cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A058017 a(n) is the smallest prime > LCM(1,...,x), where x is the n-th prime power (A000961).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 853, 2521, 27733, 360391, 720743, 12252259, 232792561, 5354228921, 26771144401, 80313433231, 2329089562843, 72201776446801, 144403552893641, 5342931457063253, 219060189739591279, 9419588158802421659, 442720643463713815201, 3099044504245996706459
Offset: 1

Views

Author

Labos Elemer, Nov 14 2000

Keywords

Comments

For the first 100 prime powers, the difference between a(n) and the LCM is either 1 or a prime.
The values of x are taken to be prime powers so that each distinct LCM occurs exactly once.

Examples

			The 6th distinct prime power is A000961(7) = 8, LCM(1,...,8) = 840 and 853 is the first prime that follows, thus a(7) = 853.

Crossrefs

Cf. A000961, A003418, A051451, A058018, A151800.

Programs

  • Mathematica
    With[{max = 50}, NextPrime[Exp[Accumulate[Join[{0}, Select[Array[MangoldtLambda, max], # > 0 &]]]]]] (* Amiram Eldar, Aug 13 2024 *)
  • PARI
    lista(nn) = {for (n=1, nn, if ((n==1) || isprimepower(n), print1(nextprime(lcm(vector(n, x, x)) + 1), ", ")));} \\ Michel Marcus, Apr 09 2015

Formula

a(n) = A151800(A051451(n)) = A051451(n) + A058018(n). - Amiram Eldar, Aug 13 2024

Extensions

Edited by Franklin T. Adams-Watters, Aug 15 2006
Offset changed to 1 and more terms from Michel Marcus, Apr 09 2015
Name corrected by Amiram Eldar, Aug 13 2024

A058018 Difference between LCM(1,...,x) and the smallest prime > LCM(1,...,x), where x is the n-th prime power (A000961).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 13, 1, 13, 31, 23, 19, 1, 41, 1, 31, 43, 1, 41, 53, 79, 59, 1, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 1, 229, 109, 139, 113, 227, 131, 191, 163, 1, 199, 151, 139, 1, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233, 283, 229, 277, 263
Offset: 1

Views

Author

Labos Elemer, Nov 14 2000

Keywords

Comments

The first value corresponds to x = 1, LCM(1) = 1.
For the first 100 prime powers, the value is either prime or 1.
The values of x are taken to be prime powers so that each distinct LCM occurs exactly once.

Examples

			The 6th distinct prime power is A000961(7) = 8, LCM(1,...,8) = 840 and 853 is the first prime that follows, thus a(7) = 853-840 = 13.

Crossrefs

Cf. A000961, A003418, A013632, A051451, A058017.

Programs

  • Mathematica
    With[{max = 250}, (NextPrime[#] - #)& /@ Exp[Accumulate[Join[{0}, Select[Array[MangoldtLambda, max], # > 0 &]]]]] (* Amiram Eldar, Aug 13 2024 *)
  • PARI
    lista(nn) = {for (n=1, nn, if ((n==1) || isprimepower(n), v = lcm(vector(n, x, x)); print1(nextprime(v+1) - v, ", ")););} \\ Michel Marcus, Apr 09 2015

Formula

a(n) = A013632(A051451(n)) = A058017(n) - A051451(n). - Amiram Eldar, Aug 13 2024

Extensions

Edited by Franklin T. Adams-Watters, Aug 15 2006
Offset set to 1 by Michel Marcus, Apr 09 2015
Name corrected by Amiram Eldar, Aug 13 2024

A058029 Primes closest to LCM(1,...,x) either above or below. Arguments x were selected from A000961 (powers of primes including primes) in order to obtain distinct values of LCM exactly once.

Original entry on oeis.org

3, 5, 11, 59, 419, 839, 2521, 27733, 360337, 720703, 12252259, 232792559, 5354228879, 26771144401, 80313433231, 2329089562799, 72201776446801, 144403552893599, 5342931457063157, 219060189739591153
Offset: 0

Views

Author

Labos Elemer, Nov 16 2000

Keywords

Links

Crossrefs

Cf. A000961, A003418, A051451.

Programs

  • Mathematica
    pcl[n_]:=Module[{lc=LCM@@Range[n],p1,p2},p1=NextPrime[lc,-1];p2= NextPrime[ lc];If[p2-lc

A076246 Totients of those numbers at which values of A051547 increase: in these consecutive terms new prime powers arise, i.e., which did not occur in neither of preceding terms.

Original entry on oeis.org

2, 4, 6, 10, 8, 16, 18, 22, 28, 46, 32, 52, 58, 54, 82, 64, 100, 102, 106, 148, 162, 166, 172, 178, 190, 196, 226, 250, 128, 256, 262, 268, 282, 292, 310, 316, 346, 358, 366, 382, 388, 466, 478, 486, 502, 508, 556, 562, 568, 586, 606, 618, 642, 652, 676, 708
Offset: 1

Views

Author

Labos Elemer, Oct 08 2002

Keywords

Examples

			8 = 2*2*2 immediately follows 10 = 2*5; 58 = 2*29 follows 52 = 2*2*13. In both cases, the latter term has a new prime factor (like 29) or an old one at a higher power (like 2*2*2).

Crossrefs

Cf. A003418, A051451, A051547, A076244, A076245.

Programs

  • Mathematica
    s0=1; s1=1; Do[s0=s1; s1=LCM[s1, EulerPhi[n]]; If[ !Equal[s0, s1], Print[n]], {n, 1, 1000}]
  • PARI
    lista(nn) = {least = 1; for (n=2, nn, nleast = lcm(least, eulerphi(n)); if (nleast > least, print1(eulerphi(n), ", ")); least = nleast;);} \\ Michel Marcus, Jul 30 2017

Formula

a(n) = phi(A076245(n + 1)). - Sean A. Irvine, Mar 25 2025

A096075 Least common multiple of first n 3-smooth numbers.

Original entry on oeis.org

1, 2, 6, 12, 12, 24, 72, 72, 144, 144, 144, 432, 864, 864, 864, 864, 1728, 1728, 5184, 5184, 5184, 10368, 10368, 10368, 10368, 10368, 31104, 62208, 62208, 62208, 62208, 62208, 62208, 124416, 124416, 124416, 373248, 373248, 373248, 373248, 746496
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2004

Keywords

Comments

Subsequence of A003586.

Examples

			The first seven 3-smooth numbers are {1, 2, 3, 4, 6, 8, 9} and their lcm is 72. - _David A. Corneth_, Jul 13 2023

Links

Crossrefs

Cf. A003418, A003586, A006899, A022330, A022331, A051451.

Programs

  • Mathematica
    seq[max_] := Module[{sm3 = Sort[Flatten[Table[2^i*3^j, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]], e2, e3}, e2 = FoldList[Max, IntegerExponent[sm3, 2]]; e3 = FoldList[Max, IntegerExponent[sm3, 3]]; 2^e2*3^e3]; seq[1000] (* Amiram Eldar, Jul 13 2023 *)

Formula

a(n) > a(n-1) iff A003586(n) is a power of 2 or of 3 (cf. A006899, A022330, A022331).

A181121 As n increases, the reciprocal of a(n) = asymptotic fraction of positive integers whose longest string of consecutive divisors is A181062(n).

Original entry on oeis.org

2, 3, 12, 15, 70, 840, 1260, 2772, 30030, 720720, 765765, 12932920, 243374040, 6692786100, 40156716600, 83181770100, 2406725881560, 144403552893600, 148414762696200, 5476504743489780, 224275908542914800
Offset: 1

Views

Author

Matthew Vandermast, Oct 07 2010

Keywords

Comments

The asymptotic average, as n increases, of n's longest string of consecutive divisors is the constant 1.787780456..., given in A064859.

Examples

			A number's longest string of consecutive divisors is a(5)=6 iff the integer is a multiple of 60 but not of 7. As n increases, the asymptotic fraction of positive integers satisfying those conditions equals 1/60 * 6/7 = 1/70. Therefore a(5) = 70.

Formula

a(n) = A051451(n) * A025473(n+1)/(A025473(n+1)-1).
If A181062(n) = 2^(e-1), then a(n) = A003418(2^e) = A051451(n+1).

A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 504, 840, 2520, 27720, 360360, 720720, 4084080
Offset: 0

Views

Author

Gus Wiseman, Oct 17 2019

Keywords

Examples

			The sequence of terms > 1 together with their nontrivial divisors begins:
    2: {}
    6: {2,3}
   12: {2,3,4,6}
   60: {2,3,4,5,6,10,12,15,20,30}
  420: {2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210}
  504: {2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,56,63,72,84,126,168,252}

Crossrefs

Positions of first appearances in A328458.
The version for all divisors is A051451.
Cf. A000005, A033676, A060775, A070824, A119313, A129308, A181063, A199970, A163870, A328448, A328449.

Programs

  • Mathematica
    dav=Table[Switch[n,1,1,_,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,1000}];
Table[Position[dav,i][[1,1]],{i,Union[dav]}]//Sort

Extensions

a(12) from Robert Israel, Mar 31 2023

A057822 Smaller of pair of twin primes whose average is lcm(1,...,m) for some m.

Original entry on oeis.org

5, 11, 59, 419, 232792559, 442720643463713815199
Offset: 1

Views

Author

Labos Elemer, Nov 08 2000

Keywords

Comments

Known values of m such that lcm(1,...,m) is a twin prime mean value are as follows: {3, 4, 5, 6, 7, 19, 20, 21, 22, 47, 48}.
No more such primes occurs below m < 2000.
No more such primes occurs below m < 30000. - Amiram Eldar, Aug 18 2024

Examples

			419 and 421 are twin primes, (419 + 421)/2 = 420 = lcm(1,2,3,4,5,6,7).

Crossrefs

Intersection of A057824 and {A049536(n)-2}.
Cf. A003418, A051451, A057706.

Programs

  • Mathematica
    Select[FoldList[LCM, Select[Range[50], PrimePowerQ]] - 1, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Aug 18 2024 *)
  • PARI
    lista(nn=50) = {for (i=1, nn, if (isprimepower(i), if (isprime(p=lcm([2..i])-1) && isprime(p+2), print1(p, ", "));););} \\ Michel Marcus, Aug 25 2019

A058021 Largest prime preceding distinct values of lcm(1,...,m): Max{p|p < A003418(A000961(n))}. To get different LCM values, the last arguments(m) of LCM were selected from A000961.

Original entry on oeis.org

-2, -2, 5, 11, 59, 419, 839, 2503, 27701, 360337, 720703, 12252197, 232792559, 5354228879, 26771144371, 80313433159, 2329089562799, 72201776446757, 144403552893599, 5342931457063157, 219060189739591153, 9419588158802421517, 442720643463713815199
Offset: 0

Views

Author

Labos Elemer, Nov 15 2000

Keywords

Examples

			lcm(1,2,3,4,5,6,7,8)=840 is the 7th term in A000961. Primes preceding it, in descending order, are  839, 829, .... So the corresponding prime in this sequence is 839. The first 2 entries come before lcm(1)=1 and lcm(1,2)=2 as 1-3 = -2 and 2-4 = -2.

Crossrefs

Cf. A000961, A003418, A051451.

A058030 Signed distance of primes from LCM(1,...,x) being closest to it. Arguments x were selected from A000961 (powers of primes including primes) in order to use distinct values of LCM exactly once. When both closest primes are in the same distance, then negative were used.

Original entry on oeis.org

1, -1, -1, -1, -1, -1, 1, 13, -23, -17, 19, -1, -1, 1, 31, -1, 1, -1, -43, -47, 59, -1, 59, -61, 113, -1, -97, 73, 73, 97, -103, 101, -89, 1, -1, 109, 139, 113, -139, 131, -139, 163, 1, -193, 151, 139, 1, 223, -167, -271, -193, -277, -181, -179, -199, -1, -193, -223, 233, -239, 229, 277, -241, -317, -1, 443, 379
Offset: 0

Views

Author

Labos Elemer, Nov 16 2000

Keywords

Crossrefs

Cf. A000961, A003418, A051451.
Previous Showing 51-60 of 64 results. Next

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