Mike Jones - cp's OEIS frontend

A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.

2, 3, 25, 7, 11, 169, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Mike Jones, Apr 30 2025

nonn

If x is an integer > 1 and p is a prime divisor of x, then a tower of x subordinate to p is an integer t such that there exists a prime divisor q of x such that q <= p, and t is the highest power of q that is a divisor of x.
If (p-1)!+1 = Product_{k} q_k^(e_k), then a(n) = Product_{k<=n} q_k^(e_k). - Sean A. Irvine, May 05 2025
Let p = prime(n). If m=p. Conjecture: a(n) = p^2 if n = 3, 6 or 103 and a(n) = p otherwise. - Chai Wah Wu, May 11 2025

			a(6) = 169 because the prime factorization of ((13 - 1)! + 1) is 13^2*2834329, and 13^2 = 169.

Cf. A007540, A060371, A383257.

a(n) = my(p=prime(n), x=(p-1)! + 1, f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i, 1] <= p, f[1, 1] = 1)); x/factorback(f); \\ Michel Marcus, Apr 30 2025

from sympy import prime, factorial
def A383578(n):
    p, c = prime(n), 1
    f = factorial(p-1)+1
    a, b = divmod(f,p)
    while not b:
        c *= p
        f = a
        a, b = divmod(f,p)
    return c # Chai Wah Wu, May 12 2025

a(n) = ((prime(n) - 1)! + 1) / A383257(n).

More terms from Michel Marcus, Apr 30 2025

A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.

Original entry on oeis.org

1, 1, 1, 103, 329891, 2834329, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

Offset: 1

Mike Jones, Apr 29 2025

nonn

If x is an integer > 1 and p is a prime divisor of x, then a tower of x subordinate to p is an integer t such that there exists a prime divisor q of x such that q <= p, and t is the highest power of q that is a divisor of x.
If (p-1)!+1 = Product_{k} q_k^(e_k), then a(n) = Product_{k>n} q_k^(e_k). - Sean A. Irvine, May 05 2025
Let p = prime(n) and k = (p-1)!+1. If mChai Wah Wu, May 11 2025

			a(6) = 2834329 because ((13 - 1)! + 1)/w = (12! + 1)/w = (13^2*2834329)/w = 2834329, where w is the product of the towers of ((13 - 1)! + 1) subordinate to 13, w equaling 13^2.

Cf. A060371, A383578.

a(n) = my(p=prime(n), f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i,1] <= p, f[1,1] = 1)); factorback(f); \\ Michel Marcus, Apr 30 2025

from sympy import prime, factorial
def A383257(n):
    p = prime(n)
    f = factorial(p-1)+1
    a, b = divmod(f,p)
    while not b:
        f = a
        a, b = divmod(f,p)
    return f # Chai Wah Wu, May 12 2025

More terms from Michel Marcus, Apr 30 2025

A382117 a(n) = sum (-1)^(((x - 1)*(y - 1))/4), where x and y are coprime positive integers, equidistant from n, such that x <= y.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 1, 4, 1, 4, 1, 4, 0, 6, 0, 8, 0, 6, 1, 8, 2, 10, 1, 8, 0, 12, 1, 12, 0, 8, 1, 16, 2, 16, 0, 12, 0, 18, 0, 16, 0, 12, 1, 20, 0, 22, 1, 16, 1, 20, 0, 24, 0, 18, 0, 24, 2, 28, 1, 16, 0, 30, 2, 32, 0, 20, 1, 32, 2, 24, 1, 24, 0, 36, 0, 36, 2, 24

Offset: 1

Mike Jones, Apr 26 2025

nonn

Sequence inspired by the law of quadratic reciprocity.

			a(4)=2 because (-1)^(((1 - 1)*(7 - 1))/4) + (-1)^(((3 - 1)*(5 - 1))/4) = (-1)^0 + (-1)^2 = 1 + 1 = 2.

Cf. A309812 (indices of zeros).

a(n) = sum(x=1, n, my(y=2*n-x); if ((x<=y) && (gcd(x, y)==1), (-1)^(((x-1)*(y-1))/4))); \\ Michel Marcus, Apr 27 2025

More terms from Alois P. Heinz, Apr 26 2025

A376158 Numbers k having two prime divisors p < q such that p! <= k <= q!.

Original entry on oeis.org

6, 10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 56, 57, 58, 60, 62, 63, 66, 68, 69, 70, 74, 75, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 129, 130, 132, 134, 136, 138, 140, 141, 142, 145

Offset: 1

Mike Jones, Sep 12 2024

nonn

Different from A330136.

			40 is in the list because 40 has at least 2 distinct prime divisors, and the smallest prime divisor of 40 is 2 and the largest prime divisor of 40 is 5, and 2! <= 40 <= 5! because 2! = 2 and 5! = 120.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A020639, A006530, A330136.

q:= n-> (s-> nops(s)>1 and min(s)!<=n and n<=max(s)!)(numtheory[factorset](n)):
select(q, [$2..150])[];  # Alois P. Heinz, Sep 20 2024

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && k >= p[[1]]! && k <= p[[-1]]!]; Select[Range[125], q] (* Amiram Eldar, Sep 20 2024 *)

A375337 a(n) = binomial(prime(n), phi(prime(n) + 1)).

Original entry on oeis.org

1, 3, 10, 35, 330, 1716, 12376, 75582, 490314, 4292145, 300540195, 17672631900, 7898654920, 960566918220, 1503232609098, 64617565719070, 109712808959985, 232714176627630544, 13413576695470557606, 5300174441392685400, 873065282167813104916, 13146145590943010676030

Offset: 1

Mike Jones, Aug 12 2024

nonn

			a(4) = 35 because binomial(prime(4), phi(prime(4) + 1)) = binomial(7, phi(8)) = binomial(7, 4) = 35.

Cf. A007318, A008331.

Map[Binomial[#, EulerPhi[# + 1]] &, Prime[Range[22]]] (* Amiram Eldar, Aug 13 2024 *)

a(n) << 2^p/sqrt(p), where p = prime(n). - Charles R Greathouse IV, Aug 12 2024

A375117 Irregular triangle of positive integers, read by rows, the elements of the n-th row being the nonzero remainders, in increasing order, when the Euclidean algorithm is applied to 2^n-1 and n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 7, 1, 2, 7, 1, 3, 1, 3, 1, 1, 2, 3, 1, 7, 1, 15, 1, 9, 1, 5, 15, 7, 1, 3, 1, 3, 6, 9, 15, 1, 6, 1, 2, 3, 1, 2, 25, 1, 2, 13, 15, 1, 3, 1, 1, 31, 1, 2, 5, 7, 1, 3, 1, 17, 9, 27, 1, 1, 2, 3, 1, 3, 4, 7, 5, 10, 15, 1, 21, 1, 1, 14, 15, 1, 3, 13, 16

Offset: 2

Mike Jones, Jul 30 2024

			The triangle begins:
    1;
    1;
    1, 3;
    1;
    3;
    1;
    1, 7;
    1, 2, 7;
    1, 3;
    1;
    3;
    1;
    3;
    ...
Row(2) is {1}, because 2^2-1 = 4-1 = 3, and 3 divided by 2 leaves a remainder of 1.
Row(4) is {1, 3}, because 2^4-1 = 16-1 = 15, and 15 divided by 4 leaves a remainder of 3, and 4 divided by 3 leaves a remainder of 1.

Cf. A049816, A014491.

row(n) = my(x=2^n-1, y=n, ok=1, list=List()); while (ok, my(z=divrem(x, y)); x = y; y = z[2]; if (y==0, ok=0, listput(list, y));); listsort(list); Vec(list); \\ Michel Marcus, Jul 31 2024

More terms from Michel Marcus, Jul 31 2024

A372781 Odd numbers k such that A001221(k) < A001221(A003958(k)).

Original entry on oeis.org

7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 121, 127, 129, 131, 137, 139, 143, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197, 199, 201, 203, 209, 211, 213, 215, 217, 223

Offset: 1

Mike Jones, Jul 04 2024

nonn

			31 is in the sequence because 31 = 31^1, so omega(31) = 1, but (31 - 1)^1 = 30^1 = 2^1 * 3^1 * 5^1, so omega(30) = 3, and 1 < 3.

