Logan J. Kleinwaks - cp's OEIS frontend

A384729 A B_2-sequence with reciprocal sum > 2.1615.

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, 352, 415, 486, 540, 651, 706, 792, 838, 928, 1046, 1134, 1228, 1358, 1407, 1512, 1624, 1869, 1938, 2087, 2170, 2367, 2480, 2608, 2765, 3033, 3080, 3232, 3567, 3605, 3797, 3950, 4267, 4505, 4677, 5064, 5290, 5480, 5655, 6059, 6507, 6892, 6967

Offset: 1

Logan J. Kleinwaks, Jun 08 2025

nonn

This is the B_2 sequence with largest reciprocal sum that is known to the author, as of the date of submission. The reciprocal sum of the first 1010 terms, which are given in the attached b-file, is 2.16150003... This already provably exceeds the reciprocal sum of the infinite sequence by R. Lewis (A046185), defined by greedy extension of a sequence of 68 terms (e.g., compute R. Lewis' sequence to 1600 terms, then bound the remainder using the result of B. Lindström on the maximum cardinality of B2-sequences with elements in [1, N]). By extending this sequence greedily after the first 1010 terms, a larger reciprocal sum can obviously be achieved.
This sequence first differs from A046185 at the 25th term.
This sequence was found by the author by using a combination of beam search and batch greedy algorithm, as part of an experiment to evaluate LLM code generation and mathematical reasoning (all code was written by the LLM, though with significant prompting).

Logan J. Kleinwaks, Table of n, a(n) for n = 1..1010
Rachel Lewis, Mian-Chowla and B2 sequences, 1999.
Bernt Lindström, An inequality for B_2-sequences, Journal of Combinatorial Theory, Volume 6, Issue 2 (1969), 211-212.
Eric W. Weisstein, B_2-Sequence.. From MathWorld--A Wolfram Web Resource.

Cf. A046185 (R. Lewis), A005282 (Mian-Chowla).

A293542 a(1)=1; for n>1, a(n) = least integer greater than a(n-1) such that the sums of the divisors of the pairwise sums of a(1),...,a(n) are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 19, 28, 56, 72, 101, 144, 202, 240, 261, 448, 511, 602, 772, 806, 1152, 1296, 1541, 1602, 2016, 2256, 2900, 3322, 3362, 3978, 4376, 5887, 6416, 7702, 8228, 8578, 11341, 11382, 13376, 13692, 16083, 16380, 16544, 17382, 22726, 24944, 26302, 27508, 30580, 33184, 34020, 37474

Offset: 1

Logan J. Kleinwaks, Oct 11 2017

nonn

			Let s(n) be the sum of the divisors of n. a(3)!=3 because s(1+3)=s(2+2)=7. a(3)=4 because s(1+1)=3, s(1+2)=4, s(1+4)=6, s(2+2)=7, s(2+4)=12, and s(4,4)=15 are all distinct.

Logan J. Kleinwaks, Table of n, a(n) for n = 1..250

Cf. A000203, A293541, A005282.

A293541 a(1)=1; for n>1, a(n) = least integer greater than a(n-1) such that the numbers of divisors of the pairwise sums of a(1),...,a(n) are all distinct.

Original entry on oeis.org

1, 3, 15, 909, 306915, 36181647, 25342642989

Offset: 1

Logan J. Kleinwaks, Oct 11 2017

			Let d(n) be the number of divisors of n.  Then a(2)!=2 because d(1+1)=d(1+2)=2. a(2)=3 because d(1+1)=2, d(1+3)=3, and d(3+3)=4 are all distinct.

Cf. A000005, A293542, A005282.

A293457 Primes that divide the numerator of the sum of the reciprocals of all smaller primes.

Original entry on oeis.org

2, 5, 19, 47, 79, 109, 3667387

Offset: 1

Logan J. Kleinwaks, Oct 09 2017

Exhaustive search finds no more terms among the first 10^7 primes.

Primes p that divide A024451(A000720(p)-1). - Antti Karttunen, Feb 08 2024

			Since 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 = 716167/510510 and 19 divides 716167, 19 is in the sequence.
Since there are no primes less than 2, the sum of their reciprocals is 0/1, and as 2 divides 0, it is therefore included as the first term of this sequence. - _Antti Karttunen_, Feb 08 2024

Cf. A000040, A000720, A024451, A120289, A369972, A369973.

lista(nn) = my(s = 0); forprime(p=2, nn, if (!(numerator(s) % p), print1(p, ", ")); s += 1/p; ); \\ Michel Marcus, Oct 09 2017, edited for the new, more inclusive definition by Antti Karttunen, Feb 08 2024

a(n) = A000040(1+A369972(n)). - Antti Karttunen, Feb 08 2024

Relaxed the definition to include 2 as the first term - Antti Karttunen, Feb 08 2024

A271058 Least m such that n equals a sum of binomial coefficients C(m,k1)+C(m,k2)+C(m,k3)+... with 0<=k1

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6, 4, 4, 4, 5, 8, 9, 5, 5, 5, 6, 11, 5, 5, 5, 6, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 19, 19, 6, 6, 6, 6, 8, 8, 6, 6, 6, 6, 6, 6, 26, 9, 9, 6, 6, 6, 29, 29, 30, 6, 6, 6, 7, 8, 10, 11, 34, 7, 7, 7, 8, 8, 37, 37, 7, 7, 7, 8, 9, 9, 9, 7, 7, 7, 8, 8

Offset: 1

Logan J. Kleinwaks, Mar 29 2016

nonn

			a(3) = 2 since 3 = C(2,0) + C(2,1) but 3 != C(1,0) or C(1,1) or C(1,0) + C(1,1).

Giovanni Resta, Table of n, a(n) for n = 1..2000

nn = 22; s = Table[Union@ Map[Total, Binomial[m, #]] &@ Rest@ Subsets@ Range[0, m], {m, nn}]; Table[SelectFirst[Range[If[n == 1, 1, Ceiling@ Log2@ n], nn], MemberQ[s[[#]], n] &], {n, 52}] /. m_ /; MissingQ@ m -> 0 (* Michael De Vlieger, Mar 30 2016, Version 10.2, values of m greater than nn show as "0". *)

A270875 Number of subsets of {1,...,n} with sum of elements equal to least common multiple of elements and at least two elements.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 5, 6, 6, 9, 9, 10, 13, 15, 15, 19, 19, 33, 35, 35, 35, 47, 47, 47, 50, 57, 57, 101, 101, 106, 108, 108, 108, 127, 127, 127, 128, 249, 249, 268, 268, 272, 358, 358, 358, 406, 406, 408, 409, 411, 411, 424, 424, 501, 502, 502, 502, 1190, 1190

Offset: 1

Logan J. Kleinwaks, Mar 24 2016

nonn

It appears that the sequence of n for which a(n)>a(n-1) has a large overlap with A175904.

			For n=3, the subsets of {1,2,3} with at least two elements have (sum,LCM) as follows: {1,2}->(3,2), {1,3}->(4,3), {2,3}->(5,6), {1,2,3}->(6,6). Only the last satisfies sum=LCM, so a(3)=1.

Cf. A270970 (similar sequence counting trivial solutions).

Table[Length[Transpose@ {Total /@ #, LCM @@@ #} /. {a_, b_} /; a != b -> Nothing &@ Rest[Subsets[Range@ n] /. {} -> Nothing]], {n, 2, 22}] (* _Michael De Vlieger, Mar 24 2016 *)

a(n) = {nb = 0; S = vector(n, k, k); for (i = 0, 2^n - 1, ss = vecextract(S, i); if (vecsum(ss) == lcm(ss), nb++);); nb - n;} \\ Michel Marcus, Mar 26 2016

a(n) = A270970(n) - n. - Michel Marcus, Mar 27 2016

More terms added using A270970 by Jinyuan Wang, May 02 2020

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User: Logan J. Kleinwaks

A384729 A B_2-sequence with reciprocal sum > 2.1615.

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A293542 a(1)=1; for n>1, a(n) = least integer greater than a(n-1) such that the sums of the divisors of the pairwise sums of a(1),...,a(n) are all distinct.

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A293541 a(1)=1; for n>1, a(n) = least integer greater than a(n-1) such that the numbers of divisors of the pairwise sums of a(1),...,a(n) are all distinct.

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A293457 Primes that divide the numerator of the sum of the reciprocals of all smaller primes.

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A271058 Least m such that n equals a sum of binomial coefficients C(m,k1)+C(m,k2)+C(m,k3)+... with 0<=k1

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A270875 Number of subsets of {1,...,n} with sum of elements equal to least common multiple of elements and at least two elements.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions