Dave Durgin - cp's OEIS frontend

A316988 The odd members of the "Almost natural numbers" (A007376).

1, 3, 5, 7, 9, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 1, 9, 1, 3, 5, 7, 9, 3, 3, 1, 3, 3, 3, 3, 3, 5, 3, 3, 7, 3, 3, 9, 1, 3, 5, 7, 9, 5, 5, 1, 5, 5, 3, 5, 5, 5, 5, 5, 7, 5, 5, 9, 1, 3, 5, 7, 9, 7, 7, 1, 7, 7, 3, 7, 7, 5, 7, 7, 7, 7, 7, 9, 1, 3, 5, 7, 9, 9, 9, 1, 9, 9, 3, 9, 9, 5, 9, 9, 7, 9, 9, 9, 1

Offset: 1

Dave Durgin, Jul 18 2018

nonn

Also odd members of A033307.

As the number of terms approaches infinity, each odd term appears equally often. - Robert G. Wilson v, Aug 01 2018

			The initial (15) terms of A007376 are 1,2,3,4,5,6,7,8,9,1,0,1,1,1,2; omitting the even terms 2,4,6,8,0,2 leaves the first (8) odd terms 1,3,5,7,9,1,1,1: a(1) through a(8).

Cf. A007376, A033307.

Select[ Flatten[ IntegerDigits /@ Range@100], OddQ] (* Robert G. Wilson v, Aug 01 2018 *)

search(n) = for(k=1, n, my(d=select(x->x%2, digits(k))); for(m=1, #d, print1(d[m], ", ")))
/* Call the function as follows to generate terms */
search(40) \\ Felix Fröhlich, Aug 01 2018

A317647 The even members of the "Almost natural numbers" (A007376).

Original entry on oeis.org

2, 4, 6, 8, 0, 2, 4, 6, 8, 2, 0, 2, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 8, 2, 0, 2, 4, 6, 8, 4, 0, 4, 4, 2, 4, 4, 4, 4, 4, 6, 4, 4, 8, 4, 0, 2, 4, 6, 8, 6, 0, 6, 6, 2, 6, 6, 4, 6, 6, 6, 6, 6, 8, 6, 0, 2, 4, 6, 8, 8, 0, 8, 8, 2, 8, 8, 4, 8, 8, 6, 8, 8, 8, 8, 0, 2, 4, 6, 8, 0, 0

Offset: 1

Dave Durgin and Robert G. Wilson v, Aug 02 2018

Also even members of A033307.

			The initial (15) terms of A007376 are 1,2,3,4,5,6,7,8,9,1,0,1,1,1,2; omitting the odd terms 1,3,5,7,9,1,1,1 leaves the first 6 even terms 2,4,6,8,0,2: a(1) through a(6).

Cf. A007376, A033307, A316988.

Select[ Flatten[ IntegerDigits /@ Range@ 100], EvenQ]

A275589 The almost-natural numbers with prime digits removed.

Original entry on oeis.org

1, 4, 6, 8, 9, 1, 0, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 9, 0, 1, 4, 6, 8, 9, 0, 1, 4, 6, 8, 9, 4, 0, 4, 1, 4, 4, 4, 4, 4, 4, 6, 4, 4, 8, 4, 9, 0, 1, 4, 6, 8, 9, 6, 0, 6, 1, 6, 6, 6, 4, 6, 6, 6, 6, 6, 8, 6, 9, 0, 1, 4, 6, 8, 9, 8, 0, 8, 1, 8, 8, 8, 4, 8

Offset: 1

Dave Durgin, Aug 02 2016

Nonprime complement of A275542.

			Write the digits of the positive integers in continuous form: 1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,...; then remove prime digits (2,3,5,7) from the sequence.

Robert Price, Table of n, a(n) for n = 1..4492

Nonprimes of A007376 and/or A033307.

Cf. A275542.

DeleteCases[Map[IntegerDigits, Range@ 120] // Flatten, k_ /; PrimeQ@ k] (* Michael De Vlieger, Aug 03 2016 *)
Table[DeleteCases[IntegerDigits[n],?PrimeQ],{n,100}]//Flatten (* _Harvey P. Dale, Dec 31 2017 *)

More terms from Robert Price, Mar 31 2017

A275542 The digits of the integers with the nonprimes removed.

Original entry on oeis.org

2, 3, 5, 7, 2, 3, 5, 7, 2, 2, 2, 2, 2, 3, 2, 2, 5, 2, 2, 7, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 5, 3, 3, 7, 3, 3, 2, 3, 5, 7, 5, 5, 5, 2, 5, 3, 5, 5, 5, 5, 5, 7, 5, 5, 2, 3, 5, 7, 7, 7, 7, 2, 7, 3, 7, 7, 5, 7, 7, 7, 7, 7, 2, 3, 5, 7, 2, 3, 5, 7

Offset: 1

Dave Durgin, Aug 01 2016

Write the digits of the positive integers one by one: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, etc. (this is A007376). Then from that sequence, remove the nonprimes, leaving a sequence that consists entirely of 2s, 3s, 5s and 7s.

			From the single-digit numbers, we obviously get the first four terms of this sequence: 2, 3, 5, 7.
10 is composite and neither of its digits is a single-digit prime, so it contributes nothing to this sequence.
11 is prime but its digits consist of two 1s, so like 10 it also contributes nothing to the sequence.
12 is composite, but its least significant digit is 2, which is a prime, and thus 12 contributes a 2 to the sequence.

Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..2401 from Robert Price)

Cf. A007376, A033307.

Flatten[Table[Select[IntegerDigits[n], PrimeQ], {n, 100}]] (* Alonso del Arte, Aug 01 2016 *)

A251599 Centers of rows of the triangular array formed by the natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723

Offset: 1

Dave Durgin, Dec 05 2014

nonn

Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - Reinhard Zumkeller, Dec 13 2014
Union of A080827 and A000982. - David James Sycamore, Aug 09 2018

			First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
               1
             2   3
           4   5   6
         7   8   9  10
      11  12  13  14  15
    16  17  18  19  20  21
  22  23  24  25  26  27  28

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David James Sycamore, A080827 & A000982
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000124, A000217, A001844.
Cf. A092942 (first differences).
Cf. A001105, A056106, A058331.
Cf. A080827, A000982.

a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
   f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
                     f (i + 1) wss
   g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
-- Reinhard Zumkeller, Dec 12 2014

a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 10 2014

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)

Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016

Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).

G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014

A229527 Start with 1, skip (sum of digits of n) numbers, accept next number.

Original entry on oeis.org

1, 3, 7, 15, 22, 27, 37, 48, 61, 69, 85, 99, 118, 129, 142, 150, 157, 171, 181, 192, 205, 213, 220, 225, 235, 246, 259, 276, 292, 306, 316, 327, 340, 348, 364, 378, 397, 417, 430, 438, 454, 468, 487, 507, 520, 528, 544, 558, 577, 597

Offset: 1

Dave Durgin, Sep 25 2013

			a(1)=1, a(2)=1+1+1=3, a(3)=3+3+1=7, a(4)=7+7+1=15, a(5)=15+1+5+1=22, a(6)=22+2+2+1=27, ...

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A007612, A004207, A160939.

a[n_] := a[n] = a[n - 1] + 1 + Plus @@ IntegerDigits@a[n - 1]; a[1] = 1; Array[a, 50] (* Robert G. Wilson v, Aug 01 2018 *)

from itertools import islice
def A229527_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a += sum(map(int,str(a)))+1
A229527_list = list(islice(A229527_gen(),40)) # Chai Wah Wu, Aug 09 2025

a(n+1) = a(n) + (sum of digits of a(n)) + 1.

A225985 List the positive numbers, remove even digits (including zeros) from each term; sequence = remaining terms.

