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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A078324 Primes of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 17, 37, 73, 101, 109, 197, 257, 401, 433, 577, 677, 1153, 1297, 1373, 1601, 1801, 2593, 2917, 3137, 3457, 3529, 3889, 4001, 4357, 5477, 7057, 8101, 8713, 8837, 9001, 10369, 12101, 13457, 14401, 15377, 15877, 16001, 16901, 17497, 17957, 18253, 18433, 20809
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2002

Keywords

Examples

			12*rad(12)+1 = 12*rad(3*2^2)+1 = 12*3*2+1 = 72+1 = 73, therefore 73 is a term.
a(33) = 10369 = 10368 + 1: A078310(1728) = (2*3)*(2^6*3^3) = 10368.

Links

Crossrefs

Intersection of A000040 and A224866.
Cf. A010051, A078310, A078325.

Programs

  • Haskell
    a078324 n = a078324_list !! (n-1)
a078324_list = filter ((== 1) . a010051') a224866_list
-- Reinhard Zumkeller, Jul 23 2013
  • Mathematica
    powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Prime[Range[2400]], powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    is(n) = isprime(n) && ispowerful(n-1); \\ Amiram Eldar, Jul 31 2022

Extensions

Missing terms 10369, 16001, 17497 and 18433 inserted by Reinhard Zumkeller, Jul 23 2013

A080404 a(n)=A007947[A055932(n)]; the sequence consists of primorial numbers;.

Original entry on oeis.org

2, 2, 6, 2, 6, 2, 6, 6, 30, 2, 6, 6, 6, 30, 2, 6, 30, 6, 6, 30, 2, 6, 30, 6, 30, 6, 210, 6, 30, 2, 30, 6, 30, 6, 30, 6, 210, 6, 30, 30, 6, 2, 30, 6, 30, 210, 6, 30, 30, 6, 30, 210, 6, 30, 30, 6, 2, 210, 30, 6, 30, 210, 6, 30, 30, 6, 210, 30, 6, 30, 210, 6, 30, 210, 30, 6, 2, 210, 30, 30, 6
Offset: 1

Views

Author

Labos Elemer, Mar 19 2003

Keywords

Links

Crossrefs

Cf. A007947, A002110, A055932.

Programs

  • Mathematica
    With[{P = FoldList[Times, Prime@ Range@ Max@ #]}, Map[P[[#]] &, #]] &@ Map[PrimeNu@ # &, Select[Range[10^4], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] (* Michael De Vlieger, Feb 06 2020 *)

A081083 Numbers n such that rad(n+1)=rad(n)+1, where rad(m)=A007947(m) is the squarefree kernel of m.

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 46, 48, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 137, 138, 141, 142, 145, 154, 157, 158, 165, 166, 173, 177, 178, 181
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 04 2003

Keywords

Comments

Nearly all terms seem to be squarefree, see A081084.

Examples

			m=46=2*23=rad(46) and rad(47)=47=46+1=rad(46)+1, therefore 46 is a term;
m=48=3*2^4, rad(48)=6 and rad(49)=rad(7*7)=7=6+1=rad(48)+1, therefore 48 is a term.

Links

Crossrefs

Cf. A007947, A049097.
Union of A007674 and A081084.

Programs

  • Mathematica
    rad[n_] := Times @@ (First/@ FactorInteger[n]); s = {}; r1= 1; Do[r2 = rad[n]; If[r2 == r1 +1, AppendTo[s, n-1]]; r1 = r2, {n,2, 182}]; s (* Amiram Eldar, Aug 22 2019 *)
  • PARI
    rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
is(n)=rad(n+1)==rad(n)+1 \\ Charles R Greathouse IV, Aug 08 2013

A099985 a(n) = rad(2n), where rad = A007947.

Original entry on oeis.org

2, 2, 6, 2, 10, 6, 14, 2, 6, 10, 22, 6, 26, 14, 30, 2, 34, 6, 38, 10, 42, 22, 46, 6, 10, 26, 6, 14, 58, 30, 62, 2, 66, 34, 70, 6, 74, 38, 78, 10, 82, 42, 86, 22, 30, 46, 94, 6, 14, 10, 102, 26, 106, 6, 110, 14, 114, 58, 118, 30, 122, 62, 42, 2, 130, 66, 134, 34, 138, 70, 142, 6
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2004

Keywords

Comments

Bisection of A007947.

Links

Crossrefs

Cf. A007947, A065463, A099984, A204455.

Programs

  • Maple
    with(numtheory): A007947 := proc(n) local i,t1,t2; t1 :=ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end: seq(A007947(2*n),n=1..78); # Emeric Deutsch, Dec 15 2004
  • Mathematica
    Table[Product[p, {p, Select[Divisors[2 n], PrimeQ]}], {n, 100}] (* Wesley Ivan Hurt, May 08 2022 *)
a[n_] := Times @@ (First /@ FactorInteger[2*n]); Array[a, 100]  (* Amiram Eldar, Nov 19 2022 *)
  • PARI
    A099985(n) = factorback(factorint(n+n)[, 1]); \\ Antti Karttunen, May 08 2022

Formula

a(n) = 2 * A204455(n).
a(n) = A007947(2n). - Wesley Ivan Hurt, May 07 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (4/5) * A065463 = 0.563553... . - Amiram Eldar, Nov 19 2022

Extensions

More terms from Emeric Deutsch, Dec 15 2004
Name changed by Wesley Ivan Hurt, May 07 2022

A121369 a(1) = a(2) = 1, a(n) = A007947(a(n-1)) + A007947(a(n-2)) for n >= 3, i.e., a(n) is the largest squarefree divisor of a(n-1) plus the largest squarefree divisor of a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 7, 9, 10, 13, 23, 36, 29, 35, 64, 37, 39, 76, 77, 115, 192, 121, 17, 28, 31, 45, 46, 61, 107, 168, 149, 191, 340, 361, 189, 40, 31, 41, 72, 47, 53, 100, 63, 31, 52, 57, 83, 140, 153, 121, 62, 73, 135, 88, 37, 59, 96, 65, 71, 136, 105, 139, 244, 261, 209, 296
Offset: 1

Views

Author

Leroy Quet, Jul 23 2006

Keywords

Comments

First terms occurring more than once: 1, 31, 37, 121, ...: a(25)=a(37)=a(44)=31, a(16)=a(55)=37, a(22)=a(50)=121. - Reinhard Zumkeller, May 05 2013

Examples

			6 is the largest squarefree divisor of a(12) = 36. 29 is the largest squarefree divisor of a(13) = 29. So a(14) = 6 + 29 = 35.

Links

Crossrefs

Cf. A007947, A121367, A121368.
Cf. A000045.

Programs

  • Haskell
    import Data.Function (on)
a121369 n = a121369_list !! (n-1)
a121369_list = 1 : 1 : zipWith ((+) `on` a007947)
                       a121369_list (tail a121369_list)
-- Reinhard Zumkeller, May 05 2013
  • Maple
    with(numtheory): A007947:= proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1], i=1..nops(t1)); end: a:=proc(n) if n=1 or n=2 then 1 else A007947(a(n-1))+A007947(a(n-2)) fi end: seq(a(n),n=1..20); # Emeric Deutsch, Jul 24 2006
  • Mathematica
    Nest[Append[#, Total@ Map[SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, Take[#, -2]]] &, {1, 1}, 64] (* Michael De Vlieger, Oct 10 2017 *)

Extensions

More terms from Emeric Deutsch, Jul 24 2006
More terms from R. J. Mathar, May 18 2007

A143702 a(n) is the minimal values of A007947((2^n)*m*(2^n-m)).

Original entry on oeis.org

2, 6, 14, 30, 30, 42, 30, 78, 182, 1110, 570, 1830, 6666, 2310, 2534, 5538, 9870, 20010, 141270, 14070, 480090, 155490, 334110, 1794858, 2463270, 2132130, 2349390
Offset: 1

Views

Author

Artur Jasinski, Nov 10 2008

Keywords

Comments

The product of distinct prime divisors of (2^n)*m*(2^n-m) is also called the radical of that number: rad((2^n)*m*(2^n-m)).
For numbers m see A143700.

Crossrefs

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

Programs

  • Mathematica
    aa = {1}; bb = {1}; rr = {2}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, 2*r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; rr (* Artur Jasinski with assistance of M. F. Hasler *)

Extensions

Name changed and a(1) added by Jinyuan Wang, Aug 11 2020

A147801 Minimal value of A007947(m*(3^n-m)) with m coprime to 3.

Original entry on oeis.org

2, 2, 10, 10, 22, 110, 278, 238, 1054, 1342, 11066, 6118, 18734, 107030, 557270, 163030, 1440430, 2195110, 11016290, 3641210, 23250370, 38188766
Offset: 1

Views

Author

M. F. Hasler, Nov 13 2008

Keywords

Comments

Related to the abc conjecture. The minima are reached for m values given in A147802.
All terms of this sequence are even, so one could also consider A147801/2 = 1, 1, 5, 5, 11, 55, 139, 119, 527, 671, 5533, 3059, 9367, 53515, 278635, 81515, ...

Crossrefs

Cf. A007947, A147298 (general case), A143702 (analog for 2^n), A147800 (analog for 5^n), A147802.

Programs

  • PARI
    A147801(n, p=3) = {my(m=n=p^n); for(x=1, (n-1)\2, x%p || next; A007947(n-x)*A007947(x)A007947((n-x)*x)); m; }

Extensions

a(17)-a(22) from Jinyuan Wang, Aug 11 2020

A147803 Least m coprime to 5 minimizing A007947(m*(5^n-m)).

Original entry on oeis.org

1, 1, 4, 49, 128, 9, 36864, 19332, 4508, 121, 2
Offset: 1

Views

Author

M. F. Hasler, Nov 13 2008

Keywords

Comments

The minima are given in A147800.
This is related to the abc conjecture: Since m is coprime to 5, it is also coprime to 5^n and thus to 5^n-m. Thus the squarefree kernel A007947(m*(5^n-m)*5^n) = 5*A007947(m*(5^n-m)).

Crossrefs

Cf. A007947, A147298 (general case), A147800 (value of minima), A143700 (analog for 2^n), A147802 (analog for 3^n), A147300 (analog for any number).

Programs

  • PARI
    A147803(n,p=5) = {my(b, m=n=p^n); for(a=1, n\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*b=a)); b; }

A172418 Numbers k that have measure of smoothness J larger than 3, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

16, 32, 64, 81, 128, 243, 256, 288, 324, 384, 432, 486, 512, 576, 625, 648, 729, 768, 864, 972, 1024, 1152, 1250, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2000, 2048, 2187, 2304, 2401, 2500, 2560, 2592, 2916, 3072, 3125, 3136, 3200, 3456, 3584, 3645
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

Subsequence of A049094.

Links

Crossrefs

Cf. A007947, A049094, A059172, A172419, A172420, A172421, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 3, AppendTo[aa, c]], {c, 2, 10000}]; aa

A172419 Numbers k that have measure of smoothness J larger than 4, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

32, 64, 128, 243, 256, 512, 729, 1024, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3125, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8192, 8748, 9216, 10240, 10368, 11664, 12288, 12500, 12800, 13122, 13824, 15552
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

Subsequence of A049094 and A172418.

Links

Crossrefs

Cf. A007947, A049094, A172418, A172420, A172421, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 4, AppendTo[aa, c]], {c, 2, 10000}]; aa
Previous Showing 51-60 of 1118 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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