A104365 -id:A104365 - cp's OEIS frontend

A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

2, 3, 7, 13, 61, 181, 13, 2521, 7561, 103, 415801, 1247401, 167, 191, 211, 127, 23, 40357, 1099944846001, 349, 41, 251, 37, 2243, 146100174169950001, 103, 53, 1217, 1156675078903494150001, 47, 2939, 251, 857, 41, 547, 13127, 47, 48563, 281, 1336484560722851, 479, 373, 2179, 577670972464621571, 17491, 1399, 97, 22893547

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104365(n)).

Tyler Busby, Table of n, a(n) for n = 1..170 (terms 1..100 from Chai Wah Wu, terms 101..165 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104367, A104358, A051301, A051342, A104358, A104365, A104367, A104368, A104369, A104370, A104371, A104372.

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1])); \\ A006530
spf(n) = if (n==1, 1, vecmin(factor(n)[,1])); \\ A020639
a(n) = spf(prod(i=2, n, gpf(i))+1); \\ Michel Marcus, Feb 21 2023

Corrected by D. S. McNeil, Dec 10 2010

A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 97, 2521, 7561, 367, 415801, 1247401, 97103, 594311, 2689891, 269, 415147, 1434493, 1099944846001, 13421, 938977307561, 1687166397251, 6121943187511, 13027211250107, 146100174169950001, 1389833, 10603380543703, 2129284819, 1156675078903494150001, 132597517693, 47172675889, 11159737, 20350106034371

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 1..76 from Amiram Eldar, terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A006530, A002583, A002585, A104350, A104359, A104365, A104366, A104368, A104369, A104370, A104371, A104372.

a[n_] := FactorInteger[1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]][[-1, 1]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A006530(A104365(n)).

Corrected by T. D. Noe, Nov 15 2006

A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A001221, A054988, A066856, A104350, A104359, A104360, A104365, A104366, A104367, A104369, A104370, A104371, A104372.

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(A104365(n)).

a(51)-a(76) from Amiram Eldar, Feb 12 2020
a(77)-a(81) from Jinyuan Wang, Apr 02 2020
Terms a(82) onward from Max Alekseyev, Oct 03 2022

A104369 Number of divisors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000005, A064144, A104350, A104361, A104365, A104366, A104367, A104368, A104370, A104371, A104372.

a[n_] := DivisorSigma[0, 1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000005(A104365(n)).

a(51)-a(76) from Amiram Eldar, Feb 12 2020

Terms a(77) onward from Max Alekseyev, Oct 03 2022

A104370 Sum of divisors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

3, 4, 8, 14, 62, 182, 1372, 2522, 7562, 38272, 415802, 1247402, 16313472, 114107904, 570257104, 1161216000, 21043021824, 57893308852, 1099944846002, 5530978026000, 39437046917604, 425165932107504, 10235889009520064, 29233062045242352, 146100174169950002, 1917846796927501440, 5805987050510562240, 39918123659008838880

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 1..76 terms from Amiram Eldar, terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000203, A104350, A104362, A104365, A104366, A104367, A104368, A104369, A104371, A104372.

a[n_] := DivisorSigma[1, 1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 30] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000203(A104365(n)).

A104371 Euler's totient of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 1152, 2520, 7560, 37332, 415800, 1247400, 16118932, 112918900, 564876900, 1109481408, 17645365584, 57890359152, 1099944846000, 5468570553600, 37559092302400, 421791599312500, 9256378099515120, 29207007622737652, 146100174169950000, 1880759745476519040, 5589847741506645552, 39852571442073216768

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 1..76 terms from Amiram Eldar, terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000010, A104350, A104363, A104365, A104366, A104367, A104368, A104369, A104370, A104372.

a[n_] := EulerPhi[1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000010(A104365(n));

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

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

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

Original entry on oeis.org

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;

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

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

A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

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A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

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A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

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A104369 Number of divisors of A104365(n) = A104350(n) + 1.

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A104370 Sum of divisors of A104365(n) = A104350(n) + 1.

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A104371 Euler's totient of A104365(n) = A104350(n) + 1.

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

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

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