A045955 -id:A045955 - cp's OEIS frontend

A057748 Number of decompositions of 2n-1 into sum of a twin lucky number and a twin even-lucky-number (from A045955, A045956).

0, 1, 2, 2, 2, 3, 4, 3, 3, 5, 4, 3, 4, 3, 2, 2, 3, 5, 5, 2, 2, 5, 4, 2, 4, 7, 7, 6, 5, 4, 4, 3, 3, 5, 6, 6, 5, 5, 6, 5, 4, 6, 8, 5, 2, 3, 4, 3, 1, 3, 7, 7, 4, 3, 5, 5, 3, 4, 7, 7, 5, 4, 4, 4, 5, 7, 7, 5, 6, 8, 6, 5, 8, 9, 8, 8, 6, 4, 3, 1, 3, 7, 6, 5, 7, 9, 8, 6, 6, 8, 10, 8, 6, 7, 7, 5, 2, 2, 7, 9, 6, 8, 11, 6

Offset: 1

Naohiro Nomoto, Oct 30 2000

Conjecture: this sequence is always positive (with n>1).

			1 is not the sum of Twin lucky number and Twin Even-Lucky-Number, so a(1)=0; 3=1+2 (one way, so a(2)=1); 5=3+2=1+4 (so a(3)=2); etc.

Jinyuan Wang, Table of n, a(n) for n = 1..5000

Cf. A000959, A057701, A045954, A045955, A045956.

Offset changed to 1 by Jinyuan Wang, Apr 07 2020

A057746 Even numbers not the sum of a pair of twin even-lucky-numbers (from A045955, A045956).

Original entry on oeis.org

2, 292, 324, 356, 372, 612, 1074, 1202, 1842

Offset: 1

Naohiro Nomoto, Oct 30 2000

Conjectured to be complete.

Cf. A045954, A045955, A045956.

seq[max_] := Module[{lst = Range[2, max, 2], i = 2, len, t}, While[i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++]; t = lst[[Position[Differences[lst], 2] // Flatten]]; Complement[Range[2, max, 2], Total /@ Tuples[Join[t, t + 2], 2]]]; seq[2000] (* Amiram Eldar, Mar 20 2024, after Robert G. Wilson v at A045954 *)

A057768 From Goldbach problem: number of decompositions of 2n-1 into sum of a prime lucky number(from A031157) and a twin even-lucky-number(from A045955, A045956).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 4, 5, 2, 3, 4, 4, 2, 2, 2, 3, 2, 1, 3, 4, 3, 4, 4, 4, 4, 5, 3, 3, 3, 2, 2, 2, 2, 4, 2, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 2, 2, 4, 4, 4, 3, 4, 3, 3, 4, 5, 4, 3, 3, 4, 3, 3, 1, 5, 4, 2, 3, 6, 4, 5, 5, 4, 4, 6, 4, 5, 3, 1, 2, 3, 4, 5, 4, 3, 5, 6, 3, 3

Offset: 0

Naohiro Nomoto, Nov 01 2000

Conjecture: this sequence is always positive (with n>2).

			1 and 3 are not the sum of a prime lucky number and a twin even-lucky-number, so a(1) = a(2) = 0; 5=3+2 (one way, so a(3)=1); 7=3+4 (so a(4)=1); 9=3+6=7+2 (so a(5)=2); etc.

Cf. A045954, A045955, A045956, A031157, A000959, A002375, A045917.

A045956 Twin even-lucky-numbers: upper terms.

Original entry on oeis.org

4, 6, 12, 20, 22, 36, 44, 52, 54, 70, 100, 102, 116, 118, 132, 140, 150, 172, 198, 214, 230, 246, 262, 300, 310, 326, 374, 388, 390, 420, 436, 438, 518, 524, 534, 548, 588, 598, 630, 644, 646, 708, 710, 716, 740, 742, 780, 870, 884, 886, 916, 964, 966, 996, 998, 1092

Offset: 1

Felice Russo

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A045954, A045955, A045957.

evenLuckies[max_] := Module[{lst = Range[2, max, 2], i = 2, len}, While[i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++]; lst]; twinUp[max_] := Module[{s = evenLuckies[max]}, s[[Position[Differences[s], 2] // Flatten]] + 2]; twinUp[1100] (* Amiram Eldar, Mar 20 2024, after Robert G. Wilson v at A045954 *)

a(n) = A045957(n) + 1 = A045955(n) + 2. - Amiram Eldar, Mar 20 2024

A045957 Twin even-lucky-numbers: middle terms.

Original entry on oeis.org

3, 5, 11, 19, 21, 35, 43, 51, 53, 69, 99, 101, 115, 117, 131, 139, 149, 171, 197, 213, 229, 245, 261, 299, 309, 325, 373, 387, 389, 419, 435, 437, 517, 523, 533, 547, 587, 597, 629, 643, 645, 707, 709, 715, 739, 741, 779, 869, 883, 885, 915, 963, 965, 995, 997, 1091

Offset: 1

Felice Russo

nonn

Numbers k such that both k-1 and k+1 are even-lucky numbers (A045954). - Amiram Eldar, Mar 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A045954, A045955, A045956.

evenLuckies[max_] := Module[{lst = Range[2, max, 2], i = 2, len}, While[i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++]; lst]; twinMid[max_] := Module[{s = evenLuckies[max]}, s[[Position[ Differences[s], 2] // Flatten]] + 1]; twinMid[1100] (* Amiram Eldar, Mar 20 2024, after Robert G. Wilson v at A045954 *)

a(n) = A045955(n) + 1 = A045956(n) - 1. - Amiram Eldar, Mar 20 2024

cp's OEIS Frontend

A057748 Number of decompositions of 2n-1 into sum of a twin lucky number and a twin even-lucky-number (from A045955, A045956).

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A057746 Even numbers not the sum of a pair of twin even-lucky-numbers (from A045955, A045956).

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A057768 From Goldbach problem: number of decompositions of 2n-1 into sum of a prime lucky number(from A031157) and a twin even-lucky-number(from A045955, A045956).

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A045956 Twin even-lucky-numbers: upper terms.

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A045957 Twin even-lucky-numbers: middle terms.

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