Michel Marcus - cp's OEIS frontend

A387216 Numbers that have at least two prime factors (counting multiplicity) congruent to 1 mod 3.

49, 91, 98, 133, 147, 169, 182, 196, 217, 245, 247, 259, 266, 273, 294, 301, 338, 343, 361, 364, 392, 399, 403, 427, 434, 441, 455, 469, 481, 490, 494, 507, 511, 518, 532, 539, 546, 553, 559, 588, 589, 602, 637, 651, 665, 676, 679, 686, 703, 721, 722, 728, 735, 741, 763, 777, 784, 793, 798

Offset: 1

Michel Marcus, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..7923
Hajrudin Fejzić, Nontrivial Solutions to a Cubic Identity and the Factorization of n^2+n+1, arXiv:2508.14937 [math.GM], 2025. See Theorem 2 p. 4.

Cf. A002476 (primes congruent to 1 mod 3), A050931 (at least one).

res := [];for n in [1..1000] do L := [ f[2] : f in Factorization(n) | f[1] mod 3 eq 1 ]; count := (#L eq 0) select 0 else &+L;if count gt 1 then Append(~res, n); end if; end for; res;
 // Vincenzo Librandi, Aug 24 2025

ff[{m_,n_}]:=Table[m,n];Select[Range[798],Count[Mod[ff/@FactorInteger[#]//Flatten,3],1]>1&] (* James C. McMahon, Aug 22 2025 *)

isok(k) = my(f=factor(k)); sum(i=1, #f~, if ((f[i,1]%3) == 1, f[i,2])) >= 2;

A386390 Numbers k such that k-1 | sigma+(k) where sigma+ is A107758.

Original entry on oeis.org

2, 6, 66, 225, 8646, 101025, 149497986, 20412000225

Offset: 1

Michel Marcus, Aug 20 2025

Cf. A107758, A055708.

a107758[n_]:=DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &];Select[Range[2,10^6],Divisible[a107758[#],#-1]&] (* James C. McMahon, Aug 21 2025 *)

isok(k) = if (k>1, !(sumdiv(k, d, if(gcd(k/d, d) == 1, sigma(d))) % (k-1)));

a(8) from Vincenzo Librandi, Aug 21 2025

A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 3, 5, 0, 9, 1, 11, 0, 3, 7, 15, 0, 17, 3, 5, 0, 21, 3, 19, 0, 17, 5, 27, 0, 29, 15, 9, 0, 15, 5, 35, 0, 11, 9, 39, 0, 41, 9, 15, 0, 45, 7, 41, 0, 15, 11, 51, 0, 27, 15, 17, 0, 57, 3, 59, 0, 25, 31, 33, 0, 65, 15, 21, 0, 69, 15, 71, 0, 19, 17, 45, 0, 77, 21

Offset: 1

Michel Marcus, Aug 19 2025

This is the phi- function in Sandor and Atanassof.

Paolo Xausa, Table of n, a(n) for n = 1..10000
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856.

Cf. A000010 (phi), A107758 (sigma+), A057723 (sigma-), A055653 (phi+).

A383954[n_] := If[n == 1, 1, Times @@ (EulerPhi[Power @@@ FactorInteger[n]] - 1)];
Array[A383954, 100] (* Paolo Xausa, Aug 19 2025 *)

a(n) = my(f=factor(n)); prod(k=1, #f~, p=f[k,1]; eulerphi(f[k,1]^f[k,2])-1);

From Amiram Eldar, Aug 19 2025: (Start)
Multiplicative with a(p^e) = (p-1)*p^(e-1) - 1.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p+1) - (p+1)/p^2) = 0.39439177573628632634... . (End)

A386949 Irregular triangle whose n-th row lists the nonzero terms of the n-th column of A386755.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Aug 10 2025

			Triangle begins:
  1;
  1, 2, 2;
  1, 3, 3, 3, 3;
  1, 2, 2, 4, 4;
  1, 5, 5, 5, 5, 5, 5;
  1, 2, 2, 3, 3, 6, 6, 6, 6;
  1, 7, 7, 7, 7, 7, 7, 7, 7;
  1, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8;
  1, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 9, 9;
  1, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 10, 10;
  ...

Cf. A386755 (original triangle), A386520 (row sums).

Cf. A027750.

orow(n) = my(v=vector(n), m=n); for(k=1, n, my(keepm = m); while(m%k, m--); if (m == 0, keepm=m, v[m] = k; m--); ); v; \\ A386755
nrow(n) = my(ok=1, k=1, last=-1, list=List(), r); while(ok, r=row(k); if ((#r >= n) && r[n], listput(list, r[n])); k++; if (#r>=n, if ((last==n) && (r[n]==0), ok = 0, last = r[n]))); Vec(list);

A386911 Numbers k such that A064945(k) = A064945(k+1).

Original entry on oeis.org

14, 14229, 194931, 8897930, 278110142

Offset: 1

Michel Marcus, Aug 07 2025

No other terms up to 10^9.

Cf. A064945.

A064945[n_] := Total[#*Range[Length[#], 1, -1]] & [Divisors[n]];q[n_]:=A064945[n]==A064945[n+1];Select[Range[200000],q[#]&] (* James C. McMahon, Aug 07 2025 *)

A386871 Least k such that A056100(k) = n or -1 if no such k exists.

Original entry on oeis.org

6, 124357252657, 4, 33, 8, 145, 9, 37063859, 16, 51, 26, 1441, 15, 2353, 34, 69, 20, 1011377103546119, 27, 7201, 25, 87, 115, 9911837, 56, 385, 58, 45, 62, 86125529, 57, 30721, 35, 123, 74, 295, 90, 15686608811, 82, 141, 86, 70561, 49, 77857739, 94, 159, 329, 34884199

Offset: 1

Michel Marcus, Aug 06 2025

Wanted terms: a(18), a(38), a(72), a(80).
a(38) = 15686608811 (found by Tomas Rokicki), a(80) = 4222433393407, and a(18), a(72) > 6*10^12. - Giovanni Resta, Aug 09 2025
a(72) > a(18). - Martin Ehrenstein, Aug 10 2025

Michel Marcus, Extension to A056100, Seqfan, Aug 05 2025.
Neil Sloane, Smallest inverse to f(n) = phi(n)*sigma(n) + 1 mod n ?, Math Fun, Aug 05 2025.

Cf. A056100, A386856 (odd bisection), A015706.

a[n_]:=Module[{k=0},Until[Mod[DivisorSigma[1, k]*EulerPhi[k] + 1, k]==n,k++];k] (* James C. McMahon, Aug 06 2025 *)

a(2) from Tomas Rokicki, Aug 05 2025
a(18) from Martin Ehrenstein, Aug 10 2025
a(19)-a(37) from Hugo Pfoertner, Aug 11 2025
a(38) from Tomas Rokicki, Aug 09 2025
More terms from Hugo Pfoertner, Aug 11 2025

A386436 Smallest k for which A385731(k) = n.

Original entry on oeis.org

1, 2, 6, 42, 1770, 47058, 547470, 8648458, 623254170

Offset: 1

Juri-Stepan Gerasimov and Michel Marcus, Jul 21 2025

Cf. A385729, A385731, A386410.

a[n_]:=Module[{k=0},Until[Length[Select[Divisors[k], Mod[-#, k]==PowerMod[-#, #, k]==PowerMod[#, #, k]&]]==n,k++];k];Array[a,7] (* James C. McMahon, Aug 06 2025 *)

a(9) from Michel Marcus, Jul 21 2025

A386856 Least k such that A056100(k) = 2*n+1 or -1 if no such k exists.

Original entry on oeis.org

6, 4, 8, 9, 16, 26, 15, 34, 20, 27, 25, 115, 56, 58, 62, 57, 35, 74, 90, 82, 86, 49, 94, 329, 60, 106, 517582, 78, 91, 122, 110, 77, 128, 111, 142, 146, 88, 427, 158, 102, 100, 265, 273, 178, 242, 95, 212, 194, 104, 202, 125, 462, 214, 218, 132, 121, 344, 138, 470, 241582

Offset: 0

Michel Marcus, Aug 05 2025

nonn

a(n) is the least k such that (sigma(k)*phi(k) + 1) mod k = 2*n+1 or -1 if no such k exists.

Michel Marcus, Table of n, a(n) for n = 0..465

Cf. A056100.

a[n_]:=Module[{k=1},While[Mod[DivisorSigma[1,k]EulerPhi[k]+1,k]!=2n+1, k++]; k]; Array[a,60,0] (* Stefano Spezia, Aug 05 2025 *)

f(n) = (sigma(n)*eulerphi(n)+1) % n; \\ A056100
a(n) = my(k=1); while (f(k) != 2*n+1, k++); k;

A386494 Primitive terms of A297305.

Original entry on oeis.org

5, 31, 89, 103, 233, 313, 317, 321, 331, 337, 347, 359, 363, 371, 383, 397, 401, 413, 417, 421, 431, 441, 443, 461, 463, 471, 479, 481, 483, 491, 493, 499, 501, 507, 511, 531, 537, 543, 551, 561, 571, 581, 603, 629, 631, 641, 643, 653, 661, 671, 677, 683, 689

Offset: 1

Michel Marcus, Jul 23 2025

nonn

Terms of A297305 that are not a multiple of a lesser term of A297305.

Titus Piezas III, On the Clustering of Sums of Quartic Sextuples, 2005.

Cf. A297305.

More terms from Jinyuan Wang, Jul 24 2025

A385810 Integers x such that sigma(x)^2 - 3*x^2 is a square.

Original entry on oeis.org

4, 6, 28, 45, 48, 60, 156, 208, 360, 496, 1170, 2016, 2520, 2925, 5733, 7605, 8128, 166617, 167580, 380160, 659044, 964080, 1085760, 1539900, 1571328, 1693440, 1778400, 2069613, 2224800, 2306304, 2410200, 2502720, 2522880, 4242420, 4311216, 4840192, 4917744, 4961484, 5331744, 5761536

Offset: 1

Michel Marcus, Jul 09 2025

nonn

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Subsequence, with (x,x,x) triples, of A385531.
Cf. A000203.
Supersequence of A000396.

fQ[x_] := IntegerQ@ Sqrt[DivisorSigma[1, x]^2 - 3*x^2]; Select[Range[2^16], fQ] (* Michael De Vlieger, Jul 09 2025 *)

PARI
```
isok(x) = issquare(sigma(x)^2-3*x^2);
```

cp's OEIS Frontend

User: Michel Marcus

A387216 Numbers that have at least two prime factors (counting multiplicity) congruent to 1 mod 3.

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A386390 Numbers k such that k-1 | sigma+(k) where sigma+ is A107758.

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A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

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A386949 Irregular triangle whose n-th row lists the nonzero terms of the n-th column of A386755.

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A386911 Numbers k such that A064945(k) = A064945(k+1).

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A386871 Least k such that A056100(k) = n or -1 if no such k exists.

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A386436 Smallest k for which A385731(k) = n.

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A386856 Least k such that A056100(k) = 2*n+1 or -1 if no such k exists.

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A386494 Primitive terms of A297305.

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