A051451 -id:A051451 - cp's OEIS frontend

A058023 a(n) is the largest prime < A051451(n) - 1.

3, 7, 53, 409, 829, 2503, 27701, 360337, 720703, 12252197, 232792501, 5354228843, 26771144371, 80313433159, 2329089562747, 72201776446757, 144403552893563, 5342931457063157, 219060189739591153, 9419588158802421517, 442720643463713815129

Offset: 3

Labos Elemer, Nov 15 2000

nonn

This is a companion to A058019.

			A051451(6) = 420, 420 - 1 = 419 is preceded by the prime 409, so a(6) = 409.

Amiram Eldar, Table of n, a(n) for n = 3..377

Cf. A000961, A003418, A051451, A058019.

NextPrime[Rest@ FoldList[LCM, Select[Range[50], PrimePowerQ]] - 1, -1] (* Amiram Eldar, Aug 27 2024 *)

Edited by N. J. A. Sloane, Aug 20 2021

A077636 Length of periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 2, 2, 4, 8, 18, 14, 36, 38, 232, 268, 110, 280, 4348, 3244, 32684, 148184, 207616, 9988, 1946132, 2154482, 13319736, 8971624, 12345748, 69705504, 159413696, 1184191340, 1183672188, 23656693528, 28963250020, 701296434876, 754283490078

Offset: 1

Labos Elemer, Nov 13 2002

			For A051451(10) = 360360, the periodic part is {3,2,1,132,1,2,3,1200} with 8 terms, so a(10) = 8.

Cf. A000961, A003285, A051451.

pp = Join[{1}, Select[Range[2, 50], Mod[ #, # - EulerPhi[ # ]] == 0 &]]; Table[ Length[ Last[ ContinuedFraction[ Sqrt[ Apply[ LCM, Table[i, {i, 1, pp[[n]]}]]]]]], {n, 1, 31}]

a(n) = A003285(A051451(n)). - Michel Marcus, Sep 30 2019

Edited and extended by Robert G. Wilson v, Nov 14 2002
a(31) from Ray Chandler, Jan 16 2009
a(32)-a(35) from Chai Wah Wu, Sep 26 2019
a(36) from Chai Wah Wu, Sep 29 2019
a(37) from Chai Wah Wu, Sep 26 2021

A077637 Largest term in periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

Original entry on oeis.org

0, 2, 4, 6, 14, 40, 56, 100, 332, 1200, 1696, 7000, 30514, 146344, 327236, 566792, 3052270, 16994324, 24033604, 146190716, 936077324, 6138269514, 42081855636, 111338124722, 810553782854, 6225981742592, 48626471887292, 68768216033362, 562892107725410, 4743013205833238

Offset: 1

Labos Elemer, Nov 13 2002

nonn

			For A051451(10) = 360360, the periodic part is {3,2,1,132,1,2,3,1200} with 1200 as largest term, so a(10) = 1200.

Chai Wah Wu, Table of n, a(n) for n = 1..37

Cf. A000961, A051451, A077636.

t={A051451(n)} Table[Max[Last[ContinuedFraction[Sqrt[Part[t, u]]]]], {u, 1, 24}]

a(25)-a(28) from Ray Chandler, Jan 16 2009

a(1) corrected and a(29)-a(30) added by Chai Wah Wu, Sep 20 2021

A058019 a(n) is the smallest prime > A051451(n)+1.

Original entry on oeis.org

5, 11, 17, 67, 431, 853, 2531, 27733, 360391, 720743, 12252259, 232792597, 5354228921, 26771144429, 80313433231, 2329089562843, 72201776446853, 144403552893641, 5342931457063253, 219060189739591279

Offset: 2

Labos Elemer, Nov 14 2000

nonn

Robert Israel, Table of n, a(n) for n = 2..375

Cf. A003418, A051451, A000961, A058023.

NextPrime[FoldList[LCM, Select[Range[50], PrimePowerQ]] + 1] (* Amiram Eldar, Aug 27 2024 *)

Edited with better definition. - N. J. A. Sloane, Aug 20 2021

A064890 Decimal expansion of the sum of reciprocals of A051451, which includes 1 and values of lcm(1,...,x), where x is a prime power (A000961).

