A342042 -id:A342042 - cp's OEIS frontend

A379901 Local maxima in A342042, in order of appearance: numbers m such that A342042(k-1) < A342042(k) = m > A342042(k+1), for k >= 2.

30, 50, 70, 90, 51, 71, 91, 52, 72, 92, 73, 93, 74, 94, 95, 96, 301, 501, 701, 901, 303, 502, 503, 702, 902, 703, 505, 704, 705, 903, 307, 507, 707, 904, 905, 906, 907, 909, 509, 710, 910, 911, 309, 510, 711, 912, 311, 511, 712, 313, 913, 512, 513, 713, 914, 515

Offset: 1

Paolo Xausa, Jan 05 2025

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

Cf. A342042, A379902 (corresponding k's), A379903, A379904.

from itertools import count, islice
def cond(s): return all(s[i+1] > s[i] for i in range(len(s)-1) if s[i] in "02468")
def A342042gen(): # generator of A342042
    seen, an = set(), 0
    while True:
        yield an
        seen.add(an)
        d = an%10
        an = minfirst = (1 - d%2)*(d+1)
        stran = str(an)
        while an in seen or not cond(stran):
            an += 1
            stran = str(an)
            if int(stran[0]) < minfirst:
                an = minfirst*10**(len(stran)-1)
def A3799012gen(): # generator of maxima, indices
    g = A342042gen()
    p = next(g)
    m = next(g)
    n = next(g)
    for k in count(2):
        if p < m > n: yield (m, k)
        p, m, n = m, n, next(g)
print([t[0] for t in islice(A3799012gen(), 56)]) # Michael S. Branicky, Jan 08 2025

A379903 Local minima in A342042, in order of appearance: numbers m such that A342042(k-1) > A342042(k) = m < A342042(k+1), for k >= 2.

Original entry on oeis.org

0, 13, 15, 17, 19, 25, 27, 29, 35, 37, 39, 47, 49, 57, 59, 69, 79, 103, 105, 107, 109, 113, 304, 115, 117, 306, 119, 125, 506, 127, 129, 133, 135, 137, 508, 139, 147, 149, 153, 155, 157, 159, 169, 173, 175, 177, 179, 193, 195, 312, 197, 199, 233, 235, 237, 514

Offset: 1

Paolo Xausa, Jan 05 2025

a(1) = 0 is included as well.

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

Cf. A342042, A379901, A379902, A379904 (corresponding k's).

from itertools import count, islice
def cond(s): return all(s[i+1] > s[i] for i in range(len(s)-1) if s[i] in "02468")
def A342042gen(): # generator of A342042
    seen, an = set(), 0
    while True:
        yield an
        seen.add(an)
        d = an%10
        an = minfirst = (1 - d%2)*(d+1)
        stran = str(an)
        while an in seen or not cond(stran):
            an += 1
            stran = str(an)
            if int(stran[0]) < minfirst:
                an = minfirst*10**(len(stran)-1)
def A3799034gen(): # generator of minima, indices
    g = A342042gen()
    p = float('inf'); m = next(g); n = next(g)
    for k in count(1):
        if p > m < n: yield (m, k)
        p, m, n = m, n, next(g)
print([t[0] for t in islice(A3799034gen(), 56)]) # Michael S. Branicky, Jan 08 2025

A377914 Indices of records in A342042.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 20, 23, 33, 45, 52, 61, 65, 71, 74, 75, 76, 77, 78, 79, 82, 85, 88, 106, 121, 136, 138, 143, 147, 153, 154, 163, 167, 181, 199, 215, 226, 240, 244, 248, 249, 258, 264, 272, 283, 295, 298, 299, 312, 319, 327, 333, 337, 351, 365, 380, 384, 385, 400, 410, 417, 421, 422, 430, 440, 443, 447, 459, 472, 484, 498, 505, 506

Offset: 1

N. J. A. Sloane, Nov 28 2024

N. J. A. Sloane, Table of n, a(n) for n = 1..2548

Cf. A342042, A377913.

A377918 a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

Original entry on oeis.org

1, 11, 77, 566, 4197, 31148, 231193, 1716043, 12737453, 94544693, 701765055, 5208903636, 38663477066, 286982552081, 2130149470506, 15811193864583, 117359769764941, 871111674250772, 6465891595866732, 47993564275737877, 356235822660837879, 2644187054283807954, 19626676300599636003

Offset: 1

Sebastian Karlsson and N. J. A. Sloane, Nov 30 2024

These are the points in the graph of A342042 where the separate paths come together.

The first differences are in A377917, which is the more fundamental sequence. To get this sequence from A377917, add an initial zero, take partial sums, and add 1 to each term.

Ray Chandler, Table of n, a(n) for n = 1..1149
Index entries for linear recurrences with constant coefficients, signature (6,10,5,-5,-9,-5,-1).

Cf. A342042, A377912, A377917.

A377918 := proc(n) local S; option remember;
S:=[1, 11, 77, 566, 4197, 31148, 231193, 1716043];
if n <= 8 then S[n] else
6*A377918(n-1)+10*A377918(n-2)+5*A377918(n-3)-5*A377918(n-4)-9*A377918(n-5)-5*A377918(n-6)-A377918(n-7); fi;
end;
[seq(A377918(i),i=1..20)];

LinearRecurrence[{6, 10, 5, -5, -9, -5, -1}, {1, 11, 77, 566, 4197, 31148, 231193, 1716043}, 25] (* Paolo Xausa, Dec 02 2024  *)

G.f. = (x^7+6*x^6+15*x^5+19*x^4+11*x^3-x^2-5*x-1)/((1-x)*(x^6+6*x^5+15*x^4+20*x^3+15*x^2+5*x-1)) (From g.f. for A377917).
Recurrence: See Maple code.
The smallest root of the denominator of the g.f. is 0.134724138401519... whose reciprocal is (say) c1 = 7.422574840... Then a(n) is asymptotically c2*c1^n, for n >= 0, where c2 = 1.3824387... This is an excellent approximation. It gives a(22) = 0.1962667617*10^20, compared with a(22) = 19626676300599636003.
This also enables us to give a formula for the lower envelope of A342042 - see that entry for details.

More terms added based on A377917. - N. J. A. Sloane, Dec 01 2024

A379902 Indices of local maxima in A342042: numbers k such that A342042(k-1) < A342042(k) > A342042(k+1), for k >= 2.

Original entry on oeis.org

14, 17, 20, 23, 27, 30, 33, 39, 42, 45, 49, 52, 58, 61, 65, 71, 79, 82, 85, 88, 94, 97, 99, 102, 106, 108, 113, 116, 118, 121, 126, 129, 133, 136, 138, 143, 147, 154, 157, 160, 163, 167, 172, 175, 178, 181, 188, 191, 194, 196, 199, 205, 209, 212, 215, 217, 221

Offset: 1

Paolo Xausa, Jan 05 2025

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

Cf. A342042, A379901 (corresponding maxima), A379903, A379904.

# uses functions/imports in A379901
print([t[1] for t in islice(A3799012gen(), 57)]) # Michael S. Branicky, Jan 08 2025

A379904 Indices of local minima in A342042: numbers k such that A342042(k-1) > A342042(k) < A342042(k+1), for k >= 2.

Original entry on oeis.org

1, 15, 18, 21, 24, 28, 31, 34, 40, 43, 46, 50, 53, 59, 62, 66, 72, 80, 83, 86, 89, 95, 98, 100, 104, 107, 109, 114, 117, 119, 122, 127, 130, 134, 137, 139, 145, 148, 155, 158, 161, 164, 168, 173, 176, 179, 183, 189, 192, 195, 197, 200, 207, 210, 213, 216, 218, 222

Offset: 1

Paolo Xausa, Jan 05 2025

a(1) = 1 is included as well.

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

Cf. A342042, A379901, A379902, A379903 (corresponding minima).

# uses functions/imports in A379903
print([t[1] for t in islice(A3799034gen(), 58)]) # Michael S. Branicky, Jan 08 2025

A377913 Records in A342042.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 50, 70, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 301, 501, 701, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959

Offset: 1

N. J. A. Sloane, Nov 28 2024

N. J. A. Sloane, Table of n, a(n) for n = 1..2548

Cf. A342042, A377914.

