A075383 -id:A075383 - cp's OEIS frontend

A075391 Product/n^n of n-th group in A075383.

1, 2, 6, 120, 180, 36288, 6720, 24710400, 227026800, 12108096000, 121080960, 197110116403200, 67365043200, 201490341212160000, 1001829765736800000, 850009620666286080000, 2144384782848000, 5991810785878910952960000

Offset: 1

Amarnath Murthy, Sep 22 2002

nonn

Cf. A075383, A075384, A075385, A075386, A075387.

A075387(n)/(n^n)

More terms from David Wasserman, Jan 17 2005

A076034 Group the natural numbers so that the n-th group contains the smallest set of n relatively prime numbers: (1), (2, 3), (4, 5, 7), (6, 11, 13, 17), (8, 9, 19, 23, 25), (10, 21, 29, 31, 37, 41), ...

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 13, 17, 8, 9, 19, 23, 25, 10, 21, 29, 31, 37, 41, 12, 35, 43, 47, 53, 59, 61, 14, 15, 67, 71, 73, 79, 83, 89, 16, 27, 49, 55, 97, 101, 103, 107, 109, 18, 65, 77, 113, 127, 131, 137, 139, 149, 151, 20, 33, 91, 157, 163, 167, 173, 179, 181, 191, 193

Offset: 1

Amarnath Murthy, Oct 01 2002

			The triangle begins:
   1;
   2, 3;
   4,  5,  7;
   6, 11, 13, 17;
   8,  9, 19, 23, 25;
  10, 21, 29, 31, 37, 41;
  ...

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A075383, A076032, A076033, A095167.

S:=[$1..1000]: Res:= NULL:
for n from 1 to 20 do
  A:= [S[1]]; R:= 1; count:= 1;
  for k from 2 while count < n do
    if andmap(t -> igcd(t,S[k])=1, A) then count:= count+1; A:= [op(A),S[k]]; R:= R,k; fi
  od;
  S:= subsop(op(map(t -> t=NULL, [R])),S);
  Res:= Res, op(A);
od:
Res; # Robert Israel, Dec 04 2022

Perl
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# See Links section.
```

More terms from David Wasserman, Jan 29 2005

Crossrefs added by Paul Tek, Oct 24 2015

A072205 a(n) = (p^2 - p + 2)/2 for p = prime(n); number of squares modulo p^2.

Original entry on oeis.org

2, 4, 11, 22, 56, 79, 137, 172, 254, 407, 466, 667, 821, 904, 1082, 1379, 1712, 1831, 2212, 2486, 2629, 3082, 3404, 3917, 4657, 5051, 5254, 5672, 5887, 6329, 8002, 8516, 9317, 9592, 11027, 11326, 12247, 13204, 13862, 14879, 15932, 16291, 18146, 18529

Offset: 1

Robert G. Wilson v, Jul 03 2002

nonn

Second terms of triple Peano sequence A071988. [Robert G. Wilson v, Jul 03 2002]
Positions of primes in A075383: A000040(n) = A075383(a(n)). [Reinhard Zumkeller, Jun 22 2009]
Number of different squares modulo p^2, for p ranging over the primes. Proof: the p multiples of p (0, p, 2p...) have the same square: 0 mod p^2. The other elements have the same square iff they are opposite: x^2 == y^2 (mod p^2) iff (x - y)(x + y) == 0 (mod p^2) iff x == y (mod p) or x == -y (mod p) or 2y == 0 (mod p). So the (p^2 - p) non-p-multiples account for (p^2 - p)/2 different squares and the p-multiples for 1 extra square, giving a total of (p^2 - p + 2)/2. [Bert Seghers, Dec 21 2011]
From Jianing Song, Apr 13 2019: (Start)
For k coprime to prime(n), k^a(n) == +-k (mod prime(n)^2).
For every integer k, k^(2a(n)) == k^2 (mod prime(n)^2). (End)

Cf. A008837, A071988, A075383.

seq[n_Integer?Positive] := Module[{fn01 = 1, fn10 = 1, fnout = 1}, Do[{fn10, fn01, fnout} = {fn10 + 1, fn01 + fn10, fn01 + fnout}, {n - 1}]; {fn10, fn01, fnout}]; Ar = Flatten[ Table[ seq[ Prime[n]], {n, 1, 50}]]; a = {}; Do[a = Append[a, Ar[[n]]], {n, 2, 150, 3}]; a

a(n)=binomial(prime(n),2)+1 \\ Charles R Greathouse IV, Jan 11 2012

[(p^2 - p + 2)/2 for p in prime_range(200)]

a(n) = A008837(n) + 1.

a(n) = A000124(A000040(n)) by definition [Bert Seghers, Jan 01 2012]

Name edited by Bert Seghers, Jan 01 2012

A371236 Triangle T(n, k), n > 0, k = 1..n, read and filled in the greedy way by rows with distinct positive integers such that for any n > 0, the terms in the n-th row are congruent modulo n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 13, 17, 21, 7, 12, 22, 27, 32, 8, 14, 20, 26, 38, 44, 10, 24, 31, 45, 52, 59, 66, 11, 19, 35, 43, 51, 67, 75, 83, 15, 33, 42, 60, 69, 78, 87, 96, 105, 16, 36, 46, 56, 76, 86, 106, 116, 126, 136, 18, 29, 40, 62, 73, 84, 95, 117, 128, 139, 150

Offset: 1

Rémy Sigrist, Mar 16 2024

Every integer appears in the sequence (as each row starts with the least missing value).

			Triangle T(n, k) begins:
                     1
                   2   4
                 3   6   9
               5  13  17  21
             7  12  22  27  32
           8  14  20  26  38  44
        10  24  31  45  52  59  66
      11  19  35  43  51  67  75  83

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A075383, A371246 and A371248 for similar sequences.

Cf. A371237 (inverse).

PARI
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See Links section.
```

