A175500 -id:A175500 - cp's OEIS frontend

A175501 a(n) = the number of divisors of A175500(n).

1, 2, 2, 3, 2, 4, 2, 6, 4, 3, 4, 4, 5, 2, 8, 2, 9, 4, 6, 2, 10, 2, 12, 6, 3, 6, 8, 4, 8, 6, 6, 7, 4, 12, 2, 5, 4, 10, 4, 9, 2, 16, 2, 15, 4, 16, 4, 18, 4, 14, 2, 20, 4, 15, 2, 18, 8, 3, 8, 8, 12, 4, 20, 2, 14, 6, 12, 8, 10, 6, 10, 8, 9, 6, 16, 6, 9, 8, 16, 8

Offset: 1

Leroy Quet, May 31 2010

nonn

Each ordered pair (a(k),a(k-1)) occurs at most once in the sequence.

Paul Tek, Table of n, a(n) for n = 1..2473

Cf. A175500.

Terms a(26) and beyond from b-file by Andrew Howroyd, Feb 05 2018

A272570 Inverse permutation to A175500.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 8, 16, 12, 18, 13, 20, 19, 22, 24, 28, 33, 35, 15, 25, 37, 39, 26, 41, 27, 43, 30, 45, 47, 49, 17, 51, 53, 62, 29, 55, 57, 64, 31, 66, 84, 82, 21, 58, 70, 88, 74, 86, 59, 90, 60, 94, 96, 92, 23, 98, 102, 76, 32, 112, 68

Offset: 1

Max Barrentine, May 03 2016

nonn

Max Barrentine, Table of n, a(n) for n = 1..693

Cf. A175500.

p = Last /@ Import["https://oeis.org/A175500/b175500.txt", "Table"]; n = First[ Complement[ Range[10^4], p]] - 1; ip = 0 Range@ n; Do[If[(v = p[[i]]) <= n, ip[[v]] = i], {i, Length@p}]; ip (* Giovanni Resta, May 21 2016 *)

ok(j, va, vs, n) = {if (vecsearch(vs, j), return (0)); for (k=1, n-1, if ((numdiv(j) == numdiv(va[k])) && (numdiv(va[k-1]) == numdiv(va[n-1])), return (0));); 1;}
findnew(va, vs, n) = {my(j = 1); my(vs = vecsort(va)); until (ok(j, va, vs, n), j++); j;}
listb(nn) = {my(va = [1]); for (n=2, nn, vs = vecsort(va); newa = findnew(va, vs, n); va = concat(va, newa);); va;}
invp(v) = {for (i=1, vecmax(v), found = 0; for (k=1, #v, if (v[k] == i, found = k; break);); if (! found, break); print1(found, ", "););}
lista(nn) = invp(listb(nn)); \\ Michel Marcus, May 04 2016

A175502 a(1) = 1. a(n) = the smallest integer not yet occurring such that each unordered pair {d(a(k)),d(a(k-1))} occurs only once at most, for 2<= k <= n, where d(m) = the number of divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 8, 10, 16, 7, 24, 9, 18, 20, 30, 14, 36, 11, 48, 15, 60, 13, 64, 21, 120, 17, 144, 22, 180, 19, 192, 25, 49, 72, 28, 80, 40, 42, 81, 32, 100, 54, 84, 90, 112, 121, 168, 44, 240, 23, 360, 26, 320, 45, 252, 56, 210, 96, 196

Offset: 1

Leroy Quet, May 31 2010

nonn

This sequence seems likely to be a permutation of the positive integers. Is it?

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A175500, A175503

More terms from Sean A. Irvine, Jun 14 2011

A287928 Lexicographically earliest sequence of distinct positive terms such that, if digsum(a(i)) = digsum(a(j)), then either i = j or digsum(a(i+1)) != digsum(a(j+1)) (where digsum is the digital sum, A007953).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 15, 14, 16, 18, 17, 19, 20, 23, 21, 24, 22, 25, 28, 26, 29, 27, 30, 32, 31, 35, 33, 36, 34, 38, 37, 39, 40, 45, 41, 44, 43, 42, 46, 47, 48, 49, 50, 54, 51, 53, 57, 52, 58, 55, 59, 56, 60, 65, 61, 66, 62, 67, 63, 64

Offset: 1

Rémy Sigrist, Jun 03 2017

This sequence is a permutation of the natural numbers, with inverse A287929.
More generally, if g is a function over the natural numbers with infinitely many distinct values, then there is a lexicographically earliest sequence of distinct positive terms, say f_g, such that, if g(f_g(i)) = g(f_g(j)), then either i = j or g(f_g(i+1)) != g(f_g(j+1)), and f_g is a permutation of the natural numbers:
- in particular, f_A007953 = a,
- and f_tau = A175500 (where tau = A000005),
- if g is injective then f_g = A000027.
Among the first 250000 terms, we have the following fixed points: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 19, 20, 25, 30, 39, 40, 46, 47, 48, 49, 50, 53, 60, 70, 76, 79, 80, 88, 89, 90, 92, 99, 100, 108, 111, 126, 193
, 675.

			For n = 1..9, a(n) = n satisfies the definition, and digsum(a(n)) = n.
Also a(10) = 10 satisfies the definition, and digsum(a(10)) = 1.
As digsum(a(10)) = digsum(a(1)), digsum(a(11)) != digsum(a(2)).
a(11) = 12 satisfies the definition.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A287928
Rémy Sigrist, Logarithmic scatterplot of the first 250000 terms
Rémy Sigrist, Illustration of the first terms
Index entries for sequences that are permutations of the natural numbers

Cf. A000005, A000027, A007953, A175500, A287929.

cp's OEIS Frontend

A175501 a(n) = the number of divisors of A175500(n).

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A272570 Inverse permutation to A175500.

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A175502 a(1) = 1. a(n) = the smallest integer not yet occurring such that each unordered pair {d(a(k)),d(a(k-1))} occurs only once at most, for 2<= k <= n, where d(m) = the number of divisors of m.

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Extensions

A287928 Lexicographically earliest sequence of distinct positive terms such that, if digsum(a(i)) = digsum(a(j)), then either i = j or digsum(a(i+1)) != digsum(a(j+1)) (where digsum is the digital sum, A007953).

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