A104372 -id:A104372 - cp's OEIS frontend

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

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

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

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.

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

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

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

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