A064955 -id:A064955 - cp's OEIS frontend

A383295 Positions of proper prime powers (A246547) in EKG-sequence.

3, 6, 8, 17, 22, 24, 31, 50, 64, 76, 112, 122, 124, 171, 232, 240, 290, 319, 359, 485, 521, 595, 696, 823, 947, 982, 1279, 1313, 1642, 1810, 1961, 2090, 2096, 2168, 2306, 2736, 3002, 3398, 3638, 3932, 4379, 4733, 4913, 5207, 6072, 6312, 6583, 6710, 7717, 7898, 9165, 9929, 10298, 11144, 11568, 11786, 12430, 14138

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Apparently also numbers k for which A265576(k) > 1 and A064413(k) is neither 2 nor 2*odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..321
Index entries for sequences related to EKG sequence.

Cf. A064413, A246547, A265576.
Setwise difference A383294 \ A064955.
Conjectured to be a subsequence of A383285.

isA383295(n) = { my(x=A064413(n)); (isprimepower(x) && !isprime(x)); };

A064425 Gaps between where primes occur in A064413.

Original entry on oeis.org

3, 5, 4, 6, 8, 5, 4, 6, 14, 4, 6, 7, 7, 8, 11, 7, 8, 13, 6, 4, 13, 9, 7, 15, 7, 8, 6, 4, 9, 20, 17, 6, 5, 15, 8, 10, 8, 9, 14, 8, 8, 19, 8, 6, 4, 20, 19, 10, 8, 6, 14, 4, 12, 13, 14, 9, 7, 13, 10, 9, 8, 24, 6, 12, 7, 23, 10, 16, 9, 6, 11, 18, 8, 17, 6, 6, 20, 10, 15

Offset: 1

N. J. A. Sloane, Oct 01 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
Index entries for sequences related to EKG sequence

Cf. A064413. First differences of A064955 and also of A064423.

Cf. A137847.

terms = 80;
ekg[s_] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++]; Append[s, k]];
EKG = Nest[ekg, {2, 4}, 12 terms];
fp[n_] := FirstPosition[EKG, Prime[n]][[1]];
Array[fp, terms] // Differences (* Jean-François Alcover, Sep 02 2018, after Robert G. Wilson v in A064413 *)

A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

Original entry on oeis.org

1, 5, 6, 11, 10, 24, 39, 22, 15, 14, 25, 69, 21, 20, 53, 130, 76, 51, 29, 28, 54, 112, 97, 50, 78, 96, 34, 33, 84, 209, 232, 38, 37, 85, 153, 44, 43, 111, 156, 179, 109, 58, 57, 142, 383, 140, 148, 352, 281, 124, 249, 299, 93, 118, 218, 62, 61, 143, 172, 68, 67, 173, 641, 696, 162, 75, 74, 210, 227, 238, 191, 535, 82, 81

Offset: 1

Antti Karttunen, May 31 2018

nonn

Permutation of A064957.

Cf. A305294 (a left inverse).

Cf. A003961, A064413, A064423, A064664, A064955, A064957.

a(n) = A064664(A003961(A064413(n))).
Other identities. For all n >= 1:
A305294(a(n)) = n.
a(A064955(n)) = A064955(1+n).
For all n >= 2, a(A064423(n)) = 1+A064955(1+n).

A382222 Smallest k such that A073734(k) = n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

Original entry on oeis.org

2, 3, 5, 8, 10, 968, 14, 17, 149, 579, 20, 11068, 28, 2126, 2406, 3070, 33, 58836, 37, 2935, 7468, 20029, 43, 50835, 321, 1065, 2220, 60390, 57, 403831, 61, 20143, 29156, 13453, 32294, 18829, 67, 2117, 56683, 65867, 74, 10242, 81, 82455, 80410, 24112, 89, 868283, 41341, 36370

Offset: 1

Scott R. Shannon, Mar 19 2025

nonn

a(630) > 1.045*10^9.

			a(6) = 968 as A064413(968) = 1014, A064413(967) = 1032, and GCD(1014,1032) = 6. No earlier pair of consecutive terms in A064413 has a GCD of 6.

Scott R. Shannon, Table of n, a(n) for n = 1..629

Cf. A064413, A073734, A064955, A382271, A064740, A073735.

If n = prime(j), j>=2, then a(n) = A064955(j).

A383294 Positions of prime powers (A246655) in EKG-sequence.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 14, 17, 20, 22, 24, 28, 31, 33, 37, 43, 50, 57, 61, 64, 67, 74, 76, 81, 89, 100, 107, 112, 115, 122, 124, 128, 134, 138, 151, 160, 167, 171, 182, 189, 197, 203, 207, 216, 232, 236, 240, 253, 259, 264, 279, 287, 290, 297, 305, 314, 319, 328, 336, 344, 359, 363, 371, 377, 381, 401, 420, 430, 438, 444

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence.

