A111273 -id:A111273 - cp's OEIS frontend

A326776 a(n) is the smallest divisor of the n-th nonprime number (A018252(n)) not already in the sequence.

1, 2, 3, 4, 9, 5, 6, 7, 15, 8, 18, 10, 21, 11, 12, 25, 13, 27, 14, 30, 16, 33, 17, 35, 36, 19, 39, 20, 42, 22, 45, 23, 24, 49, 50, 51, 26, 54, 55, 28, 57, 29, 60, 31, 63, 32, 65, 66, 34, 69, 70, 72, 37, 75, 38, 77, 78, 40, 81, 41, 84, 85, 43, 87, 44, 90, 91

Offset: 1

Rémy Sigrist, Jul 28 2019

nonn

This sequence is a permutation of the natural numbers.
Empirically:
- the subsequence with the terms satisfying a(n) <= n correspond to A093641,
- if a(n) > n, then a(n) = A018252(n),
- these two situations appear as two lines in the scatterplot of the sequence.

			The first terms, alongside the divisors of A018252(n), are:
  n   a(n)  div(A018252(n))
  --  ----  ---------------
   1     1  (1)
   2     2  (1, 2, 4)
   3     3  (1, 2, 3, 6)
   4     4  (1, 2, 4, 8)
   5     9  (1, 3, 9)
   6     5  (1, 2, 5, 10)
   7     6  (1, 2, 3, 4, 6, 12)
   8     7  (1, 2, 7, 14)
   9    15  (1, 3, 5, 15)
  10     8  (1, 2, 4, 8, 16)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A326776
Index entries for sequences that are permutations of the natural numbers

Cf. A018252, A093641, A111273.

PARI
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A309389 a(n) is the smallest positive divisor not yet in the sequence of 11*A000217(n-1); n >= 1.

Original entry on oeis.org

1, 11, 3, 2, 5, 15, 7, 4, 6, 9, 55, 22, 13, 77, 21, 8, 17, 33, 19, 10, 14, 121, 23, 12, 20, 25, 27, 18, 29, 87, 31, 16, 24, 51, 35, 30, 37, 209, 39, 26, 41, 123, 43, 86, 45, 69, 47, 44, 28, 49, 75, 34, 53, 99, 135, 70, 38, 57, 59, 66, 61, 341, 63, 32, 40, 65, 67, 134, 46, 105, 71, 36, 73, 407, 111

Offset: 1

Enrique Navarrete, Jul 27 2019

nonn

Up to n=10000, 1176 of the first 1228 odd primes appear as fixed points of a(n), i.e., 95.8%.

Conjecture: for large p prime, the odd primes (except p) appear as fixed points of b(n), where b(n) is the smallest positive divisor not yet in the sequence of p*A000217(n-1); n >= 1 (see link).

			For n = 1: a(1) = 1 is the smallest divisor of 11*0 not yet in the sequence.
For n = 23: a(23) = 23 is a fixed point and the smallest divisor of 11*253 not yet in the sequence.
For n = 73: a(73) = 73 is a fixed point and the smallest divisor of 11*2628 not yet in the sequence.

Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Cf. A000217, A111273, A309275, A309276, A309387.

A320672 a(n) is the smallest divisor of (n+2)*(n+3) not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 8, 5, 10, 11, 12, 13, 14, 15, 16, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 17, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 33, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 44, 56, 57, 58, 59, 60, 61, 62, 63, 64, 55, 66, 67, 68, 69, 70

Offset: 0

Enrique Navarrete, Oct 19 2018

nonn

a(n) is the smallest divisor of A002378(n+2) not yet in the sequence.

The numbers that are smaller than the preceding terms are: 3, 5, 9, 17, 33, 44, 55, 65, 90, 99, 143, 208, ...

			For n = 0, a(0) = 1 since 1 is the smallest divisor of 6 not yet in the sequence.
For n = 3, a(3) = 3 since 3 is the smallest divisor of 30 not yet in the sequence.

Cf. A002378, A111273, A246368.

s = {}; Do[d = Divisors[(n + 2)(n + 3)]; Do[d1 = d[[k]]; If[FreeQ[s, d1], AppendTo[s, d1]; Break[]], {k, Length[d]}], {n, 0, 100}]; s (* Amiram Eldar, Nov 14 2018 *)

toadd(n, v) = {fordiv(n, d, if (!vecsearch(v, d), return(d)); ); }
lista(nn) = {v = []; for (n = 0, nn, newt = toadd((n+2)*(n+3), v); print1(newt, ", "); v = vecsort(concat(v, newt)); ); } \\ Michel Marcus, Nov 20 2018

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A326776 a(n) is the smallest divisor of the n-th nonprime number (A018252(n)) not already in the sequence.

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A309389 a(n) is the smallest positive divisor not yet in the sequence of 11*A000217(n-1); n >= 1.

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A320672 a(n) is the smallest divisor of (n+2)*(n+3) not yet in the sequence.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

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PARI