A145605 -id:A145605 - cp's OEIS frontend

A252489 Index of the largest prime which divides n*(n+1).

1, 2, 2, 3, 3, 4, 4, 2, 3, 5, 5, 6, 6, 4, 3, 7, 7, 8, 8, 4, 5, 9, 9, 3, 6, 6, 4, 10, 10, 11, 11, 5, 7, 7, 4, 12, 12, 8, 6, 13, 13, 14, 14, 5, 9, 15, 15, 4, 4, 7, 7, 16, 16, 5, 5, 8, 10, 17, 17, 18, 18, 11, 4, 6, 6, 19, 19, 9, 9, 20, 20, 21, 21, 12, 8, 8, 6

Offset: 1

M. F. Hasler, Jan 16 2015

Yields the row of A145605 in which n appears, and also the first row of A138180 in which n appears.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002071, A145604, A138180, A145605, A002072, A074399, A061395.

A061395:= [1, seq(numtheory:-pi(max(numtheory:-factorset(n))), n=2..101)]:
zip(max,A061395[1..-2],A061395[2..-1]); # Robert Israel, Feb 12 2021

a[n_] := PrimePi[Max[FactorInteger[n][[-1, 1]], FactorInteger[n+1][[-1, 1]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 05 2023 *)

a(n)=primepi(vecmax(factor(n*(n+1))[,1]))

a(n) = pi(A074399(n)), where pi = A000720.

a(n) = max(A061395(n),A061395(n+1)). - Robert Israel, Feb 12 2021

A275156 The 108 numbers n such that n(n+1) is 17-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 39, 44, 48, 49, 50, 51, 54, 55, 63, 64, 65, 77, 80, 84, 90, 98, 99, 104, 119, 120, 125, 135, 143, 153, 168, 169, 175, 195, 220, 224, 242, 255, 272, 288, 324, 350, 351, 363, 374, 384, 440, 441, 539, 560, 594, 624, 675, 714, 728, 832, 935, 1000, 1088, 1155, 1224, 1274, 1700, 1715, 2057, 2079, 2400, 2430, 2499, 2600, 3024, 4095, 4224, 4374, 4913, 5831, 6655, 9800, 10647, 12375, 14399, 28560, 31212, 37179, 123200, 194480, 336140

Offset: 1

Jean-François Alcover, Nov 13 2016

This is the 7th row of the table A138180.

See A002071.

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492.

pMax = 17; smoothMax = 10^12; smoothNumbers[p_?PrimeQ, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand@Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

is(n)=my(t=510510); n*=n+1; while((t=gcd(n,t))>1, n/=t); n==1 \\ Charles R Greathouse IV, Nov 13 2016

A275164 The 167 numbers n such that n(n+1) is 19-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 38, 39, 44, 48, 49, 50, 51, 54, 55, 56, 63, 64, 65, 75, 76, 77, 80, 84, 90, 95, 98, 99, 104, 119, 120, 125, 132, 135, 143, 152, 153, 168, 169, 170, 175, 189, 195, 208, 209, 220, 224, 242, 255, 272, 285, 288, 323, 324, 342, 350, 351, 360, 363, 374, 384, 399, 440

Offset: 1

Jean-François Alcover, Nov 14 2016

See A002071.

The full list of 167 terms is given in the b-file (this is the 8th row of the table A138180).

Jean-François Alcover, Table of n, a(n) for n = 1..167

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492,A275156.

pMax = 19; smoothMax = 10^15; smoothNumbers[p_?PrimeQ, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand@Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

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A252489 Index of the largest prime which divides n*(n+1).

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A275156 The 108 numbers n such that n(n+1) is 17-smooth.

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A275164 The 167 numbers n such that n(n+1) is 19-smooth.

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