A045973 -id:A045973 - cp's OEIS frontend

A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

5, 7, 11, 49, 13, 77, 17, 343, 121, 91, 19, 539, 23, 119, 143, 2401, 29, 847, 31, 637, 187, 133, 37, 3773, 169, 161, 1331, 833, 41, 1001, 43, 16807, 209, 203, 221, 5929, 47, 217, 253, 4459, 53, 1309, 59, 931, 1573, 259, 61, 26411, 289, 1183, 319, 1127, 67, 9317, 247

Offset: 1

N. J. A. Sloane

			If n = 9 = 3^2, then a(n) = 11^2 = 121 (since 11 is the third prime after 3).

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045969, A045970, A045971, A045972, A045973.

f[p_, e_] := NextPrime[p, 3]^e; a[1] = 5; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

More terms from David W. Wilson

Erroneous linear recurrence deleted by Harvey P. Dale, May 07 2018

A045970 a(1)=7; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+4}^e_i.

Original entry on oeis.org

7, 11, 13, 121, 17, 143, 19, 1331, 169, 187, 23, 1573, 29, 209, 221, 14641, 31, 1859, 37, 2057, 247, 253, 41, 17303, 289, 319, 2197, 2299, 43, 2431, 47, 161051, 299, 341, 323, 20449, 53, 407, 377, 22627, 59, 2717, 61, 2783, 2873, 451, 67, 190333, 361, 3179, 403

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045971, A045972, A045973.

f[p_, e_] := NextPrime[p, 4]^e; a[1] = 7; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

More terms from David W. Wilson

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

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

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

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 *)

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

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

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

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

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

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

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

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

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

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 *)

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

More terms from David W. Wilson

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

Alex Ratushnyak, Jul 07 2013

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

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

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

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

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 *)

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

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=', ')

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

Keyword:mult added by Andrew Howroyd, Jul 23 2018

cp's OEIS Frontend

A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

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A045970 a(1)=7; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+4}^e_i.

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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.

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

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

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

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