A377915 a(n) = m means A342042(n) is the m-th term of A377912.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 14, 15, 43, 16, 17, 56, 18, 19, 67, 20, 21, 22, 44, 23, 24, 57, 25, 26, 68, 27, 29, 30, 31, 32, 45, 33, 34, 58, 35, 36, 69, 37, 38, 39, 59, 40, 41, 70, 42, 46, 47, 48, 49, 60, 50, 51, 71, 52, 53, 54, 72, 55, 61, 62, 63, 64, 73, 65, 66, 74, 75, 76, 77, 78, 201, 79, 80, 310, 81, 82, 406, 83, 84, 491, 85, 86, 87, 88, 202, 203

Offset: 1

N. J. A. Sloane, Nov 29 2024

That is, a(n) = m means A342042(n) = A377912(m).

The reason for looking at this is that A342042 is a permutation of A377912. But not all nonnegative integers appear in either sequence. The present sequence shows the numbers in A342042 in the order in which they appear in A377912. It is a permutation of the natural numbers. A377916 gives the inverse permutation.

			A342042(14) = 30 is the 28-th term of A377912, so a(14) = 38.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A342042, A377912, A377916.

PARI
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\\ See Links section.
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A377912 Numbers such that every even digit except the last is immediately followed by a strictly larger digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

Offset: 1

N. J. A. Sloane, Nov 27 2024, with thanks to Paolo Xausa for correcting the initial definition

In other words, numbers that do not contain a pair of successive digits i, j where i is even and j <= i.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025.

The complement of A347298. Cf. A342042, A377914, A377917, A377918.

A377912Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_?EvenQ, j_} /; j <= i];
Select[Range[0, 200], A377912Q] (* Paolo Xausa, Mar 17 2025 *)

def ok(n):
    s = str(n)
    return not any(s[i] in "2468" and s[i+1] <= s[i] for i in range(len(s)-1))
print([k for k in range(0, 118) if ok(k)]) # Michael S. Branicky, Nov 28 2024

Added 0 to match A342042, and replaced negative definition by a positive one. - N. J. A. Sloane, Nov 29 2024

A342043 When a digit d is even, the next digit is < d.

Original entry on oeis.org

1, 2, 11, 3, 4, 12, 13, 5, 6, 14, 15, 7, 8, 16, 17, 9, 18, 19, 21, 31, 32, 111, 33, 34, 35, 36, 37, 38, 39, 41, 42, 112, 113, 43, 51, 52, 114, 115, 53, 54, 116, 55, 56, 57, 58, 59, 61, 62, 117, 63, 64, 118, 65, 71, 72, 119, 73, 74, 121, 75, 76, 131, 77, 78, 79, 81, 82, 132, 133, 83, 84, 134, 135, 85

Offset: 1

Eric Angelini, Feb 26 2021

The sequence is always extended with the smallest positive integer not yet present that doesn't lead to a contradiction.

No digit 0 is present in the sequence (as 0, being even, would block it).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342043

Cf. A342042, A342044, A342045, A342046 and A342047 (variations on the same idea).

PARI
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See Links section.
```

cp's OEIS Frontend

A379901 Local maxima in A342042, in order of appearance: numbers m such that A342042(k-1) < A342042(k) = m > A342042(k+1), for k >= 2.

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A379903 Local minima in A342042, in order of appearance: numbers m such that A342042(k-1) > A342042(k) = m < A342042(k+1), for k >= 2.

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A377914 Indices of records in A342042.

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A377918 a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

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A379902 Indices of local maxima in A342042: numbers k such that A342042(k-1) < A342042(k) > A342042(k+1), for k >= 2.

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A379904 Indices of local minima in A342042: numbers k such that A342042(k-1) > A342042(k) < A342042(k+1), for k >= 2.

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A377913 Records in A342042.

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A377915 a(n) = m means A342042(n) is the m-th term of A377912.

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PARI

A377912 Numbers such that every even digit except the last is immediately followed by a strictly larger digit.

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Mathematica

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