A104367 -id:A104367 - cp's OEIS frontend

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

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)).

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

Original entry on oeis.org

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

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula