A059958 -id:A059958 - cp's OEIS frontend

A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

1, 9, 242, 11824, 856824, 82986080, 10097121280, 1481787433920, 254874712419200, 50305519571800960, 11209381628379724800, 2783746998856794752000, 762476362390276346060800, 228363072063685762536960000, 74247696727054926125971251200, 26044746725090717967744412672000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-2,n-k) * ((m-1)*k-1)/((m-1)*n-1), m>2, is asymptotic to (m-2) * m^(m*n-3/2) / (m-1)^((m-1)*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)
where G(x) = 1 + x*G(x)^10 is the g.f. of A059958:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...

Cf. A251580, A251700, A059968.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251668, A251669.

Table[Sum[10^k * n!/k! * Binomial[10*n-k-2,n-k] * (9*k-1)/(9*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n)); n!*polcoeff(exp(10*x*G^9)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9 + 8*G'(x)/G(x).
(2) A(x) = F(x/A(x)^9) where F(x) is the e.g.f. of A251700.
(3) A(x) = Sum_{n>=0} A251700(n)*(x/A(x)^9)^n/n! where A251700(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.
(4) [x^n/n!] A(x)^(9*n+1) = (8*n+1) * (9*n+1)^(n-1) * 10^n.
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-2,n-k) * (9*k-1)/(9*n-1) for n>=0.
Recurrence: 81*(3*n-2)*(3*n-1)*(9*n-8)*(9*n-7)*(9*n-5)*(9*n-4)*(9*n-2)*(9*n-1)*(100000000*n^9 - 1508750000*n^8 + 10158500000*n^7 - 40108637500*n^6 + 102477510000*n^5 - 175985889125*n^4 + 203494963150*n^3 - 153061617555*n^2 + 68057955478*n - 13624029912)*a(n) = 800*(1250000000000000*n^18 - 25734375000000000*n^17 + 248379687500000000*n^16 - 1494668125000000000*n^15 + 6291920187500000000*n^14 - 19707236445312500000*n^13 + 47696214907031250000*n^12 - 91443867836531250000*n^11 + 141240231848528125000*n^10 - 177729148289358906250*n^9 + 183386452781820390625*n^8 - 155416253373710737500*n^7 + 107706559814898413750*n^6 - 60246014246053412750*n^5 + 26474457002621149925*n^4 - 8675686414409435660*n^3 + 1905677176596950796*n^2 - 212632849946745072*n - 10904042717568)*a(n-1) + 10000000000*(100000000*n^9 - 608750000*n^8 + 1688500000*n^7 - 2844137500*n^6 + 3264185000*n^5 - 2692901625*n^4 + 1611256650*n^3 - 663025355*n^2 + 151278318*n + 4536)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 8 * 10^(10*n-3/2) / 3^(18*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A118478 a(n) is the smallest m such that m*(m+1) is divisible by the first n prime numbers.

Original entry on oeis.org

1, 2, 5, 14, 209, 714, 714, 62985, 367080, 728364, 64822394, 1306238010, 11182598504, 715041747420, 51913478860880, 454746157008780, 9314160363311804, 261062105979210899, 261062105979210899, 696537082207206753590, 54097844397380813592485, 286495021083846822067820

Offset: 1

Giovanni Resta, May 05 2006

nonn

a(n)*(a(n)+1)/(product of first n primes) = 1, 1, 1, 1, 19, 17, 1, 409, 604, 82, 20951, 229931, 411012, 39080794, 4382914408, ... - Robert G. Wilson v, May 13 2006 [This is now A215021. - N. J. A. Sloane, Aug 02 2012]

			a(8) = 62985 since 62985*62986 = 2*3*5*7*11*13*17*19*409, i.e., it is divisible by the first 8 prime numbers (2,3,..,19).

Jinyuan Wang, Table of n, a(n) for n = 1..30 (terms 1..25 from Robert Gerbicz)

Cf. A002110, A002378, A193314, A215021, A053670, A059958.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a118478 n = (+ 1) . fromJust $ elemIndex 0 $
            map (flip mod (a002110 n)) $ tail a002378_list
-- Reinhard Zumkeller, Jun 14 2015
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import sieve, prime, primorial
from sympy.ntheory.modular import crt
def A118478(n): return 1 if n == 1 else int(min(min(crt((m, (k:=primorial(n))//m), (0, -1))[0], crt((k//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, n//2+1) for d in combinations(sieve.primerange(prime(n)+1), l)))) # Chai Wah Wu, May 31 2022

f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[ !IntegerQ@ Sqrt[4k*p + 1], k++ ]; Floor@ Sqrt[k*p]]; Array[f, 15] (* Robert G. Wilson v, May 13 2006 *)

P=primes(25);T=1;for(n=1,25,T*=P[n];m=T;for(k=2^(n-1),2^n-1,u=binary(k); a=1;for(i=1,n,if(u[i],a*=P[i]));b=T/a;w=bezout(a,b);if(w[1]<=0,w[1]+=b); c=a*w[1]-1;m=min(m,c);w[1]=b-w[1];if(w[1]<=0,w[1]+=b);c=a*w[1];m=min(m,c)); print1(m,",")) \\ Robert Gerbicz, Aug 24 2006

a(n) = Min_{m | m*(m+1) is divisible by A002110(n)}.

