A134204 -id:A134204 - cp's OEIS frontend

A133242 Indices n such that A134204(n) < n.

12, 201, 379, 474, 588, 868, 932, 1604, 1942, 2006, 3084, 4800, 7800, 9666, 9700, 10794, 10956, 11074, 11140, 11176, 14112, 16420, 16436, 16499, 17330, 17478, 18475, 20784, 21118, 21410, 22004, 22078, 22510

Offset: 1

Robert Israel, Oct 15 2007

nonn

			The first few exceptionally small terms in A134204 that give rise to this sequence and A133243 are b(12) = 11, b(201) = 173, b(379) = 257, b(474) = 263, b(588) = 571, b(868) = 631, b(932) = 887, ..., where b(i) = A134204(i).

David Applegate, Table of n, a(n) for n = 1..6148
David Applegate, C++ Program
David Applegate, Larger file giving first 49344 terms

Corrected and extended by David Applegate, Oct 15 2007

A133243 Values of A134204(n) for n in A133242.

Original entry on oeis.org

11, 173, 257, 263, 571, 631, 887, 1579, 863, 1231, 2309, 4297, 7789, 5623, 5807, 9533, 10243, 10691, 10853, 10987, 7591, 16223, 16319, 16381, 3041, 3463, 12329, 12967, 14177, 15263, 17573, 17939, 19417

Offset: 1

Robert Israel, Oct 15 2007

nonn

David Applegate, Table of n, a(n) for n = 1..6148
David Applegate, C++ Program
David Applegate, Larger file giving first 49344 terms

Cf. A134204, A133242.

Corrected and extended by David Applegate, Oct 15 2007

A133244 Records in A134204.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 41, 67, 83, 101, 109, 233, 239, 293, 509, 733, 1009, 1259, 2027, 2393, 2749, 4591, 6359, 8447, 9901, 11093, 12577, 13649, 16249, 19081, 19949, 20593, 26711, 31219, 46141, 65831, 145861, 147689, 167891, 196831, 216607, 217717

Offset: 1

N. J. A. Sloane, Oct 15 2007

nonn

T. D. Noe, Table of n, a(n) for n = 1..123

Cf. A133245.

More terms from R. J. Mathar, Oct 19 2007

A133245 Where records occur in A134204.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 16, 20, 21, 23, 26, 29, 37, 55, 71, 79, 85, 103, 115, 191, 221, 277, 317, 335, 455, 499, 601, 639, 695, 719, 751, 815, 883, 1331, 1877, 3709, 3749, 3967, 4123, 4417, 4467, 4519, 4933, 6361, 6547, 7255, 8363, 8383, 8651, 8887

Offset: 1

N. J. A. Sloane, Oct 15 2007

nonn

T. D. Noe, Table of n, a(n) for n = 1..123

Cf. A133244.

More terms from R. J. Mathar, Oct 19 2007

Extended by T. D. Noe, Feb 01 2013

A134205 a(n) = A134204(n)+A134204(n-1).

Original entry on oeis.org

5, 8, 12, 20, 30, 36, 42, 64, 72, 60, 66, 48, 78, 126, 120, 144, 136, 126, 152, 180, 210, 198, 322, 336, 150, 286, 378, 252, 406, 390, 248, 288, 264, 170, 210, 324, 666, 760, 624, 480, 574, 630, 344, 528, 540, 506, 752, 720, 588, 750, 714, 468, 424, 594, 1100, 1064, 684

Offset: 1

Leroy Quet, Oct 14 2007

nonn

a(n) is divisible by n for every n.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A134204, A134206.

With[{nn = 57}, Total /@ Partition[#, 2, 1] &@ Fold[Append[#1, SelectFirst[Prime@ Range[2, Ceiling& Log2[nn] nn], Function[p, And[FreeQ[#1, p], Divisible[Last@ #1 + p, #2]]]]] &, {2}, Range@ nn]] (* Michael De Vlieger, Oct 16 2017 *)

More terms from Robert Israel, Oct 14 2007

A162846 The position of prime(n) in A134204.

Original entry on oeis.org

0, 1, 2, 3, 12, 4, 5, 6, 7, 10, 9, 11, 8, 34, 25, 17, 14, 15, 13, 59, 18, 19, 16, 22, 30, 20, 24, 40, 21, 28, 33, 90, 32, 27, 83, 31, 36, 42, 35, 201, 62, 43, 132, 45, 52, 63, 65, 107, 53, 60, 23, 26, 114, 38, 379, 474, 70, 51, 120, 48, 162, 29, 49, 76, 46, 66, 56, 92, 44, 82, 57

Offset: 1

T. D. Noe, Jul 19 2009

nonn

It is not known whether this sequence is defined for all n. The first 10^6 terms of A134204 contain the first 16396 primes (up to 180667).

T. D. Noe, Table of n, a(n) for n=1..16396

A232992 Let b(i) = A134204(i) and c(n) = A133242(n); a(n) is the number of primes p <= c(n) such that p is not in {b(0), b(1), ..., b(c(n)-1)}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 3, 6, 7, 6, 7, 7, 7, 6, 5, 7, 12, 11, 10, 10, 9, 10, 12, 11, 12, 11, 10, 9, 9, 8, 8, 8, 9, 8, 8, 8, 7, 10, 16, 16, 16, 19, 18, 17, 16, 15, 15, 16, 16, 17, 16, 15, 16, 16, 19, 19, 20, 20, 19, 18, 17, 16, 17, 20, 19, 20, 19, 18, 18, 19, 23, 24, 23, 25, 24, 25, 27, 26, 27, 27, 26, 25, 25

Offset: 1

N. J. A. Sloane, Dec 13 2013

nonn

Computed by David Applegate, Oct 2007.

Arises from studying the question of whether A134204 is an infinite sequence.

			Terms b(0) through b(12) of A134202 are (ignore the periods, which are just for alignment):
i:... 0, 1, 2, 3,. 4,. 5,. 6,. 7,. 8,. 9, 10, 11, 12
b(i): 2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37, 11
c(1) = 12 is the first i for which b(i)<i.
Then a(1) is the number of primes p <= 12 that are not in the set {b(0), ..., b(11)} = {2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37}.
Only p = 11 is missing, so a(1)=1.

David Applegate, Table of n, a(n) for n = 1..10000
David Applegate, C++ Program
David Applegate, Notes on programs and output
David Applegate, The first 106394 lines of output. The first 3 columns give the first 106394 terms of A133242, A133243 and A232992 (the present sequence), and establish that at least 800 million terms of A134204 exist.

Cf. A134204, A133242.

A134207 a(0) = 2; for n > 0, a(n) = the smallest prime which is > a(n-1) such that a(n-1) + a(n) is a multiple of n.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 41, 67, 73, 103, 113, 173, 191, 199, 233, 277, 281, 479, 521, 571, 617, 809, 823, 827, 863, 919, 929, 1217, 1303, 1487, 1489, 1613, 1753, 2027, 2113, 2179, 2267, 2647, 2713, 3109, 3191, 3259, 3517, 3593, 3767, 3847, 3881, 4057

Offset: 0

Leroy Quet, Oct 14 2007

nonn

			The primes that are > a(8)=41 form the sequence 43,47,53,59,61,67,71,... Of these, 67 is the smallest that when added to a(8)=41 gets a multiple of 9 -- 41+67 = 108 = 9*12. (41+p is not divisible by 9 for p = any prime which is > 41 and is < 67.) So a(9) = 67.

Eric M. Schmidt, Table of n, a(n) for n = 0..1000

Cf. A134204, A134208, A134209.

a = {2}; For[n = 1, n < 100, n++, i = 1; While[Not[Mod[a[[ -1]] + Prime[PrimePi[a[[ -1]]] + i], n] == 0], i++ ]; AppendTo[a, Prime[PrimePi[a[[ -1]]] + i]]]; a (* Stefan Steinerberger, Oct 17 2007 *)

def A134207(max) :
    res = [2]; p = 3
    for n in range(1,max+1) :
        while (res[n-1] + p) % n != 0 : p = next_prime(p)
        res.append(p); p = next_prime(p)
    return res # Eric M. Schmidt, May 23 2013

More terms from Stefan Steinerberger, Oct 17 2007

A224221 a(0)=3; for n>0, a(n) is the smallest odd prime q not already in the sequence such that the n-th prime p(n) divides a(n-1)+q. If no such prime q exists, the sequence terminates.

Original entry on oeis.org

3, 5, 7, 13, 29, 37, 41, 61, 53, 131, 43, 19, 277, 379, 137, 239, 79, 157, 331, 71

Offset: 0

Daniel Drucker and N. J. A. Sloane, Apr 05 2013

a(20) does not exist, so the sequence terminates. A134204 is a similar sequence for which the termination question is unresolved.

			After a(3)=13, to find a(4) we look for an odd prime q such that the fourth prime, 7, divides 13+q, and q=29 works, since 7 divides 13+29 = 42.
After a(19)=71 we look for an odd prime q such that p(20)=71 divides 71+q. The only candidate is q=71. Since 71 is already in the sequence, the sequence terminates.

Cf. A134204, A224222.

A224222 a(0)=3; for n>0, a(n) is the smallest prime q not already in the sequence such that the n-th prime p(n) divides a(n-1)+q. If no such prime q exists, the sequence terminates.

Original entry on oeis.org

3, 5, 7, 13, 29, 37, 2, 83, 31, 61, 113, 11, 137, 109, 149, 227, 197, 157, 331, 71

Offset: 0

Daniel Drucker and N. J. A. Sloane, Apr 05 2013

a(20) does not exist, so the sequence terminates. A134204 is a similar sequence for which the termination question is unresolved.

			After a(3)=13, to find a(4) we look for a prime q such that the fourth prime, 7, divides 13+q, and q=29 works, since 7 divides 13+29 = 42.
After a(19)=71 we look for a prime q such that p(20)=71 divides 71+q. The only candidate is q=71. Since it is already in the sequence, the sequence terminates.

Cf. A134204, A224221.

cp's OEIS Frontend

A133242 Indices n such that A134204(n) < n.

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A133243 Values of A134204(n) for n in A133242.

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A133244 Records in A134204.

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A133245 Where records occur in A134204.

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A134205 a(n) = A134204(n)+A134204(n-1).

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A162846 The position of prime(n) in A134204.

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A232992 Let b(i) = A134204(i) and c(n) = A133242(n); a(n) is the number of primes p <= c(n) such that p is not in {b(0), b(1), ..., b(c(n)-1)}.

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A134207 a(0) = 2; for n > 0, a(n) = the smallest prime which is > a(n-1) such that a(n-1) + a(n) is a multiple of n.

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A224221 a(0)=3; for n>0, a(n) is the smallest odd prime q not already in the sequence such that the n-th prime p(n) divides a(n-1)+q. If no such prime q exists, the sequence terminates.

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A224222 a(0)=3; for n>0, a(n) is the smallest prime q not already in the sequence such that the n-th prime p(n) divides a(n-1)+q. If no such prime q exists, the sequence terminates.

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