Bernardo Boncompagni - cp's OEIS frontend

User: Bernardo Boncompagni

Bernardo Boncompagni has authored 8 sequences.

A258107 Smallest number > 1 whose representation in all bases up to n consists only of 0's and 1's.

2, 3, 4, 82000

Offset: 2

Bernardo Boncompagni, May 20 2015

As with A146025, it is a plausible conjecture that there are no more terms, but this has not been proved. - Daniel Mondot, Dec 16 2016
From Devansh Singh, Mar 14 2021: (Start)
If a(n) exists then b-1|(N-Sum_{i>=0} A(i)), b-2|(N-Sum_{i>=0} A(i)*2^i), b-3|(N-Sum_{i>=0} A(i)*3^i), ... where b <= n.
If a(n) exists for n > 5 then let it be N. N = Sum_{i>=0} A(i)*b^i where A(i) is the i-th digit (0 or 1 only) of N starting from the right in base b <= n.
N = Sum_{i>=0} A(i)*b'^i + Sum_{i>=1} A(i)*(b^i - b'^i), where b' < b. If b=6 then we can see that 5|(N-Sum_{i>=0} A(i)), 4|(N-Sum_{i>=0} A(i)*2^i), 3|(N-Sum_{i>=0} A(i)*3^i). (End)

			a(4) = 4 because it is 100 in base 2, 11 in base 3 and 10 in base 4. No smaller number, except 1, can be expressed in such bases with only 0's and 1's.
a(5) = 82000: 82000 in bases 2 through 5 is 10100000001010000, 11011111001, 110001100, 10111000, containing only 0's and 1's, while all smaller numbers have a larger digit in one of those bases. For example, 12345 is 11000000111001, 121221020, 3000321, 343340. - _N. J. A. Sloane_, Feb 01 2016

Thomas Oléron Evans, Solution: Covering all the bases
Richard Green, A Curious Property of 82000
James Grime and Brady Haran, Why 82,000 is an extraordinary number, Numberphile video, 2015.

Cf. A146025.

Table[k = 2; While[Total[Total@ Drop[RotateRight[DigitCount[k, #]], 2] & /@ Range[3, n]] > 0, k++]; k, {n, 2, 5}] (* Michael De Vlieger, Aug 29 2015 *)

A126717 Least odd k such that k*2^n-1 is prime.

Original entry on oeis.org

3, 3, 1, 1, 3, 1, 3, 1, 5, 7, 5, 3, 5, 1, 5, 9, 17, 1, 3, 1, 17, 7, 33, 13, 39, 57, 11, 21, 27, 7, 213, 1, 5, 31, 3, 25, 17, 21, 3, 25, 107, 15, 33, 3, 35, 7, 23, 31, 5, 19, 11, 21, 65, 147, 5, 3, 33, 51, 77, 45, 17, 1, 53, 9, 3, 67, 63, 43, 63, 51, 27, 73, 5, 15, 21, 25, 3, 55, 47, 69

Offset: 0

Bernardo Boncompagni, Feb 13 2007

nonn

If a(n)=1 then n is a Mersenne exponent (A000043). - Pierre CAMI, Apr 22 2013
From Pierre CAMI, Apr 03 2017: (Start)
Empirically, as N increases, (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} n) tends to log(2); this is consistent with the prime number theorem as the probability that x*2^n - 1 is prime is ~ 1/(n*log(2)) if n is large enough.
For n=1 to 10000, a(n)/n < 7.5.
a(n)*2^n - 1 and a(n)*2^n + 1 are twin primes for n = 1, 2, 6, 18, 22, 63, 211, 282, 546, 726, 1032, 1156, 1321, 1553, 2821, 4901, 6634, 8335, 8529; corresponding values of a(n) are 3, 1, 3, 3, 33, 9, 9, 165, 297, 213, 177, 1035, 1065, 291, 6075, 2403, 2565, 4737, 3975, 459. (End)

			a(10)=5 because 5*2^10-1 is prime but 1*2^10-1 and 3*2^10-1 are not.

Pierre CAMI, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
Ray Ballinger, Proth Search Page
Poo-Sung Park, Multiplicative functions with f(p + q - n_0) = f(p) + f(q) - f(n_0), arXiv:2002.09908 [math.NT], 2020.

Cf. A035050, A057778, A085427, A284631.

f[n_] := Block[{k = 1}, While[ !PrimeQ[k*2^n - 1], k += 2]; k]; Table[f@n, {n, 0, 80}] (* Robert G. Wilson v, Feb 20 2007 *)

a(n) = {my(k=1); while(!isprime(k*2^n - 1), k+=2); k}; \\ Indranil Ghosh, Apr 03 2017

from sympy import isprime
def a(n):
    k=1
    while True:
        if isprime(k*2**n - 1): return k
        k+=2
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 03 2017

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Conjecture: a(n) = O(n log n). - Thomas Ordowski, Oct 15 2014

More terms from Robert G. Wilson v, Feb 20 2007

A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

Original entry on oeis.org

3, 5, 17, 41, 71, 311, 347, 659, 2381, 5879, 13397, 18539, 24419, 62297, 187907, 687521, 688451, 850349, 2868959, 4869911, 9923987, 14656517, 17382479, 30752231, 32822369, 96894041, 136283429, 234966929, 248641037, 255949949

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

			The smallest twin prime pair is 3, 5, then 5, 7 so a(1) = 3; the following pair is 11, 13 so a(2) = 5 because 11 - 5 = 6 > 5 - 3 = 2; the following pair is 17, 19: since 17 - 11 = 6 = 11 - 5 nothing happens; the following pair is 29, 31 so a(3)= 17 because 29 - 17 = 12 > 11 - 5 = 6.

Martin Raab, Table of n, a(n) for n = 1..82 (first 75 terms from Max Alekseyev)
Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes

Record gaps are given in A113274. Cf. A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 2}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k++ ]; k]; p = 3; r = 0; t = {3}; Do[q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, p]; r = q - p]; p = q, {n, 10^9}] (* Robert G. Wilson v, Oct 22 2005 *)

a(n) = A036061(n) - 2.

a(n) = A036062(n) - A113274(n).

a(22)-a(30) from Robert G. Wilson v, Oct 22 2005

Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) in Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015

A113274 Record gaps between twin primes.

Original entry on oeis.org

2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

a(n) mod 6 = 0 for each n>1.

			The first twin primes are 3,5 and 5,7 so a(0)=5-3=2. The following pair is 11,13 so a(1)=11-5=6. The following pair is 17,19 so 6 remains the record and no terms are added.

Martin Raab, Table of n, a(n) for n = 1..82 (terms up to a(72) from Alexei Kourbatov, a(73)-a(75) from Tomás Oliveira e Silva)
Richard Fischer, Maximale Lücken (Intervallen) von Primzahlenzwillingen
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013; and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053[math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes
Luis Rodriguez and Carlos Rivera, Conjecture 66. Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
Eric Weisstein's World of Mathematics, Twin Prime Constant

The smallest primes originating the sequence are given in A113275. Cf. A008407, A005250, A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k+=6]; k]; p = 5; r = 2; t = {2}; Do[ q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, q - p]; Print[{p, q - p}]; r = q - p]; p = q, {n, 10^9}]; t (* Robert G. Wilson v, Oct 22 2005 *)
DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],2,1],#[[2]]-#[[1]] == 2&][[All,2]]],GreaterEqual] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Dec 31 2022 *)

a(n) = A036063(n) + 2.
a(n) = A036062(n) - A113275(n).
From Alexei Kourbatov, Dec 29 2011: (Start)
(1) Upper bound: gaps between twin primes are smaller than 0.76*(log p)^3, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)-1.2), where a = 0.76*(log p)^2 is the average gap between twin primes near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
(End)

More terms from Robert G. Wilson v, Oct 22 2005
Corrected terms based on A036063, cross-checked with independent computations by Carlos Rivera and Richard Fischer (linked).
Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.

A113403 Primes p in prime quadruplets (p,p+2,p+6,p+8) at the end of maximal gaps in A113404.

Original entry on oeis.org

11, 101, 821, 1481, 3251, 5651, 9431, 31721, 43781, 97841, 135461, 187631, 326141, 768191, 1440581, 1508621, 3047411, 3798071, 5146481, 5610461, 9020981, 17301041, 22030271, 47774891, 66885851, 76562021, 87797861, 122231111, 132842111, 204651611, 628641701, 1749878981

Offset: 1

Bernardo Boncompagni, Oct 28 2005

nonn

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of four primes. Record (maximal) gaps between prime quadruplets are listed in A113404; see further comments there.

			Record gaps between prime quadruplets are as follows (arXiv:1309.4053, Table 4):
Initial primes: Max gap
.....5......11........6
....11.....101.......90
...191.....821......630
...821....1481......660
..2081....3251.....1170
..3461....5651.....2190
..5651....9431.....3780
...
The left column is A229907. The middle column is A113403 (this sequence); the right column is A113404.

Jud McCranie and Alexei Kourbatov, Table of n, a(n) for n = 1..71
Tony Forbes and Norman Luhn, Prime k-tuplets.
Alexei Kourbatov, Maximal gaps between prime k-tuples.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Record gaps are given in A113404. Cf. A007530, A002386.

A113404 Record gaps between prime quadruplets.

Original entry on oeis.org

6, 90, 630, 660, 1170, 2190, 3780, 6420, 8940, 9030, 13260, 16470, 24150, 28800, 29610, 39990, 56580, 56910, 71610, 83460, 94530, 114450, 157830, 159060, 171180, 177360, 190500, 197910, 268050, 315840, 395520, 435240, 440910, 513570, 536010, 539310, 557340, 635130

Offset: 1

Bernardo Boncompagni, Oct 28 2005

nonn

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=4 for quadruplets. If a gap is larger than all preceding gaps, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps. This sequence suggests that maximal gaps between prime quadruplets are O(log^5(p)). - Alexei Kourbatov, Jan 04 2012

			The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=11-5=6. The next quadruplet is (101,103,107,109), so a(2)=101-11=90. The following quadruplet is 191,193,197,199 so 90 remains the record and no terms are added.