More terms from Robert Gerbicz, Aug 24 2006

A083002 Smallest oblong number having at least n distinct prime divisors.

Original entry on oeis.org

2, 6, 30, 210, 4290, 43890, 510510, 53501910, 1487683470, 64790866140, 530514844860, 126408523110870, 3425113062060690, 660393717163700520, 26657280574571657010, 3448055881024876471350, 308480161111936386482910

Offset: 1

Jason Earls, May 30 2003

nonn

a(18) <= 32521466098360753728404190. - Donovan Johnson, Oct 05 2011

Oblong numbers are those of the form n(n+1) = A002378(n) = 2*A000217(n).

			a(4)= 210 = 2*3*5*7.

Cf. A002378, A059958.

{odf(m) = print1(0","); for(n=1,m, k=1; while(omega(k*(k+1))!=n,k++); print1(k*(k+1)",")) }

More terms from Don Reble, Jun 03 2003
a(15)-a(16) from Donovan Johnson, Apr 26 2009
a(17) from Donovan Johnson, Sep 15 2010

A069354 Lowest base with simple divisibility test for n primes; smallest B such that omega(B) + omega(B-1) = n.

Original entry on oeis.org

2, 3, 6, 15, 66, 210, 715, 7315, 38571, 254541, 728365, 11243155, 58524466, 812646121, 5163068911, 58720148851, 555409903686, 4339149420606, 69322940121436, 490005293940085, 5819629108725510, 76622240600506315

Offset: 1

Robert Munafo, Nov 19 2002

Indices of record values of primepi(n) - A181834(n) (the number of primes <= n which are not strongly prime to n). - Peter Luschny, Mar 17 2013
As pointed out by Don Reble on the SeqFan list, one has a(n) = A059958(n)+1 at least up to a(18), since so far A001221(m*(m+1)) = n (and not ">") for m = A059958(n). - M. F. Hasler, Jan 15 2014
10^13 < a(19) <= 69322940121436. - Giovanni Resta, Mar 24 2020

			a(4) = 15 because in base 15 you can test for divisibility by 4 different primes (3 and 5 directly, 2 and 7 by "casting out 14's")

Peter Luschny, Strong coprimality, November 2010.
Robert Munafo, Low bases with many divisibility tests

Cf. A001221, A181834.

A069354_list := proc(n) local i, L, Max; Max := 1; L := NULL;
for i from 2 to n do
   if nops(numtheory[factorset](i*(i-1))) = Max
   then Max := Max + 1; L := L,i fi;
od;
L end:  # Peter Luschny, Nov 12 2010

a(n) = A059958(n) + 1 for 0 < n < 19. - Robert G. Wilson v, Feb 18 2014

More terms added using data from A059958 (see there for credits) by M. F. Hasler, Jan 15 2014
a(19)-a(21) from Michael S. Branicky, Feb 11 2023
a(22) from Michael S. Branicky, Feb 23 2023

A363847 Numbers k such that Omega(m(m+1)) < Omega(k(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 2, 3, 7, 8, 15, 32, 63, 224, 255, 512, 3968, 4095, 14336, 32768, 65535, 180224, 262143, 1048575, 14680064, 16777215, 134217728, 268435455, 1073741823, 8589934592, 12884901887, 34359738368, 68719476735, 1099511627775, 4398046511103, 17592186044415, 35184372088832

Offset: 1

Amiram Eldar, Jun 24 2023

nonn

Terms a(2)-a(18) were found by Erdős and Nicolas (1978-1979).
Equivalently, numbers k such that Omega(m) + Omega(m+1) < Omega(k) + Omega(k+1), for all m < k.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 24, 26, 27, 31, 33, 34, 37, 38, 39, 40, 46, 48, 50, 51, 52, ... .

Paul Erdős and Jean-Louis Nicolas, Sur la fonction "nombre de facteurs premiers de n", Séminaire Delange-Pisot-Poitou, Théorie des nombres, Vol. 20, No. 2 (1978-1979), Talk no. 32, pp. 1-19. See p. 10.

Cf. A002378, A001222, A059958, A075088, A076550.

seq[kmax_] := Module[{o1 = 0, o2, om = 0, s = {}}, Do[o2 = PrimeOmega[k]; o = o1 + o2; If[o > om, om = o; AppendTo[s, k - 1]]; o1 = o2, {k, 2, kmax}]; s]; seq[10^5]

lista(kmax) = {my(o1 = 0, o2, om = 0); for(k = 2, kmax, o2 = bigomega(k); o = o1 + o2; if(o > om, om = o; print1(k-1, ", ")); o1 = o2); }

a(29)-a(32) from Martin Ehrenstein, Jul 08 2023

cp's OEIS Frontend

A251670 E.g.f.: exp(10*x*G(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

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A118478 a(n) is the smallest m such that m*(m+1) is divisible by the first n prime numbers.

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A083002 Smallest oblong number having at least n distinct prime divisors.

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A069354 Lowest base with simple divisibility test for n primes; smallest B such that omega(B) + omega(B-1) = n.

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A363847 Numbers k such that Omega(m*(m+1)) < Omega(k*(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).

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A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A363847 Numbers k such that Omega(m(m+1)) < Omega(k(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).