A378157 -id:A378157 - cp's OEIS frontend

A378160 The number of distinct prime factors of !n = A000166(n).

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

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A000166, A001221, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378161, A378162.

Array[PrimeNu[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(omega(s), ", "));}

a(n) = A001221(A000166(n)).

a(n) >= A001221(n-1) + 1 for n >=5.

a(82)-a(88) from Jinyuan Wang, Nov 24 2024

A378161 The number of prime factors of !n = A000166(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 3, 6, 5, 3, 3, 6, 3, 3, 5, 6, 3, 8, 5, 6, 5, 8, 5, 9, 5, 5, 11, 7, 6, 9, 4, 8, 6, 8, 4, 10, 5, 5, 7, 8, 4, 8, 2, 7, 12, 8, 6, 9, 8, 7, 8, 8, 4, 10, 10, 8, 7, 6, 6, 8, 4, 4, 8, 9, 3, 8, 6, 7, 7, 6, 7, 13, 4, 8, 8, 6, 9, 7, 4, 7, 10, 8, 4, 9, 7

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A001222, A000166, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378160, A378162.

Array[PrimeOmega[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(bigomega(s), ", "));}

a(n) = A001222(A000166(n)).

a(n) >= A001222(n-1) + 1 for n >=4, with equality if and only if n is in A301423.

a(82)-a(86) from Jinyuan Wang, Nov 24 2024

A378158 Numbers k such that lpf(!k) < lpf(k-1), where lpf(k) = A020639(k) and !k = A000166(k).

Original entry on oeis.org

20, 38, 42, 60, 90, 104, 108, 110, 114, 132, 138, 152, 164, 170, 174, 192, 194, 198, 240, 242, 258, 284, 294, 324, 338, 350, 360, 368, 390, 398, 434, 438, 450, 462, 482, 488, 500, 504, 510, 522, 524, 528, 542, 548, 564, 570, 588, 600, 602, 614, 618, 632, 642, 644, 648

Offset: 1

Amiram Eldar, Nov 18 2024

nonn

Since (k-1) | !k, we have lpf(!k) <= lpf(k-1). This sequence gives the values of k for which the inequality holds.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000166, A020639, A337986, A378157.

okQ[k_, p_] := Module[{q = 2}, While[q < p && !Divisible[k, q], q = NextPrime[q]]; q < p]; q[k_] := okQ[Subfactorial[k], FactorInteger[k-1][[1, 1]]]; Select[Range[3, 650], q]

ok(k, p) = {my(q = 2); while(q < p && k % q, q = nextprime(q+1)); q < p;}
lista(kmax) = {my(s = 1); for(k = 3, kmax, s = k * s + (-1)^k; if(ok(s, factor(k-1)[1,1]), print1(k, ", ")));}

A378159 The least prime dividing A000255(n); a(1) = 1.

Original entry on oeis.org

1, 3, 11, 53, 3, 13, 11, 3, 1468457, 11, 3, 1373, 34361893981, 3, 17713, 2130617, 3, 11, 13, 3, 11, 617, 3, 37, 11, 3, 179, 14633, 3, 76463, 97, 3, 337, 1049, 3, 11, 7237598635049, 3, 67, 11, 3, 22742406079421034331584846001936724930824184898296683, 11, 3, 13

Offset: 1

Amiram Eldar, Nov 18 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..137

Cf. A000255, A020639, A301423, A378157.

FactorInteger[#][[1, 1]] & /@ Table[Subfactorial[n + 2]/(n + 1), {n, 1, 44}]

lpf(n) = if(n == 1, 1, factor(n)[1,1]);
lista(nmax) = {my(s = 1); for(n = 1, nmax, s = (n * (n+2) * s + (-1)^n)/(n+1); print1(lpf(s), ", "));}

a(n) = A020639(A000255(n)).
a(n) = A000255(n) if and only if n+2 is a term of A301423.
a(3*n-1) = 3.

cp's OEIS Frontend

A378160 The number of distinct prime factors of !n = A000166(n).

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A378161 The number of prime factors of !n = A000166(n), counted with multiplicity.

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Author

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Mathematica

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Extensions

A378158 Numbers k such that lpf(!k) < lpf(k-1), where lpf(k) = A020639(k) and !k = A000166(k).

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Mathematica

PARI

A378159 The least prime dividing A000255(n); a(1) = 1.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula