A121805 -id:A121805 - cp's OEIS frontend

A367575 Lexicographically earliest infinite sequence of distinct positive numbers such that, when all terms are written as a product of their prime factors with specific primes as the first and last factor, the sum of the two primes adjacent to the commas between the terms equals the magnitude of the difference between the terms.

2, 6, 12, 5, 15, 9, 14, 24, 11, 33, 25, 18, 28, 13, 39, 34, 54, 48, 23, 69, 55, 45, 35, 27, 21, 7, 16, 20, 30, 26, 42, 38, 60, 29, 87, 77, 63, 49, 40, 19, 57, 51, 17, 36, 52, 56, 66, 50, 46, 72, 65, 80, 70, 74, 114, 76, 37, 111, 105, 91, 75, 68, 64, 31, 93, 85, 104, 78, 82, 126, 102, 88

Offset: 1

Scott R. Shannon, Nov 23 2023

nonn

The sequence is a prime factorization version of the 'Commas sequence', A121805. Although for many terms a following number can be chosen that is smaller than the term given in the sequence and meets the term difference requirements, all such choices ultimately lead to the sequence halting as a number is eventually reached for which no unused next number exists. See the examples for the specific factorization order for the terms.
Giving a term that can be written as p*k, where p is prime and k is either prime or composite, then p*k + 2*Gpf(k) is always a next possible term, where Gpf(k) is the greatest prime dividing k. Alternatively if a term p is prime then 3*p is a next possible term. This implies the sequence is infinite as these rules could be used to find all subsequent terms once a term larger than any previous term appears in the sequence.
Numerous values can never appear in the sequence as their only possible predecessor or successor have already appeared and did not produce the term in question. For example 3,4 and 8 can never appear as their only possible preceding terms are 3,4,8,9,12 or 15, and as these later three terms appear early in the sequence and do not produce 3,4 or 8, then these can never appear. However, unlike A367465, primes and prime powers can appear as terms, the first being 5, 9, 11, 25, 13, 23,... .

			The prime factorization of the terms, with the required prime factors in the first and last position, begins: 2, 2*3, 3*2*2, 5, 5*3, 3*3, 2*7, 3*2*2*2, 11, 11*3, 5*5, 3*2*3, 7*2*2, 13, 13*3, 2*17, 3*3*2*3, 3*2*2*2*2, 23, 23*3, 11*5, 5*3*3, 7*5, 3*3*3, 3*7, 7, 2*2*2*2, 2*2*5, 5*3*2, 2*13, 3*7*2, 2*19, 3*5*2*2, 29, 29*3, 7*11, 3*3*7, 7*7, 2*5*2*2, 19, 19*3, 3*17,...
a(7) = 14 as a(6) = 9 which is written as 3*3, and 14 = 2*7, so the two primes adjacent to the term separating comma are 3 and 2, and 3+2 = 5, which equals |14 - 9|. Note that after a(6) = 9 there are three possible numbers that would meet the difference requirement for a(7) : 3, 4, 14. Choosing 3 forces the following term to be 8, which forces the following term to be 4, but 4's only successors are 8 and 9, both of which have already been used. Likewise choosing 4 leads to a similar dead-end. This leaves 14 as the smallest choice.

Scott R. Shannon, Table of n, a(n) for n = 1..2000

Cf. A367465 (multiplication), A367504, A121805, A027746, A006530, A020639.

A367611 Numbers that are not the comma-child of any positive number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 49, 50, 51, 52, 53, 54, 62, 63, 64, 65, 74, 75, 76, 86, 87, 98

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 15 2023

A subsequence of A367600.
This 50-term sequence was found by David W. Wilson in 2007. See the Eric Angelini link.
See A367338 for definition of comma-child.

Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
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, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A367600.

A367612 gives the complement.

def ok(n): y = int(str(n)[0]); x = (n-y)%10; return n - y - 10*x < 1
print([k for k in range(1, 99) if ok(k)]) # Michael S. Branicky, Dec 15 2023

A367615 Numbers that have a comma-successor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

N. J. A. Sloane, Dec 18 2023

This is the complement to A367341.
The actual successors are given in A367340.
See A367338 for definition.

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, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A367338, A367340, A367341

fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]!={};
Select[Range[75],fQ[#]&] (* Ivan N. Ianakiev, Dec 18 2023 *)

More terms from Michael S. Branicky, Dec 18 2023

A367623 Number of comma-children of n in base 3.