Original entry on oeis.org

1, 3, 5, 7, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 3, 5, 7, 9, 3, 31, 3, 33, 3, 35, 3, 37, 3, 39, 1, 3, 5, 7, 9, 5, 51, 5, 53, 5, 55, 5, 57, 5, 59, 1, 3, 5, 7, 9, 7, 71, 7, 73, 7, 75, 7, 77, 7, 79, 1, 3, 5, 7, 9, 9, 91, 9, 93, 9, 95, 9, 97, 9, 99, 1, 11, 1, 13

Offset: 1

Dave Durgin, May 22 2013

			The natural numbers with at least one odd digit in their decimal representation are: 1, 3, 5, 7, 9, 10, 11, 12, 13, ...
By excluding their even digits, we obtain: 1, 3, 5, 7, 9, 1, 11, 1, 13, ...
Hence: a(1)=1, a(2)=3, a(3)=5, a(4)=7, a(5)=9, a(6)=1, a(7)=11, a(8)=1, a(9)=13, .... [Example corrected by _Paul Tek_, May 24 2013]

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A038573.

Cf. A014261 (duplicates removed), A226091.

a225985 n = a225985_list !! (n-1)
a225985_list = map read $ filter (not . null) $
    map (filter (`elem` "13579") . show) [0..] :: [Integer]
-- Reinhard Zumkeller, May 26 2013

FromDigits[DeleteCases[IntegerDigits[#],?EvenQ]]&/@Range[200]/. (0-> Nothing) (* _Harvey P. Dale, Apr 04 2017 *)

from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        removed  = "".join(d if d in "13579" else "" for d in str(n))
        if removed != "": yield int(removed)
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 05 2022

a(A226091(n)) = A014261(n). - Reinhard Zumkeller, May 26 2013

Definition clarified by N. J. A. Sloane, Aug 06 2022 at the suggestion of Michel Marcus

A224780 Strings of ascending digits in A007376.

Original entry on oeis.org

123456789, 12, 12, 23, 23, 34, 34, 45, 45, 56, 56, 67, 67, 78, 78, 89, 89, 12, 12, 12, 123, 12, 12, 12, 12, 12, 12, 34, 45, 56, 67, 78, 89, 12, 12, 12, 23, 23, 23, 123, 23, 234, 23, 23, 23, 23, 23, 12, 45, 12, 56, 12, 67, 12, 78, 12, 89, 12, 23, 123, 23, 23, 23, 34, 34, 34, 34, 234, 34, 345, 34, 34, 34, 34, 23, 56, 23, 67, 23, 78, 23, 89

Offset: 1

Dave Durgin, Apr 17 2013

We sample each digit in A007376 in turn; accept the longest string for which A(n+1)-A(n)=1, A(n+2)-A(n+1)=1 and so on.
We recognize only strings of length >=2 and not strings with leading zeros.
Series is infinite, but there are only 36 possible consecutive strings: 12, 123, 1234,...,123456789 (eight beginning with 1), 23, 234, 2345,...,23456789 (seven beginning with 2) and so on.

			The 1st five terms imbedded in A007376 in brackets: [123456789]1011[12]13141516171819202[12]2[23]24252627282930313[23]

A007376 := [] :
for n from 1 to 400 do
    nb := convert(n,base,10) ;
    A007376 := [op(A007376),op(ListTools[Reverse](nb))] ;
end do:
str := 1 :
while true do
    for a from 1 do
        if op(str+a,A007376) <> 1+op(str+a-1,A007376) then
            break;
        end if;
    end do:
    L := ListTools[Reverse]([op(str..str+a-1,A007376)]) ;
    if nops(L) > 1 and op(-1,L) > 0 then
        add( op(i,L)*10^(i-1),i=1..nops(L)) ;
        printf("%d,",%) ;
    end if;
    str := str+a ;
end do: # R. J. Mathar, Apr 26 2013

A222621 a(n) = (2n-1)^(2n).