A376903 Lexicographically earliest sequence of distinct positive integers with a(1) multiples of 1 followed by a(2) multiples of 2 etc.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 12, 8, 16, 20, 5, 10, 15, 25, 30, 35, 18, 24, 36, 42, 48, 54, 60, 66, 72, 7, 14, 21, 28, 49, 56, 63, 70, 77, 84, 91, 98, 32, 40, 64, 80, 88, 96, 104, 112, 27, 45, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 50, 100, 110, 120

Offset: 1

Rémy Sigrist, Oct 08 2024

This sequence combines features of Golomb's sequence (A001462) and A075383.
This sequence is a permutation of the positive integers with inverse A376904.
This sequence can also be seen as an irregular table whose n-th row contains a(n) multiples of n.

			The first terms/rows are:
  n  a(n)  n-th row
  -  ----  ---------------------------------------------
  1     1  1
  2     2  2, 4
  3     4  3, 6, 9, 12
  4     3  8, 16, 20
  5     6  5, 10, 15, 25, 30, 35
  6     9  18, 24, 36, 42, 48, 54, 60, 66, 72
  7    12  7, 14, 21, 28, 49, 56, 63, 70, 77, 84, 91, 98

Rémy Sigrist, Table of n, a(n) for n = 1..10021
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A001462, A075383, A376904 (inverse), A376905, A377093.

PARI
```
\\ See Links section.
```

from itertools import count, islice
def A376903gen(): # generator of terms
    aset, alst = {1, 2, 4}, [1, 2, 4]
    yield from [1, 2, 4]
    for n in count(3):
        nlst = []
        for k in count(n, n):
            if k not in aset:
                nlst.append(k)
                if len(nlst) == alst[n-1]:
                    break
        yield from nlst
        alst.extend(nlst)
        aset.update(nlst)
print(list(islice(A376903gen(), 100))) # Michael S. Branicky, Oct 16 2024

A337918 Rearrangement of natural numbers so that next n numbers contain n as substring.

Original entry on oeis.org

1, 2, 12, 3, 13, 23, 4, 14, 24, 34, 5, 15, 25, 35, 45, 6, 16, 26, 36, 46, 56, 7, 17, 27, 37, 47, 57, 67, 8, 18, 28, 38, 48, 58, 68, 78, 9, 19, 29, 39, 49, 59, 69, 79, 89, 10, 100, 101, 102, 103, 104, 105, 106, 107, 108, 11, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

Rémy Sigrist, Jan 29 2021

This sequence combines features of A072484 and of A075383.
This sequence first differ from A075383 for n = 67: a(67) = 120 whereas A075383(67) = 12.
This sequence is a permutation of the natural numbers with inverse A337919.

			As a triangle, the first rows are:
     1: 1
     2: 2, 12
     3: 3, 13, 23
     4: 4, 14, 24, 34
     5: 5, 15, 25, 35, 45
     6: 6, 16, 26, 36, 46, 56
     7: 7, 17, 27, 37, 47, 57, 67
     8: 8, 18, 28, 38, 48, 58, 68, 78
     9: 9, 19, 29, 39, 49, 59, 69, 79, 89
    10: 10, 100, 101, 102, 103, 104, 105, 106, 107, 108
    11: 11, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119
    12: 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 212, 312

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, Perl program for A337918
Index entries for sequences that are permutations of the natural numbers

Cf. A072484, A075383, A337919 (inverse).

Perl
```
See Links section.
```

A096782 A096780(A096780(n)).

Original entry on oeis.org

1, 2, 3, 4, 56, 11, 57, 22, 5, 8, 28, 7, 3082, 254, 79, 6, 9317, 9, 907, 12, 17, 174, 8003, 137, 23, 33, 667, 14, 677, 13

Offset: 1

Reinhard Zumkeller, Jul 09 2004

nonn

Integer permutation with inverse A096781: a(A096781(n))=A096781(a(n))=n;

A075383(a(n))=a(A075383(n))=A096780(n).

Index entries for sequences that are permutations of the natural numbers

A284038 Lexicographically earliest sequence of distinct positive terms such that A000196(n) divides a(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 9, 15, 18, 21, 24, 27, 30, 16, 20, 28, 32, 36, 40, 44, 48, 52, 5, 25, 35, 45, 50, 55, 60, 65, 70, 75, 80, 42, 54, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 7, 14, 49, 56, 63, 77, 91, 98, 105, 112, 119, 133, 140, 147, 154, 64

Offset: 1

Rémy Sigrist, May 25 2017

nonn

This is a permutation of the natural numbers, with inverse A287433; for any n > 0, n appears among the first n^2 terms.
This sequence is similar to A075383: here we have runs of length 2*k+1, there of length k, of multiples of k.
a(p^2) = p for any prime p > 3.
All fixed points belong to A006446.
Conjecturally:
- all fixed points > 3 are squares,
- if a(n) < n, then A000196(n) belongs to A007310 \ {1},
- if k belongs to A007310 \ {1}, then a(n) < n for some n such that A000196(n) = k.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A284038
Index entries for sequences that are permutations of the natural numbers

Cf. A000196, A006446, A007310, A075383, A287433 (inverse).

cp's OEIS Frontend

A075391 Product/n^n of n-th group in A075383.

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A076034 Group the natural numbers so that the n-th group contains the smallest set of n relatively prime numbers: (1), (2, 3), (4, 5, 7), (6, 11, 13, 17), (8, 9, 19, 23, 25), (10, 21, 29, 31, 37, 41), ...

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A072205 a(n) = (p^2 - p + 2)/2 for p = prime(n); number of squares modulo p^2.

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A371236 Triangle T(n, k), n > 0, k = 1..n, read and filled in the greedy way by rows with distinct positive integers such that for any n > 0, the terms in the n-th row are congruent modulo n.

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A376903 Lexicographically earliest sequence of distinct positive integers with a(1) multiples of 1 followed by a(2) multiples of 2 etc.

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PARI

Python

A337918 Rearrangement of natural numbers so that next n numbers contain n as substring.

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Perl

A096782 A096780(A096780(n)).

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A284038 Lexicographically earliest sequence of distinct positive terms such that A000196(n) divides a(n).

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