Positions of terms > 1 in A383293.
Disjoint union of A064955 and A383295.
Cf. A064413, A246655.

isA383294(n) = { my(x=A064413(n)); (isprime(x) || isprimepower(x)); };

A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 1, 5, 5, 4, 3, 10, 10, 9, 2, 1, 5, 14, 14, 13, 8, 4, 6, 11, 10, 20, 20, 19, 3, 1, 28, 28, 27, 9, 33, 33, 32, 7, 5, 14, 37, 37, 36, 2, 6, 20, 8, 10, 24, 18, 4, 15, 21, 11, 43, 43, 42, 13, 57, 57, 56, 3, 1, 28, 61, 61, 60, 12, 19, 33, 5, 67, 67, 66, 17, 9, 25, 14, 74, 74, 73, 7, 29, 34, 6, 37, 81, 81, 80, 2

Offset: 1

Antti Karttunen, May 31 2018

nonn

Cf. A064413, A064423, A064955, A064664, A064989, A305293.

a(n) = A064664(A064989(A064413(n))).
For all n >= 1, a(A305293(n)) = n.
For all n >= 2, a(A064423(n)) = a(A064955(n)) = A064955(n-1).
For all n >= 3, a(1+A064955(n)) = A064423(n-1).

A352194 It seems that the n-th prime p appears in A064413 at around position 2p; if in fact A064413(w) = p then set a(n) = 2p-w.

Original entry on oeis.org

2, 1, 0, 0, 2, -2, 1, 1, 3, 1, 1, 7, 8, 5, 5, 6, 11, 7, 6, 8, 8, 7, 6, 11, 12, 13, 9, 11, 11, 10, 18, 9, 15, 14, 19, 15, 17, 21, 20, 18, 22, 18, 19, 15, 17, 17, 21, 26, 24, 20, 22, 20, 20, 28, 27, 25, 28, 25, 24, 22, 17, 29, 33, 35, 27, 28, 33, 35, 39, 34, 36, 37, 35, 39, 34, 36, 42, 38, 36, 37, 40, 40, 48, 47, 45, 38, 41, 40

Offset: 1

N. J. A. Sloane, Mar 13 2022

sign

This is the amount by which p appears in advance of its expected position.

N. J. A. Sloane, Table of n, a(n) for n = 1..690

Cf. A008578, A064413, A064955.

a(n) = 2*A008578(n) - A064955(n-1).

A348470 a(n) = lpf(EKG(n)) = A020639(A064413(n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 2, 2, 5, 3, 2, 2, 7, 3, 2, 2, 2, 2, 11, 3, 3, 2, 5, 5, 2, 2, 13, 3, 2, 2, 2, 17, 3, 2, 2, 19, 3, 3, 2, 2, 2, 23, 3, 2, 2, 2, 2, 2, 7, 3, 2, 5, 5, 2, 2, 29, 3, 2, 2, 31, 3, 2, 2, 2, 2, 37, 3, 3, 2, 2, 2, 2, 41, 3, 3, 2, 7, 2, 2, 43, 3, 2, 5

Offset: 1

Michael De Vlieger, Dec 06 2021

nonn

Prime p_n appears first at a(A064955(n)).

Records are A008578.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^12.
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^18.
Index entries for sequences related to EKG sequence

Cf. A000040, A008578, A020639, A064413, A064955, A073734.

Map[FactorInteger[#][[1, 1]] &, Nest[Block[{k = 3}, While[Or[MemberQ[#, k], GCD[#[[-1]], k] == 1], k++]; Append[#, k]] &, {1, 2}, 84]]
(* or, faster *)
s = {1, 2}; u = 3; c[_] = 0; Set[j, 2]; Array[Set[c[#], #] &, 2]; Range[2]~Join~Reap[Do[If[PrimeQ[j], Set[u, NextPrime[u]]]; Set[k, u]; Which[And[PrimeQ[j], OddQ[j]], Set[k, 3 j], And[PrimeQ[j/2], OddQ[j/2]], Set[k, j/2], True, While[Nand[c[k] == 0, GCD[j, k] > 1], k++]]; Sow[FactorInteger[k][[1, 1]] ]; Set[c[k], i]; j = k, {i, 4, 10^4}]][[-1, -1]]

from itertools import islice, count
from math import gcd
from sympy import primefactors
def A064413gen(): # generator of terms
    yield 1
    yield 2
    l, s, b = 2, 3, set()
    for _ in count(0):
        i = s
        while True:
            if not i in b and gcd(i,l) > 1:
                yield i
                l = i
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
            i += 1
def A348470(n): return 1 if n == 1 else min(primefactors(next(islice(A064413gen(),n-1,None)))) # Chai Wah Wu, Dec 07 2021

A140418 Position of cubes in the EKG sequence (A064413).

Original entry on oeis.org

1, 8, 22, 64, 112, 199, 319, 485, 696, 958, 1279, 1649, 2090, 2612, 3241, 3932, 4733, 5608, 6583, 7707, 8926

Offset: 1

Parthasarathy Nambi, Jun 17 2008

nonn

The squares (A139476), primes (A064955) and cubes (this sequence) all appear in increasing order. Is this true for all other powers also?

			The position of 3^3 is 22.
The position of 5^3 is 112.

Cf. A064413, A064955, A139476.

cp's OEIS Frontend

A383295 Positions of proper prime powers (A246547) in EKG-sequence.

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A064425 Gaps between where primes occur in A064413.

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A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

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A382222 Smallest k such that A073734(k) = n, where A073734 is the GCD of consecutive terms of the EKG sequence A064413.

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A383294 Positions of prime powers (A246655) in EKG-sequence.

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PARI

A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

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A352194 It seems that the n-th prime p appears in A064413 at around position 2*p; if in fact A064413(w) = p then set a(n) = 2*p-w.

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A348470 a(n) = lpf(EKG(n)) = A020639(A064413(n)).

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Mathematica

Python

A140418 Position of cubes in the EKG sequence (A064413).

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A352194 It seems that the n-th prime p appears in A064413 at around position 2p; if in fact A064413(w) = p then set a(n) = 2p-w.