A176506 -id:A176506 - cp's OEIS frontend

A339361 Product of prime indices of the n-th squarefree semiprime.

2, 3, 4, 6, 8, 5, 6, 10, 7, 12, 8, 12, 9, 14, 15, 16, 10, 11, 18, 18, 12, 20, 13, 21, 14, 20, 24, 22, 15, 24, 16, 24, 27, 17, 28, 18, 26, 28, 32, 19, 30, 20, 30, 30, 21, 33, 22, 32, 36, 23, 34, 24, 36, 36, 35, 25, 38, 26, 40, 39, 27, 40, 40, 28, 42, 44, 29, 42

Offset: 1

Gus Wiseman, Dec 06 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of all squarefree semiprimes together with the products of their prime indices begins:
   6: 1 * 2 = 2
  10: 1 * 3 = 3
  14: 1 * 4 = 4
  15: 2 * 3 = 6
  21: 2 * 4 = 8
  22: 1 * 5 = 5
  26: 1 * 6 = 6
  33: 2 * 5 = 10
  34: 1 * 7 = 7
  35: 3 * 4 = 12

A001358 lists semiprimes.
A003963 gives the product of prime indices of n.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 is the sum of squarefree semiprimes of weight n.
A332765/A339114 give the greatest/least squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
A338905 groups squarefree semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A046388, A056239 (weight), A112798, A166237, A320656, A320911, A338901, A339002, A339003, A339004.

Table[Times@@PrimePi/@First/@FactorInteger[n],{n,Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]

a(n) = A003963(A006881(n)).

a(n) = A270650(n) * A270652(n).

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
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See Links section.
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a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

A178610 n-th semiprime minus difference between the prime indices of the two factors of n-th semiprime.

Original entry on oeis.org

4, 5, 9, 8, 11, 14, 19, 18, 25, 21, 30, 28, 34, 31, 35, 38, 49, 46, 53, 51, 49, 52, 62, 62, 63, 76, 70, 81, 73, 79, 89, 84, 80, 90, 91, 101, 109, 102, 116, 121, 105, 112, 117, 129, 116, 128, 123, 142, 138, 126, 147, 137, 145, 156, 144, 169, 162, 155, 167, 176, 185, 170

Offset: 1

Juri-Stepan Gerasimov, May 30 2010

nonn

a(n)=A001358(n)-A176506(n).

Corrected (153 replaced by 155) by R. J. Mathar, May 31 2010

cp's OEIS Frontend

A339361 Product of prime indices of the n-th squarefree semiprime.

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A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

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A178610 n-th semiprime minus difference between the prime indices of the two factors of n-th semiprime.

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