Original entry on oeis.org

2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 2023

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

Cf. A121805, A367355, A367356, A367606-A367609, A367622.

from sympy.ntheory.factor_ import digits
def a(k, base=3):
    m = k + base*(k%base)
    return len([m+y for y in range(1, base) if digits(m+y, base)[1] == y])
print([a(n) for n in range(1, 96)]) # Michael S. Branicky, Dec 23 2023

A367645 Lexicographically earliest sequence of distinct positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms equals the magnitude of the successive differences between the terms.

Original entry on oeis.org

1, 12, 35, 94, 135, 78, 159, 63, 30, 28, 109, 18

Offset: 1

Scott R. Shannon and Eric Angelini, Nov 25 2023

The sequence is finite; after twelve terms 18 is reached which has no following term - see A367341.

			a(6) = 78 as a(5) = 135, and |78 - 135| = 57 which can be formed from the last digit of 135 and the first digit of 78.
a(12) = 18 as a(11) = 109, and |18 - 109| = 91 which can be formed from the last digit of 109 and the first digit of 18. See A367341 for a proof that no following term can exist.

Cf. A121805, A367341, A367635.

A367646 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - GCT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b, while GCT(a,b) is the largest possible generalized Command transform (cf. A367635) where at least one digit of both a and b can be chosen.

Original entry on oeis.org

0, 1, 2, 14, 35, 78, 21, 103, 92, 53, 28, 60, 146, 132, 71, 44, 30, 73, 66, 102, 41, 17, 6, 82, 150, 129, 117, 26, 98, 29, 111, 20, 8, 16, 97, 166, 95, 164, 113, 72, 109, 88, 186, 105, 166, 115, 54, 109, 68, 164, 83, 131, 100, 89, 81, 179, 62, 158, 137, 56, 131, 70, 87, 79, 156, 65, 131, 80

Offset: 0

Scott R. Shannon, Nov 25 2023

This is a variation of A367578, where one can choose more than one digit from both a(n-2) and a(n-1) to create the largest possible step to a nonnegative number which has not previously appeared to form a(n). If all such numbers have already appeared the smallest possible forward step is chosen, which is just the standard Comma transform of a(n-2) and a(n-1).

It is conjectured that all nonnegative numbers appear in the sequence. After the first 10 million terms the only fixed points are 0, 1, 2, 29, 65, 84, 222, 377, 491, 499, and it is likely no more exist. The first number to appear twice is a(35) = a(44) = 166.

			a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
a(6) = 21 as GCT(a(4),a(5)) = GCT(35,78) = 57, so a(6) = a(5) - 57 = 21, as 21 is nonnegative and not already in the sequence.
a(13) = 132 as GCT(a(11),a(12)) = GCT(60,146) = 14, so a(13) = a(12) - 14 = 132, as 132 is nonnegative and not already in the sequence. This is the first term to differ from A367578.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 10 million terms.

Cf. A367360, A367635, A367578, A005132, A121805.

A368782 Comma transform of A366487.

Original entry on oeis.org

12, 35, 94, 15, 16, 28, 31, 34, 37, 41, 45, 55, 55, 55, 55, 61, 67, 74, 71, 89, 98, 97, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22

Offset: 1

Michael S. Branicky, Jan 05 2024

See A367360 for further information.
Let the comma sequence A121805 be known as S or C0.
A366487, the first differences of A121805, is the same as the comma transform of A121805; call it C1.
This sequence is C2 = C(C(S)), the comma transform C iterated twice.
C4 = C2, C5 = C2, ... once the first term (and the last term if the sequence is finite) are removed from the lower iterates of C.
Theorem: C^{i+2}(S) = C^i(S) for i>=2 in general and for i>=0 when all terms of S have two digits and no least significant digit is zero. See link for proof.
Remark. The lexicographically earliest sequence S with C(S) = S is A010850, all 11's.
The sequence contains 2137451 terms, with a(2137451) = 96. The next term does not exist.

Michael S. Branicky, Table of n, a(n) for n = 1..20000
Michael S. Branicky, Comma Comma Proof

Cf. A010850, A121805, A367360, A366487.

from itertools import islice, pairwise
def S(): # generator of comma sequence
    an = 1
    while True:
        yield an
        an += 10*(an%10)
        children = [an+y for y in range(1, 10) if str(an+y)[0] == str(y)]
        if not children: break
        an = children[0]
def C(g): # generator of comma transform of sequence passed as a generator
    yield from (10*(t%10) + int(str(u)[0]) for t, u in pairwise(g))
def agen(): return C(C(S()))
print(list(islice(agen(), 70))) # Michael S. Branicky, Jan 05 2024

A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

Original entry on oeis.org

1, 2, 12, 21, 22, 112, 121, 210, 120, 10, 11, 13, 110, 31, 3, 113, 131, 231, 122, 111, 211, 123, 114, 310, 43, 4, 34, 143, 41, 134, 115, 141, 51, 15, 116, 151, 61, 16, 117, 161, 71, 17, 118, 171, 81, 18, 119, 181, 91, 19, 311, 93, 190, 312, 23, 220, 32, 30, 223, 20, 132, 14, 212, 42, 24, 221