Alexei Kourbatov, Table of n, a(n) for n = 1..71
T. Forbes, Norman Luhn Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Alexei Kourbatov, Maximal gaps between prime k-tuples (graphs, more terms)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013. - From _N. J. A. Sloane_, Feb 09 2013
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
Eric Weisstein's World of Mathematics, Hardy-Littlewood Constants.
Eric Weisstein's World of Mathematics, Prime Constellation.

A229907 lists initial primes in quadruplets preceding the maximal gaps. A113403 lists the corresponding primes at the end of the maximal gaps. Cf. A008407, A007530.

DeleteDuplicates[Differences[#[[4]]&/@Select[Partition[Prime[Range[10^7]],4,1],Differences[#] == {2,4,2}&]],GreaterEqual] (* The program generates the first 29 terms of the sequence. *) (* Harvey P. Dale, Aug 04 2024 *)

From Alexei Kourbatov, Jan 04 2012: (Start)
(1) Upper bound: gaps between prime quadruplets (p, p+2, p+6, p+8) are smaller than 0.241*(log p)^5, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a*(log(p/a)-0.55), where a = 0.241*(log p)^4 is the average gap between quadruplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.241 is reciprocal to the Hardy-Littlewood 4-tuple constant 4.15118... (End)

Terms 159060 to 635130 added by Alexei Kourbatov, Jan 04 2012

A111166 Let p < q be consecutive primes; p is in the sequence if p/(q-p) is a record.

Original entry on oeis.org

2, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607

Offset: 1

Bernardo Boncompagni, Oct 21 2005

Conjecture: Except for first term, the sequence coincides with A001359. This is true for all primes < 2*10^7.

Conjecture: Except for first term, the sequence coincides with A001359. This is true for all primes < 7*10^16. Let n >= 2 be an integer, N +- 1 and M +- 1 two consecutive twin pairs where M>n*N. Finding a counterexample is the same as finding two consecutive primes P1 and P2 with n*N

The smallest prime(n) such that prime(n+1)/prime(n) is decreasing. [Thomas Ordowski, May 13 2012]

This sequence corresponds with A001359 for all terms less than 10^100. - Charles R Greathouse IV, May 14 2012

			a(0)=2 and the record is 2/(3-2)=2; a(1)<>3 because 3/(5-3)=1.5; a(1)=5 because 5/(7-5)=2.5

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

Cf. A001359.

rmax = 0; p = 2; seq = {}; Do[q = NextPrime[p]; r = p/(q-p); If[r > rmax, rmax = r; AppendTo[seq, p]]; p = q, {100}]; seq (* Amiram Eldar, Dec 24 2019 *)
DeleteDuplicates[{#[[1]],#[[2]],#[[1]]/(#[[2]]-#[[1]])}&/@Partition[Prime[Range[300]],2,1],GreaterEqual[#1[[3]],#2[[3]]]&][[;;,1]] (* Harvey P. Dale, Jun 16 2025 *)

A096356 Smallest number which can be expressed as the sum of its proper divisors in exactly n ways.

Original entry on oeis.org

1, 6, 12, 30, 112, 24, 80, 36, 228, 150, 48, 156, 160, 126, 1242, 132, 5300, 1330, 448, 1326, 108, 96, 1288, 90, 918, 84, 1026, 750, 858, 16592, 744, 72, 910, 952, 60, 696, 896, 702, 690, 760, 6966, 12464, 192, 570, 400, 6642, 546, 594, 2178, 2420, 5424, 640

Offset: 0

Bernardo Boncompagni, Aug 04 2004

nonn

All numbers in the sequence are pseudoperfect.

			a(2)=12 because 12 is the smallest number which can be expressed as the sum of its proper divisors in exactly 2 ways: 12=6+4+2 and 12=6+3+2+1.

Records are in A065218.

(* first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Count[Plus @@@ Subsets[ Drop[ Divisors[n], -1]], n]; t = Table[0, {100}]; Do[ a = f[n]; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[a, " = ", n]], {n, 2, 16600}]; t (* Robert G. Wilson v, Aug 13 2004 *)

A033630(a(n))=n; A033630(j)<>n for jR. J. Mathar, Dec 11 2006

More terms from Robert G. Wilson v, Aug 13 2004

Definition corrected by R. J. Mathar, Nov 27 2006

cp's OEIS Frontend

User: Bernardo Boncompagni

A258107 Smallest number > 1 whose representation in all bases up to n consists only of 0's and 1's.

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A126717 Least odd k such that k*2^n-1 is prime.

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A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

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A113274 Record gaps between twin primes.

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A113403 Primes p in prime quadruplets (p,p+2,p+6,p+8) at the end of maximal gaps in A113404.

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A113404 Record gaps between prime quadruplets.

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A111166 Let p < q be consecutive primes; p is in the sequence if p/(q-p) is a record.

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A096356 Smallest number which can be expressed as the sum of its proper divisors in exactly n ways.