Geoffrey Caveney - cp's OEIS frontend

A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

1, 3, 6, 4, 2, 8, 10, 5, 15, 12, 9, 18, 16, 14, 7, 21, 24, 22, 20, 25, 30, 28, 26, 13, 39, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 33, 27, 51, 17, 68, 66, 64, 62, 60, 58, 56, 49, 63, 69, 23, 92, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70

Offset: 1

Geoffrey Caveney and Michael De Vlieger, Aug 16 2025

This is a variant of A386482 that begins with 1,3 instead of 1,2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n) in red and A386482(n) in blue, n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where the latter represents powerful numbers that are not prime powers.

Cf. A386482.

Block[{c, j, k, m, p, r, nn},
  nn = 2^12; c[] := False; m[] := 1; j = 2; c[1] = c[2] = True; r = 1;
  {1}~Join~Monitor[Most@ Reap[Do[
    If[PrimePowerQ[j],
      Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@
        FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];
        If[k == 0, k = m; While[c[k*p], k++]]; k *= p,
      k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--];
        If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k] ], k++] ] ];
    If[Mod[j, 2] == Mod[k, 2], r++, Sow[r]; r = 1];
    Set[{c[k], j}, {True, k}], {n, 3, nn}] ][[-1, 1]], n] ]

A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 10, 8, 14, 7, 21, 18, 16, 20, 15, 5, 25, 30, 28, 26, 24, 22, 11, 33, 27, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 39, 13, 65, 60, 58, 56, 49, 63, 51, 17, 68, 66, 64, 62, 31, 93, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70, 55, 75, 69, 23, 92, 94, 47, 141, 138, 136, 134, 132, 130, 128

Offset: 1

N. J. A. Sloane, Aug 15 2025, based on email messages from Geoffrey Caveney

Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.
A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.
It is conjectured that every positive number appears.
It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.
It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.
[The preceding comments are from Geoffrey Caveney's emails.]
From Michael De Vlieger, Aug 15 2025: (Start)
There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.
There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.
Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

Geoffrey Caveney, Emails to N. J. A. Sloane, Aug 13 2025 - Aug 15 2025.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers that are neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Cf. A064413 (EKG), A387072 (inverse), A387073 (record high points), A387074 (indices of record high points), A387075 (first differences), A387076 (primes in order of appearance), A387077 (indices of primes), A387078 (run lengths of consecutive odd and even terms), A387080 (variant that begins with 1,3).

See also A386483, A386484, A387087-A387090.

aList[n_] := Module[{an = 2, aset = <|2 -> True|>, m}, Reap[Sow[1]; Sow[an];
Do[m = SelectFirst[Range[an - 1, 2, -1], ! KeyExistsQ[aset, #] && GCD[#, an] > 1 & ];
If[MissingQ[m], m = NestWhile[# + 1 &, an + 1, !(! KeyExistsQ[aset, #] && GCD[#, an] > 1) & ]];
aset[m] = True; an = m; Sow[an], {n - 2}]][[2, 1]]]; aList[83]  (* Peter Luschny, Aug 15 2025 *)

PARI
```
\\ See Links section.
```

from math import gcd
from itertools import count, islice
def A386482_gen(): # generator of terms
    yield 1
    an, aset = 2, {2}
    while True:
        yield an
        m = next((k for k in range(an-1, 1, -1) if k not in aset and gcd(k, an) > 1), False)
        if not m: m = next(k for k in count(an+1) if k not in aset and gcd(k, an) > 1)
        an = m
        aset.add(an)
print(list(islice(A386482_gen(), 83))) # Michael S. Branicky, Aug 15 2025

A364654 Numbers which are the sum or difference of two seventh powers.

Original entry on oeis.org

0, 1, 2, 127, 128, 129, 256, 2059, 2186, 2187, 2188, 2315, 4374, 14197, 16256, 16383, 16384, 16385, 16512, 18571, 32768, 61741, 75938, 77997, 78124, 78125, 78126, 78253, 80312, 94509, 156250, 201811, 263552, 277749, 279808, 279935, 279936, 279937, 280064, 282123, 296320

Offset: 1

Geoffrey Caveney, Jul 31 2023

nonn

Don Zagier's conjecture that the polynomial x^7 + 3y^7 is injective on rational numbers is equivalent to the non-existence of any term in this sequence that is exactly 3 times another term in this sequence.

			2059 = 3^7 - 2^7, 2315 = 3^7 + 2^7, 358061 = 6^7 + 5^7, 543607 = 7^7 - 6^7.

Mathematics Stack Exchange, Are all bivariate polynomials of degree < 7 non-injective on rational numbers?
Bjorn Poonen, Multivariable polynomial injections on rational numbers, arXiv:0902.3961 [math.NT], 2009-2010; Acta Arith. 145 (2010), no. 2, 123-127.

Cf. A001015, A247099.

T=thueinit('z^7+1);
is(n) = (n==0) || (#thue(T, n)>0); \\ Michel Marcus, Aug 01 2023

A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

Original entry on oeis.org

183783600, 367567200, 1396755360, 1745944200, 2327925600, 3491888400, 6983776800, 80313433200, 160626866400, 252706217563800, 288807105787200, 336941623418400, 404329948102080, 505412435127600, 673883246836800, 1010824870255200, 2021649740510400, 112201560598327200

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 03 2011

nonn

Infinitely many terms are superabundant (SA) A004394; the smallest is 183783600.
Infinitely many terms are colossally abundant (CA) A004490; the smallest is 367567200.
Infinitely many terms are odd (and hence neither SA nor CA); the smallest is 1058462574572984015114271643676625.
See Section 5 of "On SA, CA, and GA numbers".
For additional terms, in factored form, see "Table of proper GA1 numbers up to 10^60", where SA and CA numbers are starred * and **.

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the smallest proper GA1 number.

Amiram Eldar, Table of n, a(n) for n = 1..18408 (from the J.-L. Nicolas's table)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A000203, A001222, A067698, A197638, A197639, A201558, A216436.

Maple
```
See "Computation of GA1 numbers".
```

A197638(n) if A001222(A197638(n)) > 2

A201558 Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 1, 1, 2, 4, 1, 2, 3, 7, 7, 7, 1, 4, 7

Offset: 3

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 03 2011

nonn

The number of GA1 numbers with one (resp., two) prime factors is zero (resp., infinity).
GA1 numbers with at least three prime factors are called "proper" - see A201557.
For a(n), see Section 6.2 of "On SA, CA, and GA numbers", and below "kmax" in "Table of proper GA1 numbers up to 10^60".

			183783600 = 2^4 * 3^3 * 5^2 * 7 * 11 * 13 * 17 is the first of the a(13) = 2 GA1 numbers with 4 + 3 + 2 + 1 + 1 + 1 + 1 = 13 prime factors.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A067698, A197638, A197639, A201557.

Maple
```
See "Computation of GA1 numbers".
```

A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 18, 24, 36, 48, 60, 72, 120, 180, 240, 360, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 02 2011

nonn

Subsequence of A067698.

A member > 5040 exists iff the Riemann Hypothesis is false, in which case the sequence is infinite. In any case, 3 and 5 are the only odd members. (See Sections 1 and 4 of "On SA, CA, and GA numbers".)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A000203, A067698, A197638, A201557.

nmax = 10^6; amax = 10;
G[k_] := DivisorSigma[1, k]/(k Log[Log[k]]);
okQ[n_] := DivisorSigma[1, n] > n Exp[EulerGamma] Log[Log[n]] && AllTrue[ Range[amax], Function[a, G[n] >= G[a*n]]];
Reap[For[n = 1, n <= nmax, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

A197638 GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

Original entry on oeis.org

4, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Dec 02 2011

nonn

The members with exactly two prime divisors counted with multiplicity are 4 and 2*p, for primes p > 5. (See Section 5 of "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis".)
The smallest member with more than two prime factors is 183783600. Such GA1 numbers are called "proper" - see A201557 and "Table of proper GA1 numbers up to 10^60".
The smallest odd member is 1058462574572984015114271643676625.
See "On SA, CA, and GA numbers".

			4 is a member because G(4) > 0 > G(2) = G(4/2).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010 [math.NT], 2011-2012.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.

Cf. A000203, A196226, A197639, A201557, A201558.

Maple
```
See "Computation of GA1 numbers".
```

g[k_] := g[k] = DivisorSigma[1, k]/(k*Log[Log[k]]); okQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, i = 1; While[i <= Length[p] && g[n] >= g[n/p[[i]]], i++]; i > Length[p]]; Select[Range[2, 1000], ! PrimeQ[#] && okQ[#] &] (* T. D. Noe, Dec 03 2011 *)

g(k) = sigma(k)/(k*log(log(k)));
isga1(k)=if (isprime(k), return (0)); gk = g(k);f = factor(k); for(i=1,length(f~), if (gk < g(k/f[i,1]), return(0)););1; \\ Michel Marcus, Sep 09 2012

A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)log(log(r/p))) > sigma(r)/(rlog(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 5, 3, 3, 2, 3, 5, 3, 7, 2, 5, 5, 5, 3, 7, 7, 2

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Sep 29 2011

nonn

See comments, references, links and crossrefs in A067698.

			A067698(5) = 6 and sigma(6/2)/((6/2)*log(log(6/2))) = 14.17... > 3.42... = sigma(6)/(6*log(log(6))), so a(5) = 2.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33; see Table 1 and Lemma 11.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Cf. A067698.

A192884 Non-superabundant numbers satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

3, 5, 8, 9, 10, 16, 18, 20, 30, 72, 84

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Jul 11 2011

nonn

If another term exists, it is > 5040 and the Riemann Hypothesis is false.

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Cf. A004394 (superabundant), A091901 (Robin's inequality), A067698 (the reverse of Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality).

A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, May 30 2011

nonn

5040 is the last element in the sequence if and only if the Riemann Hypothesis is true. (See Akbary and Friggstad in A004394.)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A004394, A091901, A067698, A166981, A077006.

kmax = 10^4;
A004394 = Join[{1}, Reap[For[r = 1; k = 2, k <= kmax, k = k + 2, s = DivisorSigma[-1, k]; If[s > r, r = s; Sow[k]]]][[2, 1]]];
A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];
Intersection[A004394, A067698] (* Jean-François Alcover, Jan 28 2019 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)); \\ A067698
lista(nn) = my(r=1, t); forstep(n=2, nn, 2, t=sigma(n, -1); if(t>r && is(n), r=t; print1(n, ", "))); \\ Michel Marcus, Jan 28 2019; adapted from A004394

Equals A004394 intersect A067698.

Erroneous terms 1260 and 1680 removed by Jean-François Alcover, Jan 28 2019

cp's OEIS Frontend

User: Geoffrey Caveney

A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

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A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

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PARI

Python

A364654 Numbers which are the sum or difference of two seventh powers.

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Comments

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Crossrefs

Programs

PARI

A201557 Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A201558 Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.

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Comments

Examples

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Maple

A197639 GA2 numbers: n with G(n) >= G(a*n) for all integers a > 0, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

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Mathematica

A197638 GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

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Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)*log(log(r/p))) > sigma(r)/(r*log(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.

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A197639 GA2 numbers: n with G(n) >= G(an) for all integers a > 0, where G(k) = sigma(k)/(klog(log(k))) and sigma(k) = sum of divisors of k.

A196229 Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)log(log(r/p))) > sigma(r)/(rlog(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.