Original entry on oeis.org

1, 81, 15625, 5764801, 3486784401, 3138428376721, 3937376385699289, 6568408355712890625, 14063084452067724991009, 37589973457545958193355601, 122694327386105632949003612841, 480250763996501976790165756943041, 2220446049250313080847263336181640625

Offset: 1

Dave Durgin, Feb 26 2013

			1^2; 3^4; 5^6; 7^8 and so on.

Vincenzo Librandi, Table of n, a(n) for n = 1..190

Cf. A005408, A005843, A085535, A338168.

[(2*n - 1)^(2*n): n in [1..13]]; // Vincenzo Librandi, Feb 26 2013

Table[(2 n - 1)^(2 n), {n, 20}] (* T. D. Noe, Feb 26 2013 *)

a(n)=(2*n-1)^(2*n) \\ Charles R Greathouse IV, Aug 20 2013

a(n) = A005408(n-1)^A005843(n). - Michel Marcus, Apr 14 2018

Sum_{n>=1} 1/a(n) = A338168. - Amiram Eldar, Nov 29 2020

A182402 Total number of digits in n-th row of a triangle formed by the positive integers.

Original entry on oeis.org

1, 2, 3, 5, 10, 12, 14, 16, 18, 20, 22, 24, 26, 34, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 171, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 1

Dave Durgin, Jun 19 2012

Sequence is nonlinear at each decade transition; for example, row-5 transitions from single-digit (7) to double-digit (10) where sequence jumps (3) to (5); row-14 transitions from 2-digit (92) to 3-digit (105) where sequence jumps from (26) to (34).

The rows of nonlinearity are given by A068092. - Jon Perry, May 26 2013

			1; .................... (row 1 contains 1 digit)
2,   3; ............... (row 2 contains 2 digits)
4,   5,  6; ........... (row 3 contains 3 digits)
7,   8,  9, 10; ....... (row 4 contains 5 digits)
11, 12, 13, 14, 15; ... (row 5 contains 10 digits)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A055642, A226029 (first differences).

Cf. A068092.

a182402 n = a182402_list !! (n-1)
a182402_list = map (sum . map a055642) $ t 1 [1..] where
   t i xs = ys : t (i + 1) zs where
     (ys, zs) = splitAt i xs
-- Reinhard Zumkeller, May 26 2013

f[n_] := Length@ Flatten[ IntegerDigits[ Range[n (n - 1)/2 + 1, n (n + 1)/2]]]; Array[f, 58] (* Robert G. Wilson v, Sep 04 2013 *)

a(n) = {my(x=n*(n-1)/2+1, y=n*(n+1)/2, nx=#Str(x), ny=#Str(y), s=0); for (i=nx, ny, if (i==nx, if (i==ny, s+=(y+1-x)*i, s+=(10^i-x)*i), if (i==ny, s+=(y+1-10^(i-1))*i, s+=i*(10^(i+1)-10^i+1)););); s;} \\ Michel Marcus, Jan 26 2022

def a(n): return len("".join(str(i) for i in range(n*(n+1)//2+1, (n+1)*(n+2)//2+1)))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 26 2022

a(n) = A058183(A000217(n)) - A058183(A000217(n-1)), n >= 2. - Omar E. Pol, Jun 25 2012

Better definition from Omar E. Pol, Jun 25 2012

cp's OEIS Frontend

User: Dave Durgin

A316988 The odd members of the "Almost natural numbers" (A007376).

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A317647 The even members of the "Almost natural numbers" (A007376).

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A275589 The almost-natural numbers with prime digits removed.

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A275542 The digits of the integers with the nonprimes removed.

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A251599 Centers of rows of the triangular array formed by the natural numbers.

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A229527 Start with 1, skip (sum of digits of n) numbers, accept next number.

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A225985 List the positive numbers, remove even digits (including zeros) from each term; sequence = remaining terms.

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