Original entry on oeis.org

1, 7, 7, 0, 6, 7, 5, 2, 4, 4, 3, 2, 5, 5, 8, 0, 2, 2, 7, 9, 1, 9, 7, 9, 6, 0, 0, 7, 6, 4, 2, 6, 6, 0, 8, 0, 2, 2, 3, 3, 1, 8, 3, 7, 6, 7, 2, 7, 2, 8, 3, 3, 5, 2, 0, 5, 2, 2, 4, 5, 8, 9, 6, 4, 4, 1, 2, 2, 2, 0, 3, 3, 8, 1, 0, 2, 2, 9, 6, 1, 1, 0, 5, 6, 5, 0, 7, 0, 5, 7, 7, 5, 7, 0, 8, 0, 9, 3, 4, 0, 3, 3, 3, 0, 2

Offset: 1

Labos Elemer, Oct 11 2001

			c = 1.7706752443255802279197960076426608022331837672728335205224589644122203381...
c = 1 + (1/2) + (1/6) + (1/12) + (1/60) + (1/420) + ... = 743/420 + ... = 1.7690 + ... = 1.7706752... Compare with A064859, an analogous constant obtained from A003418, where the constant is larger than c: 1.7877805 > 1.7706752. Repeated occurrences of LCM values in A003418 is responsible for the 1.78778... - 1.77067... = 0.0171... excess.

Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 37.

Cf. A051451, A003418, A064859, A000961.

f[n_] := LCM @@ Range@ n; RealDigits[ Plus @@ (1/Union@ Array[f, 251]), 10, 111][[1]] (* Robert G. Wilson v, Jul 11 2011 *)

A077638 Sum of terms in periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

Original entry on oeis.org

0, 2, 6, 8, 18, 42, 57, 105, 372, 1344, 1800, 7291, 32524, 150567, 342906, 738854, 3298239, 20772345, 36965663, 184510241, 1433356755, 7840220998, 56906577387, 113611483212, 843530932394, 6257315565011, 60692272232438, 70311381976766, 692150332693349, 4888462119949170

Offset: 1

Labos Elemer, Nov 13 2002

nonn

			For A051451(10) = 360360, the periodic part is {3,2,1,132,1,2,3,1200} with 1344 as sum of entries, so a(10) = 1344.

Chai Wah Wu, Table of n, a(n) for n = 1..37

Cf. A000961, A051451, A077636-A077637.

t={A051451(n)} Table[Max[Last[ContinuedFraction[Sqrt[Part[t, u]]]]], {u, 1, 24}]

a(25)-a(28) from Ray Chandler, Jan 16 2009

a(1) corrected and a(29)-a(30) added by Chai Wah Wu, Sep 19 2021

A096795 Numerator of sum of reciprocals of first n prime powers; denominator=A051451(n).

Original entry on oeis.org

1, 3, 11, 25, 137, 1019, 2143, 6709, 76319, 1019867, 2084779, 36161963, 699329537, 16317371911, 82657705331, 250947687593, 7357796373397, 230420777138107, 465354165304139, 17362507669146743, 717205745892079663

Offset: 1

Reinhard Zumkeller, Aug 17 2004

			n=6: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/7 =
(420+210+140+105+84+60)/420 = 1019/420 = a(6)/A051451(6).

Cf. A000961, A024451.

A137152 Triangle read by rows: prime powers whose row products give A051451.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 7, 1, 1, 3, 1, 5, 7, 8, 1, 1, 1, 1, 5, 7, 8, 9, 1, 1, 1, 1, 5, 7, 8, 9, 11, 1, 1, 1, 1, 5, 7, 8, 9, 11, 13, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 17, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 17, 19, 1, 1, 1, 1, 5, 7, 1

Offset: 1

Mats Granvik, Jan 24 2008

Similar to tables A133232 and A133233.

			The least common multiple of the first few rows are:
lcm{1} = 1
lcm{1,2} = 2
lcm{1,2,3} = 6
lcm{1,1,3,4} = 12
lcm{1,1,3,4,5} = 60
lcm{1,1,3,4,5,7} = 420
lcm{1,1,3,1,5,7,8} = 840
lcm{1,1,1,1,5,7,8,9} = 2520
lcm{1,1,1,1,5,7,8,9,11} = 27720
Multiplying the terms in the rows produces the same result:
1 = 1
1*2 = 2
1*2*3 = 6
1*1*3*4 = 12
1*1*3*4*5 = 60
1*1*3*4*5*7 = 420
1*1*3*1*5*7*8 = 840
1*1*1*1*5*7*8*9 = 2520
1*1*1*1*5*7*8*9*11 = 27720

Cf. A051451.

A058022 Difference between lcm(1,..,n) and the largest prime before lcm(1,..,n) where n runs over the prime powers A000961, LCMs are in A051451.

Original entry on oeis.org

3, 4, 1, 1, 1, 1, 1, 17, 19, 23, 17, 43, 1, 1, 29, 41, 1, 43, 1, 43, 47, 83, 1, 83, 61, 149, 1, 97, 89, 109, 113, 103, 113, 89, 137, 1, 157, 181, 239, 139, 241, 139, 179, 233, 193, 163, 241, 173, 283, 167, 271, 193, 277, 181, 179, 199, 1, 193, 223, 239, 239, 233, 751

Offset: 1

Labos Elemer, Nov 15 2000

nonn

Note that a(1) = 3 and a(2) = 4 use -2 as the preceding prime. - Robert Israel, Nov 18 2015

			6th and 7th different values of LCM-s are 840 and 2520. Deviation of immediate preceding primes(839,2503) are:1 and 17. For n=1 LCM[1]=1 and prime=-2 is the largest with deviation 3. So the sequence starts with 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. A000961, A003418, A051451.

f:= proc(n) local m;
    m:= ilcm($1..n);
    if m < 3 then m + 2
    else  m - prevprime(m)
    fi
end proc:
A000961:= select(t -> nops(numtheory:-factorset(t))<=1, [$1..1000]):
map(f, A000961); # Robert Israel, Nov 18 2015

N=2; print1("3, 4"); for(n=3,1e3, if(isprimepower(n,&p), N*=p; print1(", "N-precprime(N-1)))) \\ Charles R Greathouse IV, Nov 18 2015

A058024 a(n) = A051451(n) - A058023(n).

Original entry on oeis.org

3, 5, 7, 11, 11, 17, 19, 23, 17, 43, 59, 37, 29, 41, 53, 43, 37, 43, 47, 83, 71, 83, 61, 149, 73, 97, 89, 109, 113, 103, 113, 89, 137, 167, 157, 181, 239, 139, 241, 139, 179, 233, 193, 163, 241, 173, 283, 167, 271, 193, 277, 181, 179, 199, 269, 193, 223, 239

Offset: 3

Labos Elemer, Nov 15 2000

nonn

			So far, all terms are primes. The analogy with fortunate numbers (A005235) is clear.

Sean A. Irvine, Table of n, a(n) for n = 3..256

Cf. A000961, A003418, A005235, A051451, A052017, A055211, A057034.

Edited by N. J. A. Sloane, Aug 20 2021

Name corrected by Sean A. Irvine, Jul 18 2022

cp's OEIS Frontend

A058023 a(n) is the largest prime < A051451(n) - 1.

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A077636 Length of periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

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A077637 Largest term in periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

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A058019 a(n) is the smallest prime > A051451(n)+1.

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A064890 Decimal expansion of the sum of reciprocals of A051451, which includes 1 and values of lcm(1,...,x), where x is a prime power (A000961).

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A077638 Sum of terms in periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.

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A096795 Numerator of sum of reciprocals of first n prime powers; denominator=A051451(n).

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A137152 Triangle read by rows: prime powers whose row products give A051451.

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A058022 Difference between lcm(1,..,n) and the largest prime before lcm(1,..,n) where n runs over the prime powers A000961, LCMs are in A051451.

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