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.

Previous Showing 11-16 of 16 results.

A045969 a(1)=6; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^e_i * Product p_{i+2}^e_i.

Original entry on oeis.org

6, 15, 35, 225, 77, 525, 143, 3375, 1225, 1155, 221, 7875, 323, 2145, 2695, 50625, 437, 18375, 667, 17325, 5005, 3315, 899, 118125, 5929, 4845, 42875, 32175, 1147, 40425, 1517, 759375, 7735, 6555, 11011, 275625, 1763, 10005, 11305, 259875, 2021, 75075
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

Crossrefs

Cf. A045965, A045966, A045967, A045968, A045970, A045971, A045972, A045973.

Programs

  • Mathematica
    f[p_, e_] := (NextPrime[p] * NextPrime[p, 2])^e; a[1] = 6; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Formula

Sum_{n>=1} 1/a(n) = -5/6 + Product_{k>=2} (1+1/(prime(k)*prime(k+1)-1)) = 0.31383788... . - Amiram Eldar, Sep 19 2023

Extensions

More terms from David W. Wilson

A045971 a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.

Original entry on oeis.org

8, 27, 125, 81, 343, 3375, 1331, 243, 625, 9261, 2197, 10125, 4913, 35937, 42875, 729, 6859, 16875, 12167, 27783, 166375, 59319, 24389, 30375, 2401, 132651, 3125, 107811, 29791, 1157625, 50653, 2187, 274625, 185193, 456533, 50625, 68921, 328509, 614125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

Crossrefs

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045972, A045973, A065483.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^(e + 2); a[1] = 8; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Formula

Sum_{n>=1} 1/a(n) = (4/5) * A065483 - 7/8 = 0.196827322859... . - Amiram Eldar, Sep 19 2023

Extensions

More terms from David W. Wilson

A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

Original entry on oeis.org

9, 25, 49, 125, 121, 1225, 169, 625, 343, 3025, 289, 6125, 361, 4225, 5929, 3125, 529, 8575, 841, 15125, 8281, 7225, 961, 30625, 1331, 9025, 2401, 21125, 1369, 148225, 1681, 15625, 14161, 13225, 20449, 42875, 1849, 21025, 17689, 75625, 2209, 207025
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

Crossrefs

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045973, A082695.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p, 2]^(e + 1); a[1] = 9; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Formula

Sum_{n>=1} 1/a(n) = (4/7) * (zeta(2)*zeta(3)/zeta(6)) - 8/9 = 0.221737646437... . - Amiram Eldar, Sep 19 2023

Extensions

More terms from David W. Wilson

A129401 a(n) is the result of replacing with its successor prime each prime in the factorization of the n-th composite number.

Original entry on oeis.org

9, 15, 27, 25, 21, 45, 33, 35, 81, 75, 63, 55, 39, 135, 49, 51, 125, 99, 105, 243, 65, 57, 77, 225, 69, 85, 189, 165, 117, 175, 87, 405, 121, 147, 95, 153, 375, 91, 297, 115, 93, 315, 111, 275, 729, 119, 195, 171, 145, 231, 675, 123, 245, 207, 143, 255, 567, 625
Offset: 1

Views

Author

Ben Paul Thurston, May 28 2007

Keywords

Comments

Each odd composite number appears in the sequence exactly once. - Jon E. Schoenfield, Jun 05 2007
Prime factors are used with multiplicity, e.g., the factors of 4 are 2 and 2, and both terms are replaced by 3, so a(1) = 3*3 = 9. - Harvey P. Dale, Mar 19 2013

Examples

			a(19) = 105 because the factorization of the 19th composite number (i.e., 30) is 2*3*5 and replacing each prime factor with the next prime results in 3*5*7 which remultiplies to 105.

Links

Crossrefs

Cf. A002808 (composite numbers), A003961.

Programs

  • Mathematica
    cnp[n_]:=Times@@(NextPrime/@Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]); With[{nn=100},cnp/@Complement[Range[2,nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, Mar 19 2013 *)
  • PARI
    lista(nn) = {forcomposite(c=1, nn, my (f = factor(c)); for (k=1, #f~, f[k,1] = nextprime(f[k,1]+1)); print1(factorback(f), ", "););} \\ Michel Marcus, Feb 26 2018

Formula

a(n) = A003961(A002808(n)). - Jon E. Schoenfield, Jun 04 2007 [edited, at the suggestion of Michel Marcus, by Jon E. Schoenfield, Feb 18 2018]
a(n) = A045965(A002808(n)). - Ivan N. Ianakiev, Feb 15 2018

Extensions

More terms from Jon E. Schoenfield, Jun 05 2007
Name edited by Jon E. Schoenfield, Feb 18 2018

A227324 Result of changing both the prime indices and the exponents in the prime factorization of n: increment odd values, decrement even values.

Original entry on oeis.org

1, 9, 4, 3, 49, 36, 25, 81, 2, 441, 169, 12, 121, 225, 196, 27, 361, 18, 289, 147, 100, 1521, 841, 324, 7, 1089, 16, 75, 529, 1764, 1369, 729, 676, 3249, 1225, 6, 961, 2601, 484, 3969, 1849, 900, 1681, 507, 98, 7569, 2809, 108, 5, 63, 1444, 363, 2209, 144
Offset: 1

Views

Author

Alex Ratushnyak, Jul 07 2013

Keywords

Comments

A self-inverse permutation on the positive integers: a(a(n)) = n.

Examples

			n = 2^3 => a(n) = 3^4 = 81.
n = 3^2 => a(n) = 2^1 = 2.

Links

Crossrefs

Cf. A003958-A003965, A011262, A011264, A045965-A045973, A151799, A151800.

Programs

  • Maple
    a:= n-> mul(ithprime(i[1])^i[2], i=map(x->map(y->y-1+2*irem(y, 2),
        [numtheory[pi](x[1]), x[2]]), ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 17 2013
  • Mathematica
    a[n_] := If[n == 1, 1, Product[{p, e} = pe; Prime[BitXor[PrimePi[p] - 1, 1] + 1]^(BitXor[e - 1, 1] + 1), {pe, FactorInteger[n]}]];
Array[a, 100] (* Jean-François Alcover, May 31 2019, after Andrew Howroyd *)
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); prime( bitxor( primepi(p)-1, 1)+1)^(bitxor(e-1, 1)+1))} \\ Andrew Howroyd, Jul 23 2018
  • Python
    primes = [2]*2
primes[1] = 3
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5, 1000000, 6):
  addPrime(n)
  addPrime(n+2)
for n in range(1,99):
  p = 1
  j = n
  i = 0
  while j>1:
    e = 0
    while j % primes[i] == 0:
      j /= primes[i]
      e+=1
    if e:
      e = ((e-1)^1) + 1
      p*= primes[i^1]**e
    i += 1
  print(str(p), end=', ')

Formula

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p-1)*(p^6 + q(p) +(p^3-1)*q(p)^2))/(p^7 - p*q(p)^2) = 0.3120270364..., where q(p) = nextprime(p) = A151800(p) if p has an odd index, and q(p) = prevprime(p) = A151799(p) otherwise. - Amiram Eldar, Sep 17 2023

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A328915 If n = Product (p_j^k_j) then a(n) = Product (nextprime(p_j)), where nextprime = A151800.

Original entry on oeis.org

1, 3, 5, 3, 7, 15, 11, 3, 5, 21, 13, 15, 17, 33, 35, 3, 19, 15, 23, 21, 55, 39, 29, 15, 7, 51, 5, 33, 31, 105, 37, 3, 65, 57, 77, 15, 41, 69, 85, 21, 43, 165, 47, 39, 35, 87, 53, 15, 11, 21, 95, 51, 59, 15, 91, 33, 115, 93, 61, 105, 67, 111, 55, 3, 119, 195, 71, 57, 145, 231
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 30 2019

Keywords

Comments

All terms are odd.

Examples

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2) * prime(3) = 3 * 5 = 15.

Links

Crossrefs

Cf. A000720, A003961, A007947, A045965, A045967, A048250, A151800, A328878, A328879.

Programs

  • Maple
    a:= n-> mul(nextprime(i[1]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 30 2019
  • Mathematica
    a[1] = 1; a[n_] := Times @@ (NextPrime[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]
  • PARI
    a(n) = my(f=factor(n)); prod(k=1, #f~, nextprime(f[k,1]+1)); \\ Michel Marcus, Oct 30 2019

Formula

If n = Product (p_j^k_j) then a(n) = Product (prime(pi(p_j) + 1)), where pi = A000720.
a(n) = A007947(A003961(n)).
Previous Showing 11-16 of 16 results.

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

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