A341356 -id:A341356 - cp's OEIS frontend

A341513 Sum of digits in A097801-base.

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Antti Karttunen, Feb 23 2021

A097801-base uses values 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*9, 2*3*5*7*9*11, 2*3*5*7*9*11*13, 2*3*5*7*9*11*13*15, ..., for its digit-positions, instead of primorials (A002110), thus up to 1889 = 2*3*5*7*9 - 1 = 9*A002110(4) - 1 its representation is identical with the primorial base A049345. Thus this sequence differs from A276150 for the first time at n=1890, where a(1890)=1, while A276150(1890)=9, as 1890 = 9*A002110(4).

			In A097801-base, where the digit-positions are given by 1 and the terms of A097801 from its term a(1) onward: 2, 6, 30, 210, 1890, 20790, 270270, 4054050, ..., number 29 is expressed as "421" as 29 = 4*6 + 2*2 + 1*1, thus a(29) = 4+2+1 = 7. In the same base, number 30 is expressed as "1000" as 30 = 1*30, thus a(30) = 1, and number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 1 also.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A097801, A341356 (most significant digit in the same base).

Cf. also A002110, A049345, A276150.

Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[Total@ IntegerDigits[#, b] &, nn + 1, 0]] (* Michael De Vlieger, Feb 23 2021 *)

A341513(n) = { my(u=0,m=2,k=3); while(n, u += n%m; n \= m; m = k; k += 2); (u); };

A341514 Number of trailing zeros in A097801-base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Antti Karttunen, Feb 25 2021

A097801-base uses values 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*9, 2*3*5*7*9*11, 2*3*5*7*9*11*13, 2*3*5*7*9*11*13*15, ..., for its digit-positions, instead of primorials (A002110), thus up to 1889 = 2*3*5*7*9 - 1 = 9*A002110(4) - 1 its representation is identical with the primorial base A049345.
From Amiram Eldar, Mar 10 2021: (Start)
The asymptotic density of the occurrences of k is 1/2 if k=0, and 2*k/(A097801(k+1)) otherwise.
The asymptotic mean of this sequence is sqrt(e*Pi/2)*erf(1/sqrt(2))/2 = 0.7053430673..., where erf(x) is the error function. (End)

			In A097801-base number 1890 = 2*3*5*7*9 is expressed as "100000", thus a(1890) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A097801, A341356, A341513.

Differs from A276084 for the first time at n=1890, as a(1890) = 5, while A276084(1890) = 4.

Block[{nn = 105, b}, b = MixedRadix@ NestWhile[Prepend[#1, 2 #2 - 1] & @@ {#, Length[#] + 1} &, {2}, Times @@ # < nn &]; Array[LengthWhile[Reverse@ IntegerDigits[#, b], # == 0 &] &, nn]] (* Michael De Vlieger, Feb 25 2021 *)

A341514(n) = { my(m=2,k=3,i=0); while(!(n%m), n /= m; m = k; k += 2; i++); (i); };

For odd n, a(n) = 0; for even n, a(n) = the largest k such that A097801(k) divides n.

cp's OEIS Frontend

A341513 Sum of digits in A097801-base.

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