A231722 -id:A231722 - 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.

A231721 Partial sums of phitorials: a(n) = A001088(1)+A001088(2)+...+A001088(n).

Original entry on oeis.org

1, 2, 4, 8, 24, 56, 248, 1016, 5624, 24056, 208376, 945656, 9793016, 62877176, 487550456, 3884936696, 58243116536, 384392195576, 6255075618296, 53220543000056, 616806151581176, 6252662237392376, 130241496125238776, 1122152167228009976, 20960365589283433976

Offset: 1

Antti Karttunen, Nov 27 2013

nonn

a(n) gives the index to the first term in each subrange of A231716. Specifically, for all n>=1, A231716(a(n)) = A007489(n).

Antti Karttunen, Table of n, a(n) for n = 1..100

Cf. A001088 ("phitorials"), A231722, A231716, A007489.

Accumulate[FoldList[Times,EulerPhi[Range[30]]]] (* Harvey P. Dale, Apr 02 2018 *)

a(n) = 1 if n=1, otherwise A001088(n)+a(n-1).

a(n) = A231722(n)+1. [Follows from the definitions]

A236856 Partial sums of A003418 starting summing from A003418(1), with a(0) = 0.

Original entry on oeis.org

0, 1, 3, 9, 21, 81, 141, 561, 1401, 3921, 6441, 34161, 61881, 422241, 782601, 1142961, 1863681, 14115921, 26368161, 259160721, 491953281, 724745841, 957538401, 6311767281, 11665996161, 38437140561, 65208284961, 145521718161, 225835151361, 2554924714161

Offset: 0

Antti Karttunen, Feb 27 2014

nonn

Similar comments about the trailing digits apply here as in A173185.
a(n) gives the position of the last element of row n in irregular tables like A238280.
From a(2)=3 onward all terms are divisible by three.
a(n) is divisible by 73 for n >= 72. Therefore a(n)/3 is prime for only 13 values of n: 3, 4, 6, 8, 9, 12, 16, 22, 23, 31, 35, 48 and 53. - Amiram Eldar, Sep 19 2022

Antti Karttunen, Table of n, a(n) for n = 0..250

One less than A173185.

Cf. A003418, A007489, A231722, A236857, A236858.

Prepend[Accumulate @ Table[LCM @@ Range[n], {n, 1, 30}], 0] (* Amiram Eldar, Sep 19 2022 *)

(define (A236856 n) (if (< n 2) n (+ (A236856 (- n 1)) (A003418 n))))

a(n) = A173185(n)-1.

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.

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A231721 Partial sums of phitorials: a(n) = A001088(1)+A001088(2)+...+A001088(n).

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A236856 Partial sums of A003418 starting summing from A003418(1), with a(0) = 0.

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

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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.