A104350 -id:A104350 - cp's OEIS frontend

A104357 a(n) = A104350(n) - 1.

0, 1, 5, 11, 59, 179, 1259, 2519, 7559, 37799, 415799, 1247399, 16216199, 113513399, 567566999, 1135133999, 19297277999, 57891833999, 1099944845999, 5499724229999, 38498069609999, 423478765709999, 9740011611329999

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..640
R. Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104358, A104359, A104360, A104361, A104362, A104363, A104365, A033312, A057588.

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; Table[A104350[n] - 1, {n, 1, 50}] (* G. C. Greubel, May 09 2017 *)
FoldList[Times,Table[FactorInteger[n][[-1,1]],{n,30}]]-1 (* Harvey P. Dale, May 28 2018 *)

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

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;

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

Original entry on oeis.org

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)).

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

A104372 Primes of the form A104350(k) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 2521, 7561, 415801, 1247401, 1099944846001, 146100174169950001, 1156675078903494150001, 750321420485151941966263672363958662088980270355720625000001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..16

Intersection of A104365 and A000040.

Cf. A104364, A104373, A088332, A018239.

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

a(14) from Amiram Eldar, Apr 09 2024

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)).

cp's OEIS Frontend

A104357 a(n) = A104350(n) - 1.

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

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

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

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A104372 Primes of the form A104350(k) + 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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