A248146 -id:A248146 - cp's OEIS frontend

A248138 Consider the partition of consecutive evil numbers (A001969) into minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists numbers of evil numbers in each block.

3, 2, 2, 3, 2, 3, 4, 2, 2, 5, 2, 5, 3, 2, 2, 2, 2, 2, 6, 4, 3, 6, 4, 7, 4, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 3, 5, 2, 3, 3, 6, 2, 4, 5, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 2, 2, 3

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 02 2014

nonn

The blocks of consecutive evil numbers of the partition are

0,3,5| 6,9| 10,12| 15,17,18| 20,23| 24,27,29| 30,33,34,36| 39,40| 43,45| 46,48,51,53,54| 57,58| 60,63,65,66,68|, etc.

Cf. A000069, A001969, A248140, A248145, A248146.

A248009 Partition of the positive odd integers into minimal blocks such that the concatenation of the numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

Original entry on oeis.org

135, 7911, 131517, 19212325272931, 33, 35373941, 43, 45, 4749, 51, 53, 5557, 596163, 65, 676971, 737577, 798183, 85, 8789, 9193, 95, 9799101103105, 107109, 111, 113, 115117119, 121123125127129, 131133, 135, 137139, 141, 143145147, 149, 151153155157, 159

Offset: 1

Vladimir Shevelev, Oct 05 2014

The numbers of the consecutive positive odd integers in the blocks of the partition are 3,3,3,7,1,4,1,1,2,1,1,2,3,1,3,3,3,1,2,2,...

			a(1)=135, since 1 and 13 are odious numbers, while 135 is evil.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A248138, A248140, A248145, A248146, A248171, A248172.

lista(nn) = {s = ""; forstep(n=1, nn, 2, ns = concat(s, Str(n)); if ((hammingweight(eval(ns)) % 2) == 0, print1(ns, ", "); s = "", s = ns););} \\ Michel Marcus, Oct 09 2014

More terms from Peter J. C. Moses, Oct 09 2014

A248140 Consider the partition of the consecutive odious numbers (A000069) into minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence gives the number of odious numbers in each block.

Original entry on oeis.org

2, 7, 3, 2, 3, 3, 4, 5, 5, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 5, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 3, 4, 9, 2, 2, 2, 2, 7, 5, 3, 2, 3, 2, 4, 4, 4, 2, 3, 4, 2, 3, 4, 3, 4, 2, 2, 3, 2, 2, 2, 9, 2, 5, 2, 5, 4, 4, 2, 4, 4, 2, 3, 3, 8, 3, 2, 2, 3, 2, 3, 2, 2, 2, 4, 2, 4, 3

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 02 2014

nonn

The blocks of consecutive odious numbers of the partition are

1,2| 4,7,8,11,13,14,16| 19,21,22|25,26| 28,31,32| 35,37,38| 41,42,44,47| 49,50,52,55,56| 59,61,62,64,67| 69,70| 73,74| 76,79|, etc.

Cf. A000069, A001969, A248138, A248145, A248146.

A248145 Consider the partition of the positive odd integers into minimal blocks such that concatenation of numbers in each block is a number of the form 3^k*prime, k>=0. Sequence lists numbers of odd integers in the blocks.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 7, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 5, 11, 7, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 348, 2, 20, 30, 453, 2, 1, 2, 3, 17, 1, 219, 1, 2, 4, 10, 1, 2, 1, 1, 46, 1303, 4, 2, 1, 2, 2, 1

Offset: 1

Vladimir Shevelev, Oct 02 2014

3^m, m>=1, is of the considered form 3^k*prime, k=m-1>=0, prime=3.

The first blocks of the partition are |1,3|,|5|,|7|,|9|,|11|,|13|,|15|,|17|,|19|,|21|,|23|,|25,27,29|,|31|,|33|,|35,37|,...

			The 12th block of partition is |25,27,29|, since we have 25=5^2, 2527=7*19^2, 252729=3^2*28081, and only the last number is of the required form. So a(12)=3.

Cf. A103899, A248146.

from gmpy2 import is_prime
from itertools import count, islice
def c(n):
    if n < 3: return False
    while n%3 == 0: n //= 3
    return n == 1 or is_prime(n)
def agen(): # generator of terms
    i = 1
    while True:
        s, an = str(i), 1
        while not c(t:=int(s)): i += 2; s += str(i); an += 1
        yield an
        i += 2
print(list(islice(agen(), 78))) # Michael S. Branicky, Oct 05 2024

a(43) and beyond from Michael S. Branicky, Oct 05 2024

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A248138 Consider the partition of consecutive evil numbers (A001969) into minimal blocks such that concatenation of numbers in each block is an odious number (A000069). Sequence lists numbers of evil numbers in each block.

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A248009 Partition of the positive odd integers into minimal blocks such that the concatenation of the numbers in each block is an evil number (A001969). Sequence lists the evil numbers obtained in this way.

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A248140 Consider the partition of the consecutive odious numbers (A000069) into minimal blocks such that concatenation of numbers in each block is an evil number (A001969). Sequence gives the number of odious numbers in each block.

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A248145 Consider the partition of the positive odd integers into minimal blocks such that concatenation of numbers in each block is a number of the form 3^k*prime, k>=0. Sequence lists numbers of odd integers in the blocks.

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Python

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