Zhicheng Wei - cp's OEIS frontend

A376926 a(n) is the number of ways n can be written as x + y with x >= y, x and y coprime, and so that the distinct prime factors of xyn are consecutive primes starting with 2.

0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 3, 1, 0, 0, 4, 0, 2, 1, 0, 0, 0, 1, 1, 0, 4, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Zhicheng Wei, Oct 10 2024

nonn

			The a(25) = 4 solutions are:
  24 + 1 = 25 and 24 * 1 * 25 = 2^3 * 3 * 5^2;
  21 + 4 = 25 and 21 * 4 * 25 = 2^2 * 3 * 5^2 * 7;
  18 + 7 = 25 and 18 * 7 * 25 = 2 * 3^2 * 5^2 * 7;
  16 + 9 = 25 and 16 * 9 * 25 = 2^4 * 3^2 * 5^2.
The a(27) = 2 solutions are:
  25 + 2 = 27 and 25 * 2 * 27 = 2 * 3^3 * 5^2;
  20 + 7 = 27 and 20 * 7 * 27 = 2^2 * 3^3 * 5 * 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055932, A346970.

f:= proc(n) local t,x,y,Pn,Px,Py,L;
   t:= 0:
   Pn:= numtheory:-factorset(n);
   for y from 1 to n/2 do
     x:= n-y;
     if igcd(x,y) > 1 then next fi;
     L:= Pn union numtheory:-factorset(x) union numtheory:-factorset(y);
     if max(L) = ithprime(nops(L)) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Nov 12 2024

a(n)={sum(k=1, n\2, if(gcd(k,n-k)==1, my(f=factor(k*(n-k)*n)[,1]~); f[#f]==prime(#f)))} \\ Andrew Howroyd, Oct 12 2024

A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.

Original entry on oeis.org

2, 9, 128, 4375, 18225, 1771561, 27295744, 1375325568, 6313843404, 289478389760

Offset: 1

Zhicheng Wei, Oct 10 2024

The terms given here can be verified by checking that the number of solutions up to a(n) equals A362593(n) and a(n) is a solution x+y.
a(2) = 9 corresponds to Catalan's conjecture (Mihăilescu's theorem).
a(4) = 4375 corresponds to the final term of A303332.

			a(3) = 128 because prime(3) = 5, and 125 + 3 = 128 with 125 and 3 coprime, and 125, 3 and 128 are all 5-smooth numbers, and no number larger than 128 has these properties.
Table x + y = a(n) is shown below (q gives abc triple quality):
  n=1: 1 + 1 = 2 (q=1),
  n=2: 8 + 1 = 9 (q=1.226)
  n=3: 125 + 3 = 128 (q=1.426)
  n=4: 4374 + 1 = 4375 (q=1.567)
  n=5: 14641 + 3584 = 18225 (q=1.267)
  n=6: 1771470 + 91 = 1771561 (q=1.395)
  n=7: 27217619 + 78125 = 27295744 (q=1.421)
  n=8: 1371299293 + 4026275 = 1375325568 (q=1.31)
  n=9: 4867359029 + 1446484375 = 6313843404 (q=1.17)
  n=10: 289478257991 + 131769  = 289478389760 (q=1.16)

Wikipedia, Abc conjecture.
Wikipedia, Catalan's conjecture.

Cf. A303332, A362593.

a(7)-a(8) from Andrew Howroyd, Oct 12 2024

a(9)-a(10) from David A. Corneth, Nov 24 2024

A376924 a(n) is the conjectured largest number such that both a(n) and a(n) - n are 7-smooth numbers. a(n) can be less than n. Otherwise, if no such number exists then a(n) = 0.

Original entry on oeis.org

4375, 8750, 13125, 17500, 21875, 26250, 30625, 35000, 39375, 43750, 3136, 52500, 1728, 61250, 65625, 70000, 5120, 78750, 6144, 87500, 91875, 6272, 2048, 105000, 109375, 3456, 118125, 122500, 2430, 131250, 4000, 140000, 9408, 10240, 153125, 157500, 32805, 12288, 5184, 175000

Offset: 1

Zhicheng Wei, Oct 10 2024

nonn

			a(31) = 4000 since both 4000 and 4000 - 31 = 3969 are 7-smooth numbers, and there is conjectured no number k larger than 4000 such that both k and k - 31 are 7-smooth numbers.

Cf. A002473.

M(v,u,lim)={vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))}
Gen(lim,k)={my(v=[1]); forprime(p=2, k, v=M(v, vector(logint(lim, p)+1, e, p^(e-1)), lim)); v}
lista(n,lim=10^30)={my(v=vector(n), G=Gen(lim,7)); for(i=1, #G, my(t=G[i], k=i+1); while(k<=#G && G[k]-t<=n, v[G[k]-t]=G[k]; k++)); v} \\ Andrew Howroyd, Oct 11 2024

a(23) onwards from Andrew Howroyd, Oct 11 2024

Running Andrew Howroyd's PARI program with limit=10^100 does not change the terms in DATA. - Hugo Pfoertner, Oct 13 2024

A369183 a(n) = n - A329004(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6

Offset: 1

Zhicheng Wei, Jan 15 2024

nonn

			a(15) = 3 because the largest Ramanujan's largely composite number below 15 is 12, and 15-12=3.

Cf. A067128, A329004.

dmax = 0; nmax = 1; seq = {}; Do[If[(d = DivisorSigma[0, n]) >= dmax, dmax = d; nmax = n]; AppendTo[seq,n- nmax], {n, 1, 102}];seq (* James C. McMahon, Jan 28 2024 *)

A369182 Numbers k such that d(k) is greater than d(k-2), d(k-1), d(k+1) and d(k+2) where d(k) is the number of divisors of k.

Original entry on oeis.org

12, 24, 30, 36, 48, 60, 66, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 165, 168, 176, 180, 192, 204, 210, 216, 220, 224, 228, 234, 240, 252, 256, 260, 264, 270, 276, 280, 288, 294, 300, 312, 315, 320, 324, 330, 336, 357, 360, 364, 368, 372

Offset: 1

Zhicheng Wei, Jan 15 2024

nonn

			30 is a term since d(28) = 6, d(29) = 2, d(30) = 8, d(31) = 2, d(32) = 6 and 6 < 8 > 6, 2 < 8 > 2.

Cf. A000005.

Subsequence of A075027.

SequencePosition[Table[DivisorSigma[0, n], {n, 1, 400}], {a_, b_, c_, d_, e_} /; Max[a, b, d, e] < c][[;; , 1]] + 2 (* Amiram Eldar, Jan 16 2024 *)

A369155 Numbers k such that d(k) < d(k - 1) and d(k) < d(k + 1), and d(k) is also a record for this type of number where d(k) is the number of divisors of k.

Original entry on oeis.org

5, 9, 51, 153, 351, 3249, 6579, 19551, 47151, 122451, 246975, 393471, 3292289, 10792495, 15270849, 25770879, 58967271, 60642945, 242340175, 481701375, 5122147185, 6644739375, 6971026699, 21061868751, 92330654625, 213089528575, 1159484186575, 1305664357375

Offset: 1

Zhicheng Wei, Jan 14 2024

Numbers in A075025 with record number of divisors.

			351 is a term in this sequence because d(351) = 8, d(350) = 12, and d(352) = 12, so 351 is a number that has fewer divisors than each of its neighbors, but no number below 351 has that property and has at least 8 divisors.

Cf. A000005. Subsequence of A075025.

lista(kmax) = my(d1 = numdiv(1), d2 = numdiv(2), d3, dm = 0); for(k = 3, kmax, d3 = numdiv(k); if(d2 < d1 && d2 < d3 && d2 > dm, print1(k-1, ", "); dm = d2); d1 = d2; d2 = d3); \\ Amiram Eldar, Jan 16 2024

a(9)-a(20) from Michel Marcus, Jan 15 2024
a(21)-a(22) from Amiram Eldar, Jan 16 2024
a(23)-a(28) from Martin Ehrenstein, Feb 08 2024

cp's OEIS Frontend

User: Zhicheng Wei

A376926 a(n) is the number of ways n can be written as x + y with x >= y, x and y coprime, and so that the distinct prime factors of x*y*n are consecutive primes starting with 2.

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A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.

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A376924 a(n) is the conjectured largest number such that both a(n) and a(n) - n are 7-smooth numbers. a(n) can be less than n. Otherwise, if no such number exists then a(n) = 0.

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A369183 a(n) = n - A329004(n).

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A369182 Numbers k such that d(k) is greater than d(k-2), d(k-1), d(k+1) and d(k+2) where d(k) is the number of divisors of k.

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A369155 Numbers k such that d(k) < d(k - 1) and d(k) < d(k + 1), and d(k) is also a record for this type of number where d(k) is the number of divisors of k.

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PARI

Extensions

A376926 a(n) is the number of ways n can be written as x + y with x >= y, x and y coprime, and so that the distinct prime factors of xyn are consecutive primes starting with 2.