A104350 -id:A104350 - cp's OEIS frontend

A104373 Numbers m such that (A104350(m)-1, A104350(m)+1) is a twin prime pair.

3, 4, 5, 6, 9, 11

Offset: 1

Reinhard Zumkeller, Mar 06 2005

No more terms < 2000. - David Wasserman, Apr 24 2008

a(7) > 5000, if it exists. - Amiram Eldar, Apr 08 2024

			a(5)=9: A104350(9) = 2*3*2*5*3*7*2*3 = 7560, A000040(959) = 7559 = 7560-1, A000040(960) = 7561 = 7560+1.

Cf. A104350, A104357, A104364, A104365, A104372.

Cf. A001359, A006512, A014574.

Position[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]], ?(PrimeQ[# - 1] && PrimeQ[# + 1] &)] // Flatten (* _Amiram Eldar, Apr 08 2024 *)

A104354 Euler's totient of A104350(n).

Original entry on oeis.org

1, 1, 2, 4, 16, 48, 288, 576, 1728, 8640, 86400, 259200, 3110400, 21772800, 108864000, 217728000, 3483648000, 10450944000, 188116992000, 940584960000, 6584094720000, 72425041920000, 1593350922240000, 4780052766720000, 23900263833600000, 310703429836800000

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..640
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000010, A048855, A104350.

EulerPhi[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 30]]] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = my(f=factor(n)[, 1]); f[#f];
a(n) = eulerphi(prod(i=2, n, gpf(i))); \\ Michel Marcus, Nov 12 2023

a(n) = A000010(A104350(n)).

a(1) prepended by Michel Marcus, Nov 12 2023

A104356 Smallest m such that A104350(m) has exactly n trailing zeros in decimal representation.

Original entry on oeis.org

1, 5, 10, 15, 20, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Reinhard Zumkeller, Mar 06 2005

A104355(a(n)) = n and A104355(m) < n for m < a(n).

a(n) = if n=0 then 1 else (if n<5 then 5*n else 2^n).

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A046670 Partial sums of A006530.

Original entry on oeis.org

1, 3, 6, 8, 13, 16, 23, 25, 28, 33, 44, 47, 60, 67, 72, 74, 91, 94, 113, 118, 125, 136, 159, 162, 167, 180, 183, 190, 219, 224, 255, 257, 268, 285, 292, 295, 332, 351, 364, 369, 410, 417, 460, 471, 476, 499, 546, 549, 556, 561, 578, 591, 644, 647, 658, 665, 684

Offset: 1

N. J. A. Sloane

Handbook of Number Theory, D. S. Mitrinovic et al., Kluwer, Section IV.1.

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]

Cf. A006530, A046669, A104350, A360559.

a046670 n = a046670_list !! (n-1)
a046670_list = scanl1 (+) a006530_list -- Reinhard Zumkeller, Jun 15 2013

Accumulate[Prepend[Table[FactorInteger[n][[-1,1]],{n,2,100}],1]] (* Harvey P. Dale, Jun 11 2011 *)

gpf(n)=if(n<4,n,n=factor(n)[,1];n[#n])
a(n)=sum(k=1,n,gpf(k)) \\ Charles R Greathouse IV, Feb 19 2014

a(n) = Pi^2/12 * n^2/log n + O(n^2/log^2 n). [See Mitrinovic et al.] - Charles R Greathouse IV, Feb 19 2014

More terms from James Sellers

A072486 a(1) = 1, a(n) = a(n-1) times smallest prime factor of n.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 840, 1680, 5040, 10080, 110880, 221760, 2882880, 5765760, 17297280, 34594560, 588107520, 1176215040, 22348085760, 44696171520, 134088514560, 268177029120, 6168071669760, 12336143339520, 61680716697600

Offset: 1

Amarnath Murthy, Jul 13 2002

nonn

a(n) is the product of the smallest divisors > 1 of all the numbers from 2 to n.

			a(5) = 60. a(6) = 1*2*3*2*5*2 = 120. Also a(6) = a(5) * 2 = 120.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A020639, A104350.

a072486 n = a072486_list !! (n-1)
a072486_list = scanl1 (*) a020639_list
-- Reinhard Zumkeller, Apr 10 2014

a = {1}; Do[AppendTo[a, a[[n - 1]] FactorInteger[n][[1, 1]]], {n, 2, 25}]; a (* Michael De Vlieger, Aug 25 2015 *)
nxt[{n_,a_}]:={n+1,a*FactorInteger[n+1][[1,1]]}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, May 03 2018 *)

al(n)=local(v=vector(n));v[1]=1;for(k=2,n,v[k]=v[k-1]*factor(k)[1,1]);v  \\ Franklin T. Adams-Watters, Jan 13 2012

A072486(n)=prod(k=2,n,factor(k)[1,1])  \\ M. F. Hasler, Jan 13 2012

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 25 2003

A330043 Product of largest prime power factors of numbers <= n.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 360, 2520, 20160, 181440, 907200, 9979200, 39916800, 518918400, 3632428800, 18162144000, 290594304000, 4940103168000, 44460928512000, 844757641728000, 4223788208640000, 29566517460480000, 325231692065280000, 7480328917501440000, 59842631340011520000

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Partial products of A034699.

Cf. A000720, A000961, A001221, A034699, A048803, A104350, A210208, A246655, A284521.

a[n_] := Product[Max[#[[1]]^#[[2]] & /@ FactorInteger@k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

A001221(a(n)) = A000720(n).

cp's OEIS Frontend

A104373 Numbers m such that (A104350(m)-1, A104350(m)+1) is a twin prime pair.

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A104354 Euler's totient of A104350(n).

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A104356 Smallest m such that A104350(m) has exactly n trailing zeros in decimal representation.

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A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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A046670 Partial sums of A006530.

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A072486 a(1) = 1, a(n) = a(n-1) times smallest prime factor of n.

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A330043 Product of largest prime power factors of numbers <= n.

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