A335600 -id:A335600 - cp's OEIS frontend

A335854 The digital-root sandwiches sequence (see Comments lines for definition).

11, 9, 99, 999, 9999, 9990, 1, 112, 12, 3, 125, 33, 4, 15, 8, 337, 44, 5, 154, 88, 2, 37, 24, 49, 55, 6, 14, 38, 81, 22, 53, 7, 92, 48, 495, 552, 66, 71, 47, 387, 813, 227, 531, 77, 79, 26, 483, 45, 152, 86, 64, 715, 471, 376, 83, 52, 73, 51, 87, 75, 792, 261, 43, 74, 56, 121, 863, 642, 759, 41, 436, 58, 385, 29

Offset: 1

Carole Dubois and Eric Angelini, Jun 26 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. A digital-root sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the digital root of their sum. The pair [1951, 2020] would then produce the DR-sandwich 132. Please note that the pair [2020, 1951] would produce the genuine DR-sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 119, 999, 999, 999, 999, 011, 121, 231, ...
The first one (119) is visible between a(1) = 11 and a(2) = 9; we get the sandwich by inserting the digital root of the sum 1 + 9 = 10 (which is 1) between 1 and 9.
The second sandwich (999) is visible between a(2) = 9 and a(3) = 99; we get the sandwich by inserting the digital root of the sum 9 + 9 = 18 (which is 9) between 9 and 9.
The third sandwich (999) is visible between a(3) = 99 and a(3) = 999; we get the sandwich by inserting the digital root of the sum 9 + 9 = 18 (which is 9) between 9 and 9.
(...)
The sixth sandwich (011) is visible between a(6) = 9990 and a(7) = 1; we get the sandwich by inserting the digital root of the sum 0 + 1 = 1 (which is 1) between 0 and 1; etc.
The successive sandwiches rebuild, digit after digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..428

Cf. A335600 (first definition of such a sandwich) and A010888 (digital root of n).

A335886 The heavy sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

1, 2, 22, 4, 228, 44, 8, 28, 3, 24, 43, 288, 16, 282, 433, 6, 241, 64, 36, 2881, 61, 222, 84, 31, 86, 612, 21, 66, 41, 23, 6122, 166, 12, 221, 68, 412, 318, 863, 662, 42, 1666, 244, 122, 3186, 2216, 6124, 216, 683, 242, 63, 864, 83, 18, 62, 842, 2161, 224, 4126, 361, 226, 366, 48, 26, 3663, 622, 126, 32, 484

Offset: 1

Eric Angelini and Carole Dubois, Jun 28 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the product of those two digits. The pair [1951, 2020] would then produce the sandwich 122. Please note that the pair [2020, 1951] would produce the genuine sandwich 001 (we keep the leading zeros: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 122, 242, 284, 482, 8324, ...
The first one (122) is visible between a(1) = 1 and a(2) = 2; we get the sandwich by inserting the product 2 between 1 and 2.
The second sandwich (242) is visible between a(2) = 2 and a(3) = 22; we get this sandwich by inserting the product 4 between 2 and 2.
The third sandwich (284) is visible between a(3) = 22 and a(4) = 4; we get this sandwich by inserting the product 8 between 2 and 4.
The fourth sandwich (482) is visible between a(4) = 4 and a(5) = 228; we get this sandwich by inserting the product 8 between 4 and 2.
The fifth sandwich (8324) is visible between a(5) = 228 and a(6) = 44; we get this sandwich by inserting the product 32 between 8 and 4; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..611

Cf. A335600 (the "poor" sandwich sequence).

A336324 The power sandwiches sequence, version 1 (see Comments lines for definition).

Original entry on oeis.org

1, 2, 22, 4, 221, 6, 44, 16, 21, 66, 640, 9, 64, 41, 166, 42, 1666, 46, 65, 660, 19, 9100, 7, 76, 96, 642, 5, 641, 11, 6409, 6421, 1640, 964, 646, 656, 657, 77, 6601, 193, 8, 74, 20, 48, 990, 17, 78, 23, 54, 3, 765, 31, 441, 9646, 6566, 225, 55, 777, 661, 111, 669, 100, 776, 966, 1110, 194, 12, 9666

Offset: 1

Carole Dubois and Eric Angelini, Jul 17 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit R of a(n), the leftmost digit L of a(n+1) and, in between, L^R. The pair [1951, 2020] would then produce the power sandwich 122. Please note that the pair [2020, 1951] would produce the power and genuine sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 122, 242, 2164, 4162, 166, 640964, ...
The first one (122) is visible between a(1) = 1 and a(2) = 2; we get the sandwich by inserting 2^1 = 2.
The second sandwich (242) is visible between a(2) = 2 and a(3) = 22; we get this sandwich by inserting 2^2 = 4 between 2 and 2.
The third sandwich (2164) is visible between a(3) = 22 and a(4) = 4; we get this sandwich by inserting 4^2 = 16 between 2 and 4; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..541

Cf. A336325 (same idea, but between L and R we insert R^L instead of L^R), A335600 (poor sandwiches), A335854 (digital-root sandwiches), A335886 (heavy sandwiches).

A336325 The power sandwiches sequence, version 2 (see Comments lines for definition).

Original entry on oeis.org

1, 11, 111, 1111, 112, 6, 4, 66, 12, 9, 64, 440, 96, 666, 125, 129, 95, 3, 14, 41, 642, 5, 6400, 964, 665, 6666, 15, 51, 93, 8, 7, 420, 48, 99, 512, 53, 33, 142, 56, 411, 62, 32, 55, 156, 2, 5600, 94, 40, 966, 515, 625, 6661, 531, 25, 511, 936, 561, 88, 20, 97, 152, 77, 240, 1400, 481, 34, 21, 772, 89, 9590

Offset: 1

Carole Dubois and Eric Angelini, Jul 17 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit R of a(n), the leftmost digit L of a(n+1) and, in between, R^L. The pair [1951, 2020] would then produce the power sandwich 112. Please note that the pair [2020, 1951] would produce the power and genuine sandwich 001 (we keep the leading zeros: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 111, 111, 111, 111, 2646, 612964,...
The first one (111) is visible between a(1) = 1 and a(2) = 11; we get the sandwich by inserting 1^1 = 1 between 1 and 1.
The second sandwich (111) is visible between a(2) = 11 and a(3) = 111; we get this sandwich by inserting 1^1 = 1 again between 1 and 1.
(...)
The fifth sandwich (2646) is visible between a(5) = 112 and a(6) = 6; we get this sandwich by inserting 2^6 = 64 between 2 and 6; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..1192

Cf. A336324 (same idea, but between L and R we insert L^R instead of R^L), A335600 (poor sandwiches), A335854 (digital-root sandwiches), A335886 (heavy sandwiches).

A336894 The empty sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

1, 2, 22, 220, 3, 33, 330, 4, 44, 440, 5, 55, 550, 6, 66, 660, 7, 77, 770, 8, 88, 880, 9, 99, 990, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68

Offset: 1

Eric Angelini and Carole Dubois, Aug 07 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, some combination c of those two digits (see A335600 for instance). The pair [1951, 2020] would then produce the sandwich 1c2. Please note that the pair [2020, 1951] would produce the genuine sandwich 0c1 (we keep the leading zero: these are sandwiches after all, not integers).
In this sequence we don't insert anything between the two "slices of bread": there is no c, the sandwiches are empty.
Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 12, 22, 22, 03, 33, 33, 04,...
The 1st one (12) is visible between a(1) = 1 and a(2) = 2.
The 2nd one (22) is visible between a(2) = 2 and a(3) = 22.
The 3rd one (22) is visible between a(3) = 22 and a(4) = 220.
The 4th one (03) is visible between a(4) = 220 and a(5) = 3; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..113

Cf. A335600.

A336895 The prime sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

1, 22, 2, 3, 225, 33, 7, 25, 11, 331, 37, 71, 72, 5, 19, 112, 3312, 9, 373, 17, 13, 77, 24, 15, 54, 31, 94, 712, 53, 32, 59, 99, 6, 133, 67, 177, 113, 73, 777, 92, 4, 8, 315, 89, 549, 731, 10, 194, 103, 7210, 75, 310, 93, 21, 135, 91, 27, 991, 316, 61, 371, 313, 96, 714, 917, 151, 131, 57

Offset: 1

Carole Dubois and Eric Angelini, Aug 07 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the smallest prime p not yet inserted in a sandwich. The pair [1951, 2020] would then produce the sandwich 1p2. Please note that the pair [2020, 1951] would produce the genuine sandwich 0p1 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 122, 232, 253, 372, 5113, 3137,...
The 1st one (122) is visible between a(1) = 1 and a(2) = 22 (insert 2).
The 2nd one (232) is visible between a(2) = 22 and a(3) = 2 (insert 3).
The 3rd one (253) is visible between a(3) = 2 and a(4) = 3 (insert 5).
The 4th one (372) is visible between a(4) = 3 and a(5) = 225 (insert 7).
The 5th one (5113) is visible between a(5) = 225 and a(6) = 33 (insert 11); etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..1964

Cf. A336894 (empty sandwiches), A335600 (poor sandwiches).

