A104350 -id:A104350 - 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.

A104355 Number of trailing zeros in decimal representation of A104350(n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI link
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000079, A027868, A080193, A104350, A104356, A122840.

IntegerExponent[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]], 10] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = valuation(prod(k = 2, n, gpf(k)), 10); \\ Amiram Eldar, Apr 08 2024

\\ See link. David A. Corneth, Apr 08 2024

a(A104356(n)) = n and a(m) < n for m < A104356(n);

a(n) = A122840(A104350(n)). - Reinhard Zumkeller, Mar 10 2013

A104351 Number of digits in decimal representation of A104350(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 13, 13, 14, 15, 16, 17, 18, 19, 19, 20, 22, 22, 24, 24, 25, 26, 27, 28, 29, 30, 32, 32, 34, 35, 36, 37, 38, 40, 41, 42, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 54, 55, 57, 58, 59, 59, 60, 61, 63, 64, 66, 67, 69, 69, 71, 72

Offset: 1

Reinhard Zumkeller, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A034886, A055642, A104350.

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

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = logint(prod(k = 2, n, gpf(k)), 10) + 1; \\ Amiram Eldar, Apr 08 2024

a(n) = A055642(A104350(n)).

A104352 Number of divisors of A104350(n).

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 48, 64, 96, 192, 240, 480, 720, 960, 1200, 2400, 2880, 5760, 7200, 9600, 14400, 28800, 33600, 40320, 60480, 69120, 86400, 172800, 201600, 403200, 483840, 645120, 967680, 1161216, 1306368, 2612736, 3919104, 5225472, 5971968

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000005, A104350, A104353, A027423.

DivisorSigma[0, FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 50]]] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = numdiv(prod(k = 2, n, gpf(k))); \\ Amiram Eldar, Apr 08 2024

a(n) = A000005(A104350(n)).

Offset changed to 1 and a(1) prepended by Amiram Eldar, Apr 08 2024

A104353 Sum of divisors of A104350(n).

Original entry on oeis.org

1, 3, 12, 28, 168, 546, 4368, 9360, 28800, 148800, 1785600, 5401440, 75620160, 538793640, 2711348640, 5603453856, 100862169408, 303420079872, 6068401597440, 30380907997440, 213199354368000, 2362959510912000, 56711028261888000, 170288884313856000

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. A000203, A104350, A104352, A062569.

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

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = sigma(prod(k = 2, n, gpf(k))); \\ Amiram Eldar, Apr 08 2024

a(n) = A000203(A104350(n)).

a(14), a(21) corrected by Georg Fischer, Feb 28 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

A104355 Number of trailing zeros in decimal representation of A104350(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A104351 Number of digits in decimal representation of A104350(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula