A033035 -id:A033035 - cp's OEIS frontend

A155721 Positions of parity change in A033035.

4, 20, 84, 340, 1364, 5460, 21844, 87380, 349524, 1398100, 5592404, 22369620, 89478484, 357913940, 1431655764, 5726623060, 22906492244, 91625968980, 366503875924, 1466015503700, 5864062014804, 23456248059220, 93824992236884

Offset: 1

Zak Seidov, Jan 25 2009

nonn

Numbers n such that A033035(n)=(7/8)*(9^n - 1)=7*A002452(n).

			A033035(n=1..4) all are odd, then A033035(n=5..20) all are even, then A033035(n=21..84) all are odd, then A033035(n=85..340) all are even, etc.

A002452, A033035, A080674.

a(n=1,2..)=(4/3)*(4^n-1)=A080674(n).

a(n) = 5*a(n-1) - 4*a(n-2) for n > 2. G.f.: 4*x/((x - 1)*(4*x - 1)). - Chai Wah Wu, Aug 30 2019

A351894 Numbers that contain only odd digits in their factorial-base representation.

Original entry on oeis.org

1, 3, 9, 21, 33, 45, 81, 93, 153, 165, 201, 213, 393, 405, 441, 453, 633, 645, 681, 693, 873, 885, 921, 933, 1113, 1125, 1161, 1173, 1353, 1365, 1401, 1413, 2313, 2325, 2361, 2373, 2553, 2565, 2601, 2613, 2793, 2805, 2841, 2853, 3753, 3765, 3801, 3813, 3993, 4005

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms above 1 are odd multiples of 3.

			3 is a term since its factorial-base presentation, 11, has only odd digits.
21 is a term since its factorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..18056 (terms below 11!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A016945, A351893.
Subsequence: A007489
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!, 2], AllTrue[fctBaseDigits[#], OddQ] &]

A056760 Integers with exactly 2 prime divisors such that the cube of the number of divisors exceeds the number.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172

Offset: 1

Labos Elemer, Aug 16 2000

Numbers with 8 prime divisors also occur among cases satisfying relation d^3>n.

Prime divisors are counted without multiplicity. - Harvey P. Dale, May 14 2012

			The sequence is finite and almost surely complete. Between 270000 and 17000000 no more cases were found. The last 3 entries are: 165888, 186624, 248832. E.g. k = 1024*343 = 248832, with 66 divisors and d^3 = 287496 > 248832.

Donovan Johnson, Table of n, a(n) for n = 1..254 (complete sequence)

Cf. A000005, A033033-A033035, A034884.

Select[Range[180],PrimeNu[#]==2&&DivisorSigma[0,#]^3>#&] (* Harvey P. Dale, May 14 2012 *)

Integers k = (p^w)*(q^u) such that d(k)^3 > k, where d(k) = A000005(k).

A363242 Numbers whose primorial-base representation contains only odd digits.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 111, 159, 171, 249, 261, 309, 321, 369, 381, 669, 681, 729, 741, 789, 801, 1089, 1101, 1149, 1161, 1209, 1221, 1509, 1521, 1569, 1581, 1629, 1641, 1929, 1941, 1989, 2001, 2049, 2061, 2559, 2571, 2619, 2631, 2679, 2691, 2979, 2991, 3039

Offset: 1

Amiram Eldar, May 23 2023

All the terms above 1 are odd multiples of 3.

The partial sums of the primorials (A143293) are terms, since the primorial-base representation of A143293(n) is n+1 1's.

			3 is a term since its primorial-base presentation, 11, has only odd digits.
21 is a term since its primorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..14620 (terms below A002110(8) = 19#)
Index entries for sequences related to primorial base.

Cf. A002110, A049345, A328849.
Subsequence: A143293.
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary), A351894 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[1, nmax, 2], AllTrue[prmBaseDigits[#], OddQ] &]]

is(n) = {my(p = 2); if(n < 1, return(0)); while(n > 0, if((n%p)%2 == 0, return(0)); n \= p; p = nextprime(p+1)); return(1);}

cp's OEIS Frontend

A155721 Positions of parity change in A033035.

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A351894 Numbers that contain only odd digits in their factorial-base representation.

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A056760 Integers with exactly 2 prime divisors such that the cube of the number of divisors exceeds the number.

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A363242 Numbers whose primorial-base representation contains only odd digits.

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