A104357 -id:A104357 - cp's OEIS frontend

A104359 Greatest prime factor of A104357(n) = A104350(n) - 1.

1, 5, 11, 59, 179, 1259, 229, 7559, 37799, 415799, 17569, 71437, 18979, 62597, 1135133999, 1646947, 445771, 277021, 5499724229999, 2217247573, 721381, 46313123, 29220034833989999, 16347569521, 5464930609, 4939567, 319699160368361, 2605998587146349, 178974179, 15701603, 116318025830291273, 126202964557

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Tyler Busby, Table of n, a(n) for n = 2..159 (terms 2..74 from Amiram Eldar, terms 75..145 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A006530, A002582, A002584, A104350, A104357, A104358, A104360, A104361, A104362, A104363, A104364, A104367.

a[n_] := FactorInteger[-1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]][[-1, 1]]; Array[a, 50, 2] (* Amiram Eldar, Feb 12 2020 *)

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1])); \\ A006530
a(n) = gpf(prod(i=2, n, gpf(i))-1); \\ Michel Marcus, Feb 21 2023

a(n) = A006530(A104357(n)).

A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 5, 11, 59, 179, 1259, 11, 7559, 37799, 415799, 71, 227, 5981, 9067, 1135133999, 11717, 61, 79, 5499724229999, 97, 1543, 31, 29220034833989999, 8937119, 181, 401, 124759, 443851, 31, 2141, 3082663, 8191, 37230797, 1697, 1408101540804746673385499999, 10613, 73, 59, 107, 79, 617, 163, 173

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104357(n)).

Tyler Busby, Table of n, a(n) for n = 2..181 (terms 2..120 from Chai Wah Wu, terms 121..160 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104357, A104359, A104360, A104361, A104362, A104363, A104364, A104366, A054415, A057713.

Typo in data corrected by Gionata Neri, Oct 20 2017

A104360 Number of distinct prime factors of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 3, 3, 1, 3, 3, 3, 1, 2, 4, 4, 2, 2, 5, 4, 2, 3, 2, 3, 1, 3, 4, 3, 5, 3, 4, 4, 5, 1, 3, 4, 2, 2, 2, 3, 1, 2, 3, 2, 3, 1, 4, 1, 3, 4, 6, 4, 4, 2, 6, 1, 5, 4, 4, 1, 2, 5, 7, 4, 5, 3, 4, 4, 5, 4, 5, 2, 4, 4, 5, 3, 3, 3, 2, 5, 2, 5, 4, 7, 2, 5, 3, 2, 6, 3, 4, 2, 3, 3, 3, 5, 4, 3, 5, 2

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A001221, A054989, A066877, A104350, A104357, A104358, A104359, A104361, A104362, A104363, A104364, A104368.

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; PrimeNu[Table[A104350[n] - 1, {n, 2,50}]] (* G. C. Greubel, May 10 2017 *)

a(n) = A001221(A104357(n)).

a(51)-a(74) from Amiram Eldar, Feb 12 2020
More terms from Jinyuan Wang, Apr 02 2020
Terms a(90) onward from Max Alekseyev, Oct 03 2022

A104361 Number of divisors of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 4, 2, 4, 8, 8, 2, 8, 8, 8, 2, 4, 16, 16, 4, 4, 32, 16, 4, 8, 4, 8, 2, 8, 16, 8, 32, 8, 16, 16, 32, 2, 8, 16, 4, 4, 4, 8, 2, 4, 8, 4, 8, 2, 16, 2, 8, 16, 64, 16, 16, 4, 64, 2, 32, 16, 16, 2, 4, 32, 128, 16, 32, 8, 16, 16, 32, 16, 32, 4, 16, 16, 32, 8, 8, 8, 4, 32, 4, 32, 16, 128, 4, 32, 8, 4

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000005, A064145, A104350, A104357, A104358, A104359, A104360, A104362, A104363, A104364, A104369.

a[n_] := DivisorSigma[0, -1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 50, 2] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000005(A104357(n)).

a(51)-a(74) from Amiram Eldar, Feb 12 2020

Terms a(75) onward from Max Alekseyev, Oct 03 2022

A104362 Sum of divisors of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 6, 12, 60, 180, 1260, 2760, 7560, 37800, 415800, 1265040, 16287864, 113538360, 567638664, 1135134000, 19298936664, 58868650320, 1113894381120, 5499724230000, 39112247205360, 423754918508832, 10054207233388032, 29220034833990000, 146100190526456640, 1915895635570469280, 5712343370808883200, 39885667247556843120

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145 (terms for n = 2..74 from Amiram Eldar)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000203, A104350, A104357, A104358, A104359, A104360, A104361, A104363, A104364, A104370.