Cf. A001221, A003958, A005408, A138889.

q:= n-> (f-> n::odd and f(n) nops(ifactors(k)[2])):
select(q, [$1..333])[];  # Alois P. Heinz, Jul 04 2024

f[p_, e_] := (p - 1)^e; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[3, 1000, 2], PrimeNu[#] < PrimeNu[s[#]] &] (* Amiram Eldar, Jul 04 2024 *)

A362239 Primes such that all composite numbers up to the next prime have the same number of distinct prime divisors.

Original entry on oeis.org

2, 3, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 71, 97, 101, 107, 137, 149, 157, 179, 191, 197, 223, 227, 239, 269, 281, 311, 347, 419, 431, 461, 499, 521, 569, 599, 617, 641, 643, 659, 673, 739, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151

Offset: 1

Mike Jones, Apr 12 2023

nonn

			19 is a term because 19 is a prime and each of the composite numbers up to the next prime (20, 21, and 22) has exactly 2 distinct prime divisors.

A001359 is a subsequence.

Cf. A001221 (omega).

q[p_] := Length[Union[Table[PrimeNu[c], {c, Range[p + 1, NextPrime[p] - 1]}]]] <= 1; Select[Prime[Range[200]], q] (* Amiram Eldar, May 18 2023 *)

isok(p)=if(isprime(p), my(q=nextprime(p+1), t=omega(p+1)); for(i=p+2, q-1, if(omega(i)<>t, return(0))); 1, 0) \\ Andrew Howroyd, Apr 12 2023

More terms from Andrew Howroyd, Apr 12 2023

A356137 Positive integers m such that the fractional part of the geometric mean of the sequence s(m) does not exceed the fractional part of the arithmetic mean of s(m), where s(m) is the sequence 1 + 1/1, 2 + 1/2, ..., m + 1/m.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 13, 14, 16, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 48, 54, 56, 61, 62, 64, 69, 70, 72, 78, 80, 86, 88, 92, 94, 96, 100, 102, 108, 110, 115, 116, 118, 124, 126, 132, 134, 138, 140, 146, 148, 154, 156, 161, 162, 164, 170, 172, 178, 180

Offset: 1

Mike Jones, Jul 27 2022

nonn

The idea is to take note of when the fractional parts of the geometric mean and arithmetic mean "follow suit" with respect to the celebrated geometric mean <= arithmetic mean inequality.

			2 is a term because the geometric mean of 1 + 1/1 and 2 + 1/2 is the geometric mean of 2 and 2.5, which is a bit less than 2.24, whereas the arithmetic mean of 2 and 2.5 is 2.25, and 0.24 <= 0.25.
4 is not a term because the geometric mean is 2.90..., whereas the arithmetic mean is 3.02..., and 0.90 > 0.02.

Wikipedia, Fractional part
Wikipedia, Inequality of arithmetic and geometric means

Cf. A356142/A102928 (the arithmetic mean of s(n)).

max=180; a={}; s[m_]:=m+1/m; For[m=1,m<=max,m++,If[FractionalPart[Mean[Table[s[k],{k,m}]]] >= FractionalPart[GeometricMean[Table[s[k],{k,m}]]],AppendTo[a,m]]]; a (* Stefano Spezia, Jul 27 2022 *)

isok(m) = my(v=vector(m, k, k+1/k)); frac(sqrtn(vecprod(v), m)) <= frac(vecsum(v)/m); \\ Michel Marcus, Jul 28 2022

More terms from Stefano Spezia, Jul 27 2022

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User: Mike Jones

A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.

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A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.

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A382117 a(n) = sum (-1)^(((x - 1)*(y - 1))/4), where x and y are coprime positive integers, equidistant from n, such that x <= y.

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A376158 Numbers k having two prime divisors p < q such that p! <= k <= q!.

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A375337 a(n) = binomial(prime(n), phi(prime(n) + 1)).

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A375117 Irregular triangle of positive integers, read by rows, the elements of the n-th row being the nonzero remainders, in increasing order, when the Euclidean algorithm is applied to 2^n-1 and n.

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PARI

Extensions

A372781 Odd numbers k such that A001221(k) < A001221(A003958(k)).

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Maple

Mathematica

A362239 Primes such that all composite numbers up to the next prime have the same number of distinct prime divisors.

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