A367366 a(n) = smallest k such that the commas sequence (cf. A121805) with initial term k contains n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 10, 2, 25, 26, 27, 28, 29, 30, 31, 32, 30, 21, 1, 3, 37, 38, 39, 40, 41, 42, 43, 40, 31, 20, 13, 4, 49, 50, 51, 52, 53, 54, 50, 41, 32, 10, 14, 60, 5, 62, 63, 64, 65, 60, 51, 42, 30, 70, 2, 15, 6, 74, 75

Offset

1,2

Comments

Every k >= 1 appears in this sequence exactly A330128(k) times. So there are 2137453 1's, 194697747222394 2's, 2 3's, 209534289952018960 6's, and so on.

a(n) is the most remote ancestor of n in the comma-successor graph.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube

Example

All terms n in A121805 have a(n) = 1, all n in A139284 have a(n) = 2, all n in A366492 have a(n) = 4, and so on.

Prog

(Python)
def comma_predecessor(n): # A367614(n)
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    kk = k + 10*x + y-1
    return k if k > 0 and int(str(kk)[0]) != y-1 else -1
def a(n):
    an = n
    while (cp:=comma_predecessor(an)) > 0: an = cp
    return an
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Dec 18 2023

Crossrefs

Cf. A121805, A139284, A330128, A366492, A367338, A367341, A367617.

Sequence in context: A331375 A179622 A367617 * A285757 A136687 A086706

Adjacent sequences: A367363 A367364 A367365 * A367367 A367368 A367369

Keyword

nonn,base

Author

N. J. A. Sloane, Dec 05 2023

Status

approved