A137671 -id:A137671 - cp's OEIS frontend

A137674 Numbers m such that A137671(m) = 1.

1, 2, 5, 13, 35, 84, 198, 414, 960, 1898, 4263, 8556, 18473, 37490, 79526, 159956, 338873, 680306, 1438941, 2869908, 6057397, 12038460, 25300258, 50460684, 105275708, 211424744, 437486956, 879869316, 1810901110, 3642407379, 7495644753, 15038540180, 30957576741

Offset: 1

Reinhard Zumkeller, Feb 05 2008

Rémy Sigrist, C program for A137674.

Cf. A000225, A137671, A137672, A137673.

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b[1] = 1; b[n_] := b[n] = Count[Array[s, n - 1], s[n - 1]]; s[n_] := s[n] = DigitCount[b[n], 2, 1]; Select[Range[10000], b[#] == 1 &] (* Amiram Eldar, Jul 27 2023 *)

A137671(a(n)) = 1.
A137672(a(n)) = 1.
For n > 1: A137671(a(n)-1) = 2^(n-1)-1 = A000225(n-1), A137672(a(n)-1) = n-1, and a(n) = A137673(A137671(a(n)-1)) + 1.

More terms from Rémy Sigrist, Jun 28 2019

A137672 Number of ones in binary representation of A137671(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 1, 3, 2, 1, 4, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 4, 2, 4, 1, 2, 2, 3, 1, 3, 2, 3, 2, 4, 2, 3, 3, 2, 4, 2, 4, 3, 3, 3, 4, 1, 2, 5, 1, 3, 2, 1, 3, 3, 3, 4, 2, 2, 2, 3, 3, 4, 2, 2, 3, 4, 3, 5, 1, 4

Offset: 1

Reinhard Zumkeller, Feb 05 2008

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

Cf. A000120, A137671, A137674.

b[1] = 1; b[n_] := b[n] = Count[Array[a, n-1], a[n-1]]; a[n_] := a[n] = DigitCount[b[n], 2, 1]; Array[a, 100] (* Amiram Eldar, Jul 27 2023 *)

a(n) = A000120(A137671(n)).
a(A137674(n)) = 1.
a(A137674(n)-1) = n-1.

A137673 Smallest m such that A137671(m) = n.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 12, 14, 15, 20, 22, 28, 30, 32, 34, 41, 45, 47, 48, 50, 51, 54, 59, 62, 63, 67, 69, 71, 74, 76, 83, 87, 93, 94, 95, 99, 100, 107, 110, 113, 114, 116, 119, 128, 133, 135, 140, 142, 143, 150, 153, 155, 156, 161, 163, 166, 167, 170, 183, 186, 188, 191

Offset: 1

Reinhard Zumkeller, Feb 05 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..2000

Cf. A137671, A137674.

b[1] = 1; b[n_] := b[n] = Count[Array[s, n - 1], s[n - 1]]; s[n_] := s[n] = DigitCount[b[n], 2, 1]; seq[len_, nmax_] := Module[{t = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = b[n]; If[t[[i]] == 0, c++; t[[i]] = n]; n++]; t]; seq[100, 1000] (* Amiram Eldar, Jul 27 2023 *)

A137671(a(n)) = n and A137671(m) < n for m < a(n).

A137674(n) = a(A137671(A137674(n)-1)) + 1.

A356348 a(0) = 0; for n > 0, a(n) is the number of preceding terms having the same digit sum as a(n-1).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 11, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 12, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 13, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 14, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 15, 6, 7, 6, 8, 6, 9, 6, 10, 16, 7, 8, 7, 9, 7, 10, 17, 8, 9

Offset: 0

Scott R. Shannon, Oct 15 2022

			a(21) = 2 as a(20) = 11 which has a digit sum of 2, and there has been two previous terms with a digit sum of two: a(3) = 2 and a(20) = 11.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of n=0..1000000.

Cf. A007953, A137671 (base 2), A342585.

nn = 94; c[] = 0; a[0] = k = 0; c[0] = 1; Do[a[n] = c[k]; k = Total@ IntegerDigits[a[n]]; c[k]++, {n, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 15 2022 *)

from itertools import islice
from collections import Counter
def sd(n): return sum(map(int, str(n)))
def agen(): # generator of terms
    an, s, inventory = 0, 0, Counter([0])
    while True:
        yield an; an = inventory[s]; s = sd(an); inventory[s] += 1
print(list(islice(agen(), 95))) # Michael S. Branicky, Oct 21 2022

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A137674 Numbers m such that A137671(m) = 1.

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A137672 Number of ones in binary representation of A137671(n).

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A137673 Smallest m such that A137671(m) = n.

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A356348 a(0) = 0; for n > 0, a(n) is the number of preceding terms having the same digit sum as a(n-1).

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