A161189 -id:A161189 - cp's OEIS frontend

A339013 Class number m containing n in a partitioning of the natural numbers into classes B_m by William J. Keith.

2, 3, 2, 4, 2, 4, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2

Offset: 1

Kevin Ryde, Nov 19 2020

a(n)=m when n is in class B_m. Keith's residues formula in lemma 1 is equivalent to requiring that n-1 in factorial base representation ends in m-2 nonzero digits, so m = A339012(n-1) + 2.
a(n)=m iff n mod m! is among certain residue classes determined by m. The residues for A339012 are rows of A227157 and here add +1 to each residue (mod m!).  For example 3 or 5 (mod 24) in A339012 becomes here 4 or 6 (mod 24).
The frequency of appearance of the term k = 2, 3, ... in this sequence is 1/(k*(k-1)). - Amiram Eldar, Feb 15 2021

Kevin Ryde, Table of n, a(n) for n = 1..10080
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A005408 (class B_2), A016933 (class B_3).

Cf. A161189 (class number in partition A_k), A339012.

a[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; m]; Array[a, 30] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde's PARI code *)

a(n) = n--; my(b=2,r); while([n,r]=divrem(n,b);r!=0, b++); b;

a(n) = A339012(n-1) + 2.
a(n) = m iff n == 1 + Sum_{j=1..m-2} d[j]*j! (mod m!) with d[j] in ranges 1 <= d[j] <= j. [Keith, section 2.1 lemma 1]
a(n)=2 iff n mod 2 = 1. [Keith section 4 residues].
a(n)=3 iff n mod 6 = 2.
a(n)=4 iff n mod 24 = 4 or 6.
a(n)=5 iff n mod 120 = any of 10, 12, 16, 18, 22, 24.

A143028 A sequence of asymptotic density zeta(2) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 69, 70, 72, 73, 74, 77, 80, 81, 82, 84, 85, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100, 101, 102, 105

Offset: 1

William J. Keith, Jul 17 2008, Jul 18 2008

nonn

x is an element of this sequence if when m>1 is the least natural number such that the least positive residue of x mod m! is no more than (m-2)!, floor[x/(m!)] is not congruent to m-1 mod m. The sequence is made up of the residue classes 1 mod 4; 2 and 8 mod 18; 4, 6, 28, 30, 52 and 54 mod 96, etc. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

A161189(n) = 2 if n is a term of this sequence. Similarly A161189(n) = 3, 4, 5, ... if n is in A143029, A143030, ...; such that the number system is partitioned into relative densities tending to (zeta(2) - 1), (zeta(3) - 1), ... such that Sum_{k>=2} (zeta(k) - 1) = 1.0. This implies that the density of 2's in A161189 tends to (zeta(2) - 1) = (Pi^2/6 - 1) = 0.644934... . - Gary W. Adamson, Jun 07 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143029, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A339013.

Cf. A161189. - Gary W. Adamson, Jun 07 2009

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[100], f[#] == 2 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143029 A sequence of asymptotic density zeta(3) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

3, 11, 14, 19, 27, 32, 35, 43, 51, 59, 67, 68, 75, 76, 78, 83, 86, 91, 99, 107, 115, 122, 123, 131, 139, 140, 147, 155, 163, 171, 172, 174, 176, 179, 187, 194, 195, 203, 211, 219, 227, 230, 235, 243, 248, 251, 259, 267, 268, 270, 275, 283, 284, 291, 299, 302

Offset: 1

William J. Keith, Jul 17 2008, Jul 18 2008

nonn

x is an element of this sequence if when m is the least natural number such that the least positive residue of x mod m! is no more than (m-2)!, floor(x/(m!)) is congruent to m-1 mod m and floor(x/(m*(m!))) is not congruent to m-1 mod m. The sequence is made up of the residue classes 3 mod 8; 14 and 32 mod 54; 76, 78, 172, 174, 268 and 270 mod 384, etc. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[300], f[#] == 3 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

a(n) = 2*a(n-1) + 3. [Obviously wrong, R. J. Mathar, Jul 14 2016]

G.f.: 1/(exp(x)-1). [Apparently not, R. J. Mathar, Jul 14 2016]

A143030 A sequence of asymptotic density zeta(4) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

7, 23, 39, 50, 55, 71, 87, 103, 104, 119, 135, 151, 167, 183, 199, 212, 215, 231, 247, 263, 266, 279, 295, 311, 327, 343, 359, 364, 366, 374, 375, 391, 407, 423, 428, 439, 455, 471, 487, 503, 519, 535, 536, 551, 567, 583, 590, 599, 615, 631, 647, 663, 679

Offset: 1

William J. Keith, Jul 18 2008

nonn

x is an element of this sequence if when m is the least natural number such that the least positive residue of x mod m! is no more than (m-2)!, floor(x/(m!)) and floor(x/(m*(m!))) are congruent to m-1 mod m, but floor(x/((m^2)*(m!))) is not. The sequence is made up of the residue classes 7 (mod 16); 50 and 104 (mod 162); 364, 366, 748, 750, 1132 and 1134 (mod 1536), etc. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[700], f[#] == 4 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143031 A sequence of asymptotic density zeta(5) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

15, 47, 79, 111, 143, 158, 175, 207, 239, 271, 303, 320, 335, 367, 399, 431, 463, 495, 527, 559, 591, 623, 644, 655, 687, 719, 751, 783, 806, 815, 847, 879, 911, 943, 975, 1007, 1039, 1071, 1103, 1130, 1135, 1167, 1199, 1231, 1263, 1292, 1295, 1327, 1359

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143032, A143033, A143034, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[1400], f[#] == 5 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143032 A sequence of asymptotic density zeta(6) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

31, 95, 159, 223, 287, 351, 415, 479, 482, 543, 607, 671, 735, 799, 863, 927, 968, 991, 1055, 1119, 1183, 1247, 1311, 1375, 1439, 1503, 1567, 1631, 1695, 1759, 1823, 1887, 1940, 1951, 2015, 2079, 2143, 2207, 2271, 2335, 2399, 2426, 2463, 2527, 2591, 2655

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143031, A143033, A143034, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[2700], f[#] == 6 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143033 A sequence of asymptotic density zeta(7) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

63, 191, 319, 447, 575, 703, 831, 959, 1087, 1215, 1343, 1454, 1471, 1599, 1727, 1855, 1983, 2111, 2239, 2367, 2495, 2623, 2751, 2879, 2912, 3007, 3135, 3263, 3391, 3519, 3647, 3775, 3903, 4031, 4159, 4287, 4415, 4543, 4671, 4799, 4927, 5055, 5183, 5311

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143031, A143032, A143034, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[5300], f[#] == 7 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143034 A sequence of asymptotic density zeta(8) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

127, 383, 639, 895, 1151, 1407, 1663, 1919, 2175, 2431, 2687, 2943, 3199, 3455, 3711, 3967, 4223, 4370, 4479, 4735, 4991, 5247, 5503, 5759, 6015, 6271, 6527, 6783, 7039, 7295, 7551, 7807, 8063, 8319, 8575, 8744, 8831, 9087, 9343, 9599, 9855, 10111

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143031, A143032, A143033, A143035, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[10^4], f[#] == 8 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143035 A sequence of asymptotic density zeta(9) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

255, 767, 1279, 1791, 2303, 2815, 3327, 3839, 4351, 4863, 5375, 5887, 6399, 6911, 7423, 7935, 8447, 8959, 9471, 9983, 10495, 11007, 11519, 12031, 12543, 13055, 13118, 13567, 14079, 14591, 15103, 15615, 16127, 16639, 17151, 17663, 18175

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143031, A143032, A143033, A143034, A143036, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[2*10^4], f[#] == 9 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

A143036 A sequence of asymptotic density zeta(10) - 1, where zeta is the Riemann zeta function.

Original entry on oeis.org

511, 1535, 2559, 3583, 4607, 5631, 6655, 7679, 8703, 9727, 10751, 11775, 12799, 13823, 14847, 15871, 16895, 17919, 18943, 19967, 20991, 22015, 23039, 24063, 25087, 26111, 27135, 28159, 29183, 30207, 31231, 32255, 33279, 34303, 35327

Offset: 1

William J. Keith, Jul 18 2008

nonn

Made up of a collection of mutually exclusive residue classes modulo multiples of factorials. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A143028, A143029, A143030, A143031, A143032, A143033, A143034, A143035, A161189, A339013.

f[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[4*10^4], f[#] == 10 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

cp's OEIS Frontend

A339013 Class number m containing n in a partitioning of the natural numbers into classes B_m by William J. Keith.

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A143028 A sequence of asymptotic density zeta(2) - 1, where zeta is the Riemann zeta function.

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A143029 A sequence of asymptotic density zeta(3) - 1, where zeta is the Riemann zeta function.

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A143030 A sequence of asymptotic density zeta(4) - 1, where zeta is the Riemann zeta function.

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A143031 A sequence of asymptotic density zeta(5) - 1, where zeta is the Riemann zeta function.

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A143032 A sequence of asymptotic density zeta(6) - 1, where zeta is the Riemann zeta function.

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A143033 A sequence of asymptotic density zeta(7) - 1, where zeta is the Riemann zeta function.

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A143034 A sequence of asymptotic density zeta(8) - 1, where zeta is the Riemann zeta function.

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A143035 A sequence of asymptotic density zeta(9) - 1, where zeta is the Riemann zeta function.

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