A000142 := proc(n) RETURN(n!) ; end: A006530 := proc(n) local i, t1, t2, t3, t4; if n = 1 then RETURN(1) ; else t1 := numtheory[divisors](n); t2 := convert(t1, list); t3 := sort(t2); t4 := nops(t3); for i from 1 to t4 do if isprime(t3[t4+1-i]) then RETURN(t3[t4+1-i]); fi; od; RETURN(1); fi ; end: A104350 := proc(n) local k,resul ; resul := 1 ; for k from 1 to n do resul := resul*A006530(k) ; od ; RETURN(resul) ; end: A104357 := proc(n) A104350(n)-1 ; end: A104362 := proc(n) numtheory[sigma](A104357(n)) ; end: for n from 2 to 30 do printf("%d,",A104362(n)) ; od ; # R. J. Mathar, Oct 30 2006

a[n_] := DivisorSigma[1, Product[FactorInteger[k][[-1, 1]], {k, 1, n}]-1]; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Feb 10 2018 *)

a(n) = A000203(A104357(n));

a(p) = A104350(p) for primes p.

Corrected and extended by R. J. Mathar, Oct 30 2006

A104363 Euler's totient of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 4, 10, 58, 178, 1258, 2280, 7558, 37798, 415798, 1229760, 16144536, 113488440, 567495336, 1135133998, 19295619336, 56915913600, 1085995965600, 5499724229998, 37888326510336, 423202615117920, 9425816095466520, 29220034833989998, 146100157813443360, 1882777893068160000, 5683471349506454400, 39885027849235856880

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145 (terms for n = 2..74 from Amiram Eldar)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000010, A104357, A104358, A104359, A104360, A104361, A104362, A104364, A104371.

a[n_] := EulerPhi[-1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 50, 2] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000010(A104357(n)).

A104350 Partial products of largest prime factors of numbers <= n.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 1260, 2520, 7560, 37800, 415800, 1247400, 16216200, 113513400, 567567000, 1135134000, 19297278000, 57891834000, 1099944846000, 5499724230000, 38498069610000, 423478765710000, 9740011611330000

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Partial Products of A006530: a(n) = Product_{k=1..n} A006530(k).
a(n) = a(n-1)*A006530(n) for n>1, a(1) = 1;
A020639(a(n)) = A040000(n-1), A006530(a(n)) = A007917(n) for n>1.
A001221(a(n)) = A000720(n), A001222(a(n)) = A001477(n-1).
A007947(a(n)) = A034386(n).
a(n) = A000142(n) / A076928(n). [Corrected by Franklin T. Adams-Watters, Oct 30 2006]
In decimal representation: A104351(n) = number of digits of a(n), A104355(n) = number of trailing zeros of a(n).
A104357(n) = a(n) - 1, A104365(n) = a(n) + 1.

Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publ. Inst. Elie Cartan, Vol. 13, Nancy, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 1..641
Romeo Meštrović, 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.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000142, A002110, A006530, A007947, A020639, A046670, A072486, A076928, A104351, A104355, A104357, A104365.

Cf. A001620, A084945.

a104350 n = a104350_list !! (n-1)
a104350_list = scanl1 (*) a006530_list
-- Reinhard Zumkeller, Apr 10 2014

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; Table[A104350[n], {n, 30}] (* G. C. Greubel, May 09 2017 *)
FoldList[Times,Table[FactorInteger[n][[-1,1]],{n,30}]] (* Harvey P. Dale, May 25 2023 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=prod(i=2,n,gpf(i)) \\ Charles R Greathouse IV, Apr 29 2015

first(n)=my(v=vector(n,i,1)); forfactored(k=2,n, v[k[1]]=v[k[1]-1]*vecmax(k[2][,1])); v \\ Charles R Greathouse IV, May 10 2017

log(a(n)) = c * n * log(n) + c * (1-gamma) * n + O(n * exp(-log(n)^(3/8-eps))), where c is the Golomb-Dickman constant (A084945) and gamma is Euler's constant (A001620) (Tenenbaum, 1990). - Amiram Eldar, May 21 2021

More terms from David Wasserman, Apr 24 2008

A104365 a(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 1261, 2521, 7561, 37801, 415801, 1247401, 16216201, 113513401, 567567001, 1135134001, 19297278001, 57891834001, 1099944846001, 5499724230001, 38498069610001, 423478765710001, 9740011611330001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A006530, A006862, A038507, A104366, A104367, A104368, A104369, A104370, A104371, A104357.

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

a(n) = (a(n-1) - 1) * A006530(n) + 1 for n>1, a(1) = 0;

A104364 Primes of the form A104350(k) - 1.

Original entry on oeis.org

5, 11, 59, 179, 1259, 7559, 37799, 415799, 1135133999, 5499724229999, 29220034833989999, 1408101540804746673385499999, 43673268652925265723884051023987499999

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26

Intersection of A104357 and A000040.

Cf. A104372, A104373, A055490, A057705.

Select[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]] - 1, PrimeQ] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); if(n == 1, 1, p[#p]);}
lista(nmax) = {my(r = 1); for(k = 1, nmax, r * = gpf(k); if(isprime(r-1), print1(r-1, ", ")));} \\ Amiram Eldar, Apr 08 2024

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A104359 Greatest prime factor of A104357(n) = A104350(n) - 1.

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A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

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A104360 Number of distinct prime factors of A104357(n) = A104350(n) - 1.

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A104361 Number of divisors of A104357(n) = A104350(n) - 1.

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A104362 Sum of divisors of A104357(n) = A104350(n) - 1.

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A104363 Euler's totient of A104357(n) = A104350(n) - 1.

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A104350 Partial products of largest prime factors of numbers <= n.

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A104365 a(n) = A104350(n) + 1.

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