A054684 -id:A054684 - cp's OEIS frontend

A256785 Numbers n such that digitsum(n) is a whole number when n is represented in the fractional base 1.5 = 3/2.

1, 5, 11, 14, 20, 21, 22, 23, 26, 29, 30, 31, 33, 34, 38, 39, 40, 41, 45, 46, 51, 52, 53, 56, 57, 58, 60, 61, 65, 69, 70, 71, 74, 78, 79, 83, 84, 85, 87, 88, 89, 90, 91, 95, 101, 105, 106, 110, 111, 112, 113, 116, 117, 118, 122, 126, 127, 132, 133, 135, 136, 140, 146, 149, 155, 159, 160, 161, 164, 165, 166, 168, 169, 173, 174, 175

Offset: 1

Anthony Sand, Apr 10 2015

Base 1.5 requires three digits: 1, 0 and H = 0.5. For example:
= 1 = 1 * 1.5^0
= 1H = 1 * 1.5^1 + 0.5 * 1.5^0 = 1.5 + 0.5
= 1H0 = 1 * 1.5^2 + 0.5 * 1.5^1 = 2.25 + 0.75
= 1H1 = 1 * 1.5^2 + 0.5 * 1.5^1 + 1 * 1.5^0 = 2.25 + 0.75 + 1
= 1H0H = 1 * 1.5^3 + 0.5 * 1.5^2 + 0.5 * 1.5^0 = 3.375 + 1.125 + 0.5
= 1H10 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 = 3.375 + 1.125 + 1.5
= 1H11 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 + 1 * 1.5^0 = 3.375 + 1.125 + 1.5 + 1
The sequence above lists the n for which digsum(n,base=1.5) is a whole number.

			The sequence begins with 1, 5 and 11, because:
digsum(1,b=1.5) = 1
digsum(5,b=1.5) = 2 = digsum(1H0H) = 1 + 0.5 + 0.5
digsum(11,b=1.5) = 4 = digsum(1H11H) = 1 + 0.5 + 1 + 1 + 0.5
The digsums are all whole numbers. However, 2, 3 and 4 are excluded because:
digsum(2,b=1.5) = 1.5 = digsum(1H) = 1 + 0.5
digsum(3,b=1.5) = 1.5 = digsum(1H0) = 1 + 0.5 + 0
digsum(4,b=1.5) = 2.5 = digsum(1H1) = 1 + 0.5 + 1
The digsums are not whole numbers.

Anthony Sand, Table of n, a(n) for n = 1..1000
Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019

Cf. A007953, A054683, A054684.

{ b=3/2; dmx=30; d=vector(dmx); nmx=1000; n=0; ni=0; while(ni0, di++; d[di]=nn-floor(nn/b)*b; nn\=b; ); s=0; for(i=1,di,s+=d[i]); if(floor(s)==s, ni++; write("digsum.txt",ni," ",n)); ); }

A358270 Numbers whose sum of digits is even and that have an even number of even digits.

Original entry on oeis.org

11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 1001, 1003, 1005, 1007, 1009, 1010, 1012, 1014, 1016, 1018, 1021, 1023, 1025, 1027, 1029, 1030

Offset: 1

Bernard Schott, Nov 06 2022

There are only terms with an even number of digits, and precisely, there exist A137233(2*k) terms with 2*k digits.
The conditions separately are A054683 for even sum of digits, and A356929 for even number of even digits, so that this sequence is their intersection.
The opposite conditions, an odd sum of digits, and an odd number of odd digits, are the same and are A054684.

			26 is a term since 2+6 = 8 (even) and 26 has two even digits.
39 is a term since 3+9 = 12 (even) and 39 has zero even digits.
1012 is a term since 1+0+1+2 = 4 (even) and 1012 has two even digits.

Intersection of A054683 and A356929.
Cf. A001637 (even length), A179081 (digit sum mod 2).
Cf. A014263, A054684, A137233.

Select[Range[1000], EvenQ[Plus @@ IntegerDigits[#]] && EvenQ[Plus @@ DigitCount[#, 10, Range[0, 8, 2]]] &] (* Amiram Eldar, Nov 06 2022 *)

a(n) = n*=2; n += 100^logint(110*n,100) \ 11; n - sumdigits(n)%2; \\ Kevin Ryde, Nov 10 2022

def ok(n): s = str(n); return sum(map(int, s))%2 == sum(1 for d in s if d in "02468")%2 == 0
print([k for k in range(1031) if ok(k)]) # Michael S. Branicky, Nov 06 2022

from itertools import count, islice, chain
def A358270_gen(): # generator of terms
    return filter(lambda n:not (len(s:=str(n))&1 or sum(int(d) for d in s)&1), chain.from_iterable((range(10**l,10**(l+1)) for l in count(1,2))))
A358270_list = list(islice(A358270_gen(),61)) # Chai Wah Wu, Nov 11 2022

a(n) = t - A179081(t) where t = A001637(2*n). - Kevin Ryde, Nov 10 2022

cp's OEIS Frontend

A256785 Numbers n such that digitsum(n) is a whole number when n is represented in the fractional base 1.5 = 3/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI