A231721 -id:A231721 - cp's OEIS frontend

A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

1, 3, 5, 9, 11, 21, 23, 33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119, 153, 155, 165, 167, 177, 179, 189, 191, 201, 203, 213, 215, 225, 227, 237, 239, 633, 635, 645, 647, 657, 659, 669, 671, 681, 683, 693, 695, 705, 707, 717, 719, 873, 875

Offset: 1

Antti Karttunen, Nov 12 2013

A001088(n+1) gives the number of terms x in sequence for which A084558(x)=n.

Because totatives (the reduced residue set) of each natural number k form a multiplicative group of integers modulo same k, it means that taking e.g. inverses of each digit modulo same k or multiplying them (again modulo k) by some member of that set keeps the set closed, and thus applying these operations to each digit modulo i+1 (2 for the least significant digit in factorial base, 3 for the next, and so on) yield only digits allowed in this sequence, and thus they induce various permutations of this sequence. These can be further "normalized" to be permutations of natural numbers with a suitable ranking function, which is to be submitted later.

			This can be viewed as an irregular table, where row n has A001088(n+1) elements, starts from position A231721(n) and ends at position A231722(n+1):
1;
3, 5;
9, 11, 21, 23;
33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119;
...

Antti Karttunen, Rows 1..7, flattened
Wikipedia, Totative

Positions of ones in A231715.
The first term of each row: A007489(n) = a(A231721(n)).
The last term of each row: A033312(n+1) = a(A231722(n+1)).
Subset of A227157.
Cf. also A007623, A000010, A001088, A084558.

A231722 Partial sums of A001088 starting from its second term; a(1)=0, a(n) = A001088(2)+...+A001088(n).

Original entry on oeis.org

0, 1, 3, 7, 23, 55, 247, 1015, 5623, 24055, 208375, 945655, 9793015, 62877175, 487550455, 3884936695, 58243116535, 384392195575, 6255075618295, 53220543000055, 616806151581175, 6252662237392375, 130241496125238775, 1122152167228009975, 20960365589283433975

Offset: 1

Antti Karttunen, Nov 27 2013

nonn

a(n+1) gives the index to the last term in each row of A231716. Specifically, for all n>=1, A231716(a(n+1)) = A033312(n+1).
a(n) = natural number which is written as the n-th repunit in "totient phi number system": 0, 1, 10, 11, 100, 101, 110, 111, 200, 201, 210, 211, 300, 301, 310, 311, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 1200, ..., and so on. Note how the 1st, the 3rd, the 7th and 23rd terms of this list are 1, 11, 111, and 1111.
In this number system the i-th digit from right (the least significant digit = digit_0) may contain integers in range 0..A000010(i+3)-1, and the value of the number is obtained as sum_{i=0..} digit_i * A001088(i+2).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

One less than A231721.

Cf. A000010 (Euler's totient function phi), A001088 (its partial products, "phitorials"), A231716, A033312.

with(numtheory): A231722:=n->add(product(phi(k), k=1..i), i=2..n): seq(A231722(n), n=1..20); # Wesley Ivan Hurt, Aug 09 2014

Table[Sum[Product[EulerPhi[k], {k, i}], {i, 2, n}], {n, 20}] (* Wesley Ivan Hurt, Aug 09 2014 *)

a(n) = sum(i=2, n, prod(k=1, i, eulerphi(k))); \\ Michel Marcus, Aug 09 2014

Scheme
```
(define (A231722 n) (- (A231721 n) 1))
```

a(n) = A231721(n)-1. [The terms are one less than the partial sums of "phitorials", A001088, cumulatively summed from their first term]

A173185 Partial sums of A003418.

Original entry on oeis.org

1, 2, 4, 10, 22, 82, 142, 562, 1402, 3922, 6442, 34162, 61882, 422242, 782602, 1142962, 1863682, 14115922, 26368162, 259160722, 491953282, 724745842, 957538402, 6311767282, 11665996162, 38437140562, 65208284962, 145521718162, 225835151362, 2554924714162

Offset: 0

Jonathan Vos Post, Feb 12 2010

From Antti Karttunen, Feb 27 2014: (Start)
For all n >= 4, a(n) mod 10 = 2 (as A003418(5) = 60, the first multiple of ten in that sequence).
For all n >= 24, a(n) mod 100 = 62 (as A003418(25) = 26771144400, the first multiple of one hundred in that sequence).
Cf. also A236856.
a(n-1) gives the position of the first element of row n in irregular tables like A238280.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2297 (first 251 terms from Antti Karttunen)

One more than A236856.

Cf. A002944, A003418, A003422, A102910, A093880, A133233, A231721, A236857, A236858, A238280.

b:= proc(n) b(n):= `if`(n=0, 1, ilcm(n, b(n-1))) end:
a:= proc(n) a(n):= `if`(n<0, 0, a(n-1) +b(n)) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 2018

Table[If[n == 0, 1, LCM @@ Range[n]], {n, 0, 50}] // Accumulate (* Jean-François Alcover, Jan 03 2022 *)

a(n) = sum(k=0, n, lcm(vector(k, i, i))); \\ Michel Marcus, Mar 13 2018

(define (A173185 n) (if (< n 1) 1 (+ (A173185 (- n 1)) (A003418 n))))

a(n) = Sum_{i=0..n} A003418(i).

Missing term a(9)=3922 inserted by Antti Karttunen, Feb 27 2014

A319688 Sum of digits when n is represented in phitorial (A001088) base.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6

Offset: 0

Antti Karttunen, Oct 02 2018

			For n = 9, its phitorial representation is "102" as 9 = 1*A001088(2) + 0*A001088(3) + 2*A001088(4) = 1*1 + 0*2 + 2*4. Thus a(9) = 1+0+2 = 3.
For n = 577, its phitorial representation is "300001" as 577 = 1*A001088(2) + 3*A001088(7) = 1*1 + 3*192, thus a(577) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..18432

Cf. A000010, A001088 (gives the positions of ones), A231721, A231722.

Cf. also A000120, A034968, A276150.

With[{max = 7}, bases = EulerPhi[Range[max, 1, -1]]; nmax = Times @@ bases - 1; a[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Array[a, nmax, 0]] (* Amiram Eldar, Jul 29 2023 *)

A319688(n) = { my(s=0, i=3, d, b); while(n, b = eulerphi(i); d = (n%b); s += d; n = (n-d)/b; i++); (s); };

Starting from i=3, compute the remainder when n is divided by phi(i), and then continue iterating with n -> floor(n/phi(i)), and i -> i+1, until n is zero. a(n) is the sum of remainders encountered in process.

For all n >= 0, a(A231722(n)) = n.

cp's OEIS Frontend

A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = du*u! + ... + d2*2! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A231722 Partial sums of A001088 starting from its second term; a(1)=0, a(n) = A001088(2)+...+A001088(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

A173185 Partial sums of A003418.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

A319688 Sum of digits when n is represented in phitorial (A001088) base.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.