):
  addPrime(p)
  addPrime(p+2)

from itertools import accumulate, count, islice
from operator import mul
from sympy import prime
def A225841_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate((prime(n) for n in count(1)), mul))) if m % (i+1) == 0)
A225841_list = list(islice(A225841_gen(),6)) # Chai Wah Wu, Feb 23 2022

cp's OEIS Frontend

A257466 Smallest prime number p such that p + pps(1), p + pps(2), ..., p + pps(n) are all prime but p + pps(n+1) is not, where pps(n) is the partial primorial sum (A060389(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A000918 a(n) = 2^n - 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A070826 One half of product of first n primes A000040.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A203008 (n-1)-st elementary symmetric function of the first n odd primes; a(0) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs