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.

Showing 1-6 of 6 results.

A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 9, 26, 37, 50, 17, 28, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 126, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 1445, 1522, 401, 1682, 1765, 1850, 969, 676, 2117, 2210, 289, 344, 501, 2602, 1353
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2002

Keywords

Comments

A112526(a(n) - 1) = 1, see also A224866. - Reinhard Zumkeller, Jul 23 2013
Increase each exponent in the prime factorization by one, then add 1 to the new product. - M. F. Hasler, Jan 22 2017

Links

Crossrefs

Smallest, greatest factor: A078311, A078312, number of factors: A078313, A078314, min, max exponent: A078315, A078316, number, sum of divisors: A078317, A078318, sum of prime factors: A078319, A078320, Euler's totient: A078321, squarefree kernel: A078322, arithmetic derivative: A078323.
Cf. primes: A078324, squarefree: A078325, squareful: A078326.

Programs

  • Haskell
    a078310 n = n * a007947 n + 1
-- Reinhard Zumkeller, Jul 23 2013, Oct 19 2011
  • Maple
    a:= n-> 1+n*mul(i[1], i=ifactors(n)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 22 2017
  • Mathematica
    A078310[n_] := n*Times @@ FactorInteger[n][[All, 1]] + 1; Array[A078310, 50] (* G. C. Greubel, Apr 25 2017 *)
  • PARI
    rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=n*rad(n)+1 \\ Charles R Greathouse IV, Jul 09 2013
  • PARI
    a(n)={n=factor(n);n[,2]+=vectorv(matsize(n)[1],i,1);factorback(n)+1} \\ M. F. Hasler, Jan 22 2017
  • PARI
    a(n)=prod(k=1,matsize(n=factor(n))[1],n[k,1]^(n[k,2]+1))+1 \\ M. F. Hasler, Jan 22 2017

Formula

a(n) = A064549(n)+1.

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

A224866 Numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 9, 10, 17, 26, 28, 33, 37, 50, 65, 73, 82, 101, 109, 122, 126, 129, 145, 170, 197, 201, 217, 226, 244, 257, 289, 290, 325, 344, 362, 393, 401, 433, 442, 485, 501, 513, 530, 577, 626, 649, 676, 677, 730, 785, 801, 842, 865, 901, 962, 969, 973, 1001
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 23 2013

Keywords

Comments

A078310 in natural order.

Links

Crossrefs

Subsequences: A078325, A078324.
Cf. A001694, A078310, A007947.

Programs

  • Haskell
    a224866 n = a224866_list !! (n-1)
a224866_list = [x | x <- [2..] , let x' = x - 1, let k = a007947 x',
                    let (y,m) = divMod x' k, m == 0, a007947 y == k]
  • Mathematica
    powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[1001], powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    is(n) = n>1 && ispowerful(n-1) \\ Charles R Greathouse IV, Aug 08 2013, corrected by Amiram Eldar, Jul 31 2022

Formula

a(n) = A001694(n) + 1.

A078322 a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 3, 26, 37, 10, 17, 14, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 42, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 85, 1522, 401, 58, 1765, 370, 969, 26, 2117, 2210, 17, 86, 501, 2602, 1353, 2810, 65
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2002

Keywords

Comments

a(n) = A007947(A078310(n)).

Examples

			a(25) = rad(25*rad(25)+1) = rad(25*rad(5^2)+1) = rad(25*5+1) = rad(125+1) = rad(126) = rad(2*3*3*7) = 2*3*7 = 42.

Links

Crossrefs

Cf. A007947, A078310, A078325.

Programs

  • Haskell
    a078322 = a007947 . a078310
-- Reinhard Zumkeller, Oct 19 2011
  • Maple
    rad:= n-> mul(i, i=numtheory[factorset](n)):
a:= n-> rad(n*rad(n)+1):
seq(a(n), n=1..70);  # Alois P. Heinz, May 04 2017
  • Mathematica
    rad[n_] := Times @@ FactorInteger[n][[All, 1]]; Table[ rad[n*rad[n] + 1], {n, 1, 54}] (* Jean-François Alcover, Dec 03 2012 *)
  • PARI
    rad(n)=vecprod(factor(n)[,1])
a(n)=rad(n*rad(n)+1) \\ Charles R Greathouse IV, Jul 09 2013

A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

Original entry on oeis.org

9, 289, 676, 9801, 12168, 235225, 332929, 465125, 1825201, 11309769, 384199201, 592192225, 4931691076, 5425069448, 13051463049, 221322261601, 443365544449, 865363202001, 8192480787001, 11968683934832, 13325427460801, 15061377048201, 28821995554248
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2002

Keywords

Comments

a(n) = u*rad(u) = v*rad(v)+1 for appropriate u, v, where rad(n) = A007947(n) is the squarefree kernel.
Also numbers n such that n(n-1) is a powerful number. - Charles R Greathouse IV, Aug 08 2013

Links

Crossrefs

Cf. A078310, A064549, A078315, A078325, A060355, A078326.

Programs

Formula

a(n) = A060355(n)+1.

Extensions

a(22)-a(23) from Donovan Johnson, Jul 29 2011

A359749 Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 15, 16, 24, 25, 26, 27, 31, 32, 35, 36, 48, 63, 64, 71, 72, 81, 100, 107, 108, 120, 121, 124, 125, 127, 128, 143, 144, 168, 169, 195, 196, 199, 200, 215, 216, 224, 225, 242, 243, 255, 256, 287, 289, 323, 342, 361, 391, 392, 399, 400, 431, 432, 440
Offset: 1

Views

Author

Amiram Eldar, Jan 13 2023

Keywords

Comments

Either k or k+1 is a powerful number (A001694). Except for k=8, are there terms k such that both k and k+1 are powerful (i.e., terms that are also in A060355)? None of the terms A060355(n) for n = 2..39 is in this sequence.
A002496(k)-1, A078324(k)-1, A078325(k)-1, and A049533(k)^2 are terms for all k >= 1.

Examples

			3 is a term since 3 has the exponent 1 in its prime factorization, and 3 + 1 = 4 = 2^2 has a different exponent in its prime factorization, 2.

Links

Crossrefs

Subsequences: A000079, A000225 \ {0}, A075408, A359747.
Cf. A001694, A002496, A049533, A060355, A078324, A078325.

Programs

  • Mathematica
    q[n_] := UnsameQ @@ Join @@ (Union[FactorInteger[#][[;; , 2]]]& /@ (n + {0, 1})); Join[{1}, Select[Range[400], q]]
  • PARI
    lista(nmax) = {my(e1 = [], e2); for(n = 2, nmax, e2 = Set(factor(n)[,2]); if(setintersect(e1, e2) == [], print1(n-1, ", ")); e1 = e2); }
Showing 1-6 of 6 results.

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