Offset: 1

Scott R. Shannon, Jan 19 2024

The sequence is conjectured to be a permutation of the positive integers. The fixed points begin 1, 2, 10, 11, 2863, 3164, 3545, 3947, 6835, 6947, 7052, ... although it is likely there are infinitely more.

			a(3) = 12 as the comma transform of 1 and 2 is 12.
a(6) = 112 as the comma transform of 21 and 22 is 12, but 12 has already appeared so the next lowest unused number to contain '12' is 112.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first one million terms. The green line is a(n) = n.

Cf. A367360, A121805, A367362.

A374725 The "multiplicative comma sequence": the lexicographically earliest sequence of positive integers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive ratios between the terms.

Original entry on oeis.org

1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 233846052, 5846151300

Offset: 1

Nicholas M. R. Frieler, Jul 17 2024

A more formal definition can be given as follows: a(1) = 1; for n > 1, let x be the least significant digit of a(n-1); then a(n) = a(n-1) * (10*x + y), with y being the most significant digit of a(n). Choose the smallest such y if such a y exists. If no such y exists, the sequence ends. We also restrict y to being a nonzero digit.
The sequence is given in its entirety as there is no possible next term after 5846151300.
Choosing other values for a(1) yields finite sequences up to a(1) = 10000 as long as a(1) is not of the form 1...0 otherwise the sequence is constant and infinite. For example, if a(1) = 120, then a(2) = 120 because 120 * 01 = 120.

			Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first ratios between the terms of the sequence:
Sequence: 1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 233846052, 5846151300
Ratios:    11, 11,  11,   11,    11,     11,      11,       12,        25
For example: a(9) = 233846052 = 12 * 19487171 = 12 * a(8)

Cf. A121805.

a[1] = 1; a[n_] := a[n] = For[x = Mod[a[n - 1], 10]; y = 1, y <= 9, y++, an = a[n - 1]*(10*x + y); If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 10]

A347353 Lexicographically earliest sequence of distinct positive terms such that the rightmost digit of a(n) concatenated with the leftmost digit of a(n+1) form an integer that is the sum of the digits of a(n) and a(n+1).

Original entry on oeis.org

1, 19, 1899999999, 29, 1799999999, 119, 1889999999, 128, 169999999, 1299, 178999999, 1389, 179899999, 1398, 15999999, 13999, 16899999, 14899, 16989999, 14989, 16998999, 14998, 1499999, 149999, 1589999, 158999, 1598999, 159899, 1599899, 159989, 1599989, 159998, 139999, 1599999, 148999

Offset: 1

Eric Angelini and Carole Dubois, Aug 28 2021

In other words, the two digits squeezing a comma form the sum of the digits of the two terms squeezing the comma. This sequence is finite and has 1309 terms, the last one being 191.

			a(1), a(2) = 1, 19 and 11 is 1 + (1+9);
a(2), a(3) = 19, 1899999999 and 91 is (1+9) + (1+8+9+9+9+9+9+9+9+9);
a(3), a(4) = 1899999999, 29 and 92 is (1+8+9+9+9+9+9+9+9+9) + (9+2);
a(4), a(5) = 29, 1799999999 and 91 is (2+9) + (1+7+9+9+9+9+9+9+9+9); etc.

Cf. A121805, A139284.

a[1]=1;a[n_]:=a[n]=(t=1;While[!IntegerQ[m=Min@Union@Flatten@Table[l=FromDigits[{Last[s=IntegerDigits@a[n-1]],k}]-Total@s;Complement[FromDigits/@Flatten[Permutations/@Select[Sort/@IntegerPartitions[l,t,Range@9],First@#==k&],1],Array[a,n-1]],{k,9}]],t++];m);Array[a,35] (* Giorgos Kalogeropoulos, May 05 2022 *)

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A367611 Numbers that are not the comma-child of any positive number.

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A367615 Numbers that have a comma-successor.

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A367623 Number of comma-children of n in base 3.

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A367645 Lexicographically earliest sequence of distinct positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms equals the magnitude of the successive differences between the terms.

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A368782 Comma transform of A366487.

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A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

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A374725 The "multiplicative comma sequence": the lexicographically earliest sequence of positive integers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive ratios between the terms.

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A347353 Lexicographically earliest sequence of distinct positive terms such that the rightmost digit of a(n) concatenated with the leftmost digit of a(n+1) form an integer that is the sum of the digits of a(n) and a(n+1).

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