A336904 The natural sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

11, 2, 22, 223, 23, 4, 235, 44, 6, 25, 7, 448, 66, 9, 2510, 77, 1, 14, 8, 12, 661, 3, 99, 1420, 15, 771, 61, 117, 141, 88, 81, 91, 220, 612, 13, 32, 29, 92, 310, 24, 152, 5, 71, 26, 6127, 17, 28, 112, 98, 830, 813, 19, 132, 20, 33, 62, 34, 133, 53, 236, 293, 79, 238, 30, 39, 2440, 124, 155, 42, 714

Offset: 1

Carole Dubois and Eric Angelini, Aug 07 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the smallest natural number N not yet inserted into a sandwich. The pair [1951, 2020] would then produce the natural sandwich 1N0. Please note that the pair [2020, 1951] would produce the genuine sandwich 0N1 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 112, 222, 232, 342, 354, 462,...
The 1st one (112) is visible between a(1) = 11 and a(2) = 2; we get the sandwich by inserting 1 between 1 and 2.
The 2nd sandwich (222) is visible between a(2) = 2 and a(3) = 22; we get this sandwich by inserting 2 between 2 and 2.
The 3rd sandwich (232) is visible between a(3) = 22 and a(4) = 223; we get this sandwich by inserting 3 between 2 and 2;
The 4th sandwich (342) is visible between a(4) = 223 and a(5) = 23; we get this sandwich by inserting 4 between 3 and 2; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A335600.

A336690 The rich sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

9, 10, 101, 1, 12, 11, 2, 123, 113, 22, 3, 13, 4, 135, 225, 33, 41, 37, 44, 5, 15, 7, 25, 8, 337, 414, 371, 14, 49, 55, 6, 151, 27, 79, 251, 38, 81, 137, 114, 47, 31, 21, 48, 491, 45, 51, 16, 67, 1132, 71, 479, 112, 143, 816, 812, 17, 814, 84, 710, 313, 215, 481, 24, 154, 510, 512, 161, 26, 78, 129, 715, 4910

Offset: 1

Eric Angelini and Carole Dubois, Jul 31 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the sum of those two digits. The pair [1951, 2020] would then produce the (rich) sandwich 132. (Why rich? Because a poor sandwich would insert the absolute difference of the digits instead of their sum - that is 112 in this example). Please note that the pair [2020, 1951] would produce the rich and genuine sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).

Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.

			The first successive sandwiches are: 9101, 011, 121, 121, 231, 132,...
The first one (9101) is visible between a(1) = 9 and a(2) = 10; we get the sandwich by inserting the sum 10 between 9 and 1.
The second sandwich (011) is visible between a(2) = 10 and a(3) = 101; we get this sandwich by inserting the sum 1 between 0 and 1.
The third sandwich (121) is visible between a(3) = 101 and a(4) = 1; we get this sandwich by inserting the sum 2 between 1 and 1; etc.
The successive sandwiches rebuild, digit by digit, the starting sequence.

Cf. A335600 (the poor sandwiches sequence).

A336874 The self-sandwiches sequence (see Comments lines for definition).

Original entry on oeis.org

11, 2, 21, 212, 22, 222, 1, 2221, 112, 12, 122, 1221, 221, 12212, 121, 1121, 1210, 220, 110, 111, 113, 223, 114, 224, 115, 225, 226, 116, 227, 228, 117, 229, 2211, 3, 31, 312, 32, 23, 2123, 2231, 2122, 2120, 118, 119, 22111, 1111

Offset: 1

Eric Angelini and Carole Dubois, Aug 06 2020

Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the single digit d of the sequence itself not been yet duplicated inside a sandwich. The pair [1951, 2020] would then produce the sandwich 1d2. Please note that the pair [2020, 1951] would produce the genuine sandwich 0d1 (we keep the leading zero: these are sandwiches after all, not integers).
Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.
The authors are unable to compute more terms than the ones proposed here and ask the readers' indulgence.

			The first successive sandwiches are: 112, 212, 122, 222, 212,...
The 1st one (112) is visible between a(1) = 11 and a(2) = 2; we get the sandwich by inserting the 1st digit of the sequence itself, 1.
The 2nd sandwich (212) is visible between a(2) = 2 and a(3) = 21; we get this sandwich by inserting inserting the 2nd digit of the sequence itself, 1.
The 3rd sandwich (122) is visible between a(3) = 21 and a(4) = 212; we get this sandwich by inserting the 3rd digit of the sequence itself, 2.
The 4th sandwich (222) is visible between a(4) = 212 and a(5) = 22; we get this sandwich by inserting the 4th digit of the sequence itself, 2. Etc..
The successive sandwiches rebuild, digit by digit, the starting sequence.

Cf. A335600 (first sequence of this kind, linked to many others).

cp's OEIS Frontend

A335854 The digital-root sandwiches sequence (see Comments lines for definition).

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A335886 The heavy sandwiches sequence (see Comments lines for definition).

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A336324 The power sandwiches sequence, version 1 (see Comments lines for definition).

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A336325 The power sandwiches sequence, version 2 (see Comments lines for definition).

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A336894 The empty sandwiches sequence (see Comments lines for definition).

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A336895 The prime sandwiches sequence (see Comments lines for definition).

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A336904 The natural sandwiches sequence (see Comments lines for definition).

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A336690 The rich sandwiches sequence (see Comments lines for definition).

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A336874 The self-sandwiches sequence (see Comments lines for definition).

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