Arkadiusz Wesolowski - cp's OEIS frontend

A383267 Decimal expansion of (4/11)^(1/3).

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

Offset: 0

Arkadiusz Wesolowski, Apr 21 2025

In the standard cosmology, the temperature of the free-streaming neutrinos which formed the cosmic neutrino background is (4/11)^(1/3) of the relic photon temperature after the electron-positron annihilation in the early universe (assuming that all electrons and positrons annihilated into photons).

			0.713765855503608170671899991762661247590796547589038066915626752084583...

E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley, Redwood City, CA, 1990, p. 503 Appendix A.
R. E. Lopez, S. Dodelson, A. Heckler and M. S. Turner, Precision detection of the cosmic neutrino background, Physical Review Letters 82 (1999) 3952-3955, p. 3952.
Steven Weinberg, Gravitation and Cosmology Principles and Applications of the General Theory of Relativity, John Wiley, New York, 1972, p. 537.

Wikipedia, Cosmic neutrino background.

Cf. A111728.

SetDefaultRealField(RealField(106)); n:=(4/11)^(1/3); Reverse(Intseq(Floor(10^105*n)));

RealDigits[(4/11)^(1/3),10,105][[1]] (* Stefano Spezia, Apr 25 2025 *)

PARI
```
(4/11)^(1/3)
```

Equals 1/A111728 = A005480/A010583.

A375459 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squared primes, or 0 if no such magic square exists.

Original entry on oeis.org

21, 18, 15, 51, 195, 435, 76035

Offset: 0

Arkadiusz Wesolowski, Aug 15 2024

a(9) = 0.
a(n) = A268855(n+1) for n = 0, 1, 2, and 5.
a(7) and a(8) - if nonzero, they are greater than 10^10.

Carlos Rivera, Puzzle 79. The Chebrakov's - by two months - challenge, The Prime Puzzles and Problems Connection.
Arkadiusz Wesolowski, Examples of these magic squares
Index entries for sequences related to magic squares

Cf. A001248, A268855, A375361.

A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

Original entry on oeis.org

25, 65, 75, 85, 125, 145, 169, 175, 185, 195, 205, 221, 225, 255, 265, 275, 289, 305, 325, 365, 375, 377, 425, 435, 445, 455, 475, 481, 485, 493, 505, 507, 525, 533, 545, 555, 565, 575, 585, 595, 615, 625, 629, 663, 675, 685, 689, 697, 715, 725, 745, 765, 775

Offset: 1

Arkadiusz Wesolowski, Aug 13 2024

nonn

Odd numbers k such that k^2 can be expressed as the arithmetic mean of two distinct perfect squares in more than one way. For example, 25^2 = (5^2 + 35^2)/2 = (17^2 + 31^2)/2.

Let x be a squared integer which is the central element of a 3 X 3 magic square in which seven (or more) of the entries are squared integers. If the greatest common divisor of all nine entries is 1, then the square root of x is a composite number that is divisible only by primes congruent to 1 mod 4. For example, sqrt(A221669(5)) = 425 is both in A004613 and in this sequence.

			65 is in this sequence because 65 has two prime factors of the form 4*k + 1, namely 5 = 4*1 + 1 and 13 = 4*3 + 1.

Cf. A004613, A008844, A221669, A221670, A268855, A309812, A375459.

f:=func; nopf:=func; sum:=func; [n: n in [1..775 by 2] | sum(n) gt 1];

isok(n) = my(v=Vec(factor(n))); n%2&&sum(t=1, omega(n), if((v[1]%4)[t]==1, v[2][t]))>1;

isok(n) = my(t); if(n%2, for(k=sqrtint(n^2-1)+2, sqrtint(2*n^2-1), if(issquare(2*n^2-k^2)&&t++>1, return(1)))); 0;

A368340 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.

Original entry on oeis.org

3, 1, 5, 17, 19, 7, 15, 1, 17, 9, 197, 49, 51, 127, 11, 577, 35, 1, 37, 721, 13, 199, 24335, 97, 99, 649, 485, 15, 19603, 31, 63, 1, 65, 33, 251, 17, 3699, 57799, 53, 161, 163, 55, 10405, 77617, 19, 1151, 2143295, 4801, 99, 1, 101, 5201, 32080051, 1351, 21, 127

Offset: 1

Arkadiusz Wesolowski, Dec 21 2023

			For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).

Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.

Cf. A000217, A002350, A069129, A368339.

pellsolve(n)={if(issquare(n/2), return(1), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(x))};

a(n) = A002350(8*n).
a(n) = sqrt(8*n*A368339(n)^2 + 1).
a(A000217(n)) = 2*n + 1, n >= 1.

A368339 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

Original entry on oeis.org

1, 0, 1, 3, 3, 1, 2, 0, 2, 1, 21, 5, 5, 12, 1, 51, 3, 0, 3, 57, 1, 15, 1794, 7, 7, 45, 33, 1, 1287, 2, 4, 0, 4, 2, 15, 1, 215, 3315, 3, 9, 9, 3, 561, 4137, 1, 60, 110532, 245, 5, 0, 5, 255, 1557945, 65, 1, 6, 48, 455, 14127, 11, 11, 207480, 20, 29427, 285, 1

Offset: 1

Arkadiusz Wesolowski, Dec 21 2023

			For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).

Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.

Cf. A000217, A002349, A268856, A368340.

pellsolve(n)={if(issquare(n/2), return(0), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(y))};

a(n) = A002349(8*n).
a(n) = sqrt((A368340(n)^2 - 1)/(8*n)).
a(A000217(n)) = 1, n >= 1.

A364413 Odd numbers m such that for every k >= 1, m*2^k + 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

Original entry on oeis.org

189035277393779, 212050850472529, 618127765127603, 777947701660121, 1171304921532749, 1358735367828947, 1834310020939021, 2357654372323739, 2638037471052913, 3025664372930897, 3935005074246167, 4688754513654559, 4996748200142999, 5425272498782051, 5455203077891285

Offset: 1

Arkadiusz Wesolowski, Jul 23 2023

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34560

Cf. A076336, A076335, A180247, A291360, A364412.

For n > 34560, a(n) = a(n-34560) + 10014447295554878022.

A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

Original entry on oeis.org

144323411864333, 175321252530209, 190779128601685, 316031956469111, 389882208980861, 450590081221877, 2420018284798363, 2715458757443051, 3161282469971861, 3366332338600025, 3643757921262355, 4380746955320089, 4409682697067321, 5089175909950511, 5281690092088615

Offset: 1

Arkadiusz Wesolowski, Jul 23 2023

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34560

Cf. A101036, A076335, A180247, A291360, A364413.

For n > 34560, a(n) = a(n-34560) + 10014447295554878022.

A364186 Primes p such that p divides 2^((p-1)/x) - 1, where x is the smallest odd prime factor of p - 1.

Original entry on oeis.org

31, 43, 109, 127, 157, 223, 229, 251, 277, 283, 307, 397, 431, 433, 439, 457, 499, 601, 641, 643, 691, 727, 733, 739, 811, 911, 919, 953, 971, 997, 1013, 1021, 1051, 1069, 1093, 1103, 1163, 1181, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1709

Offset: 1

Arkadiusz Wesolowski, Jul 15 2023

nonn

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Supersequence of A014752.

Cf. A059858, A270802, A360652.

[p: p in PrimesUpTo(1709) | #Factorization(p-1) ge 2 and Modexp(2, Truncate((p-1)/Factorization(p-1)[2][1]), p) eq 1];

forprime(p=2, 1709, v=Vec(factor(p-1))[1]; if(#v>1, t=0; e=v[2]; x=floor(p^(1/e))+1; until(x==p||t==2, if(Mod(x, p)^e==2, t++); x++); if(t==2, print1(p, ", "))));

isok(p) = my(v=Vec(factor(p-1))[1]); isprime(p) && #v>1 && Mod(2, p)^((p-1)/v[2])==1;

A363286 Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 97, 101, 103, 107, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 313, 317, 347, 349, 359, 367, 373, 379, 383, 389, 401, 409, 419

Offset: 1

Arkadiusz Wesolowski, May 25 2023

nonn

An odd prime p belongs to this sequence if and only if A001917(A000720(p)) is equal to 1 or 2.

Cf. A001917, A014664.

[p: p in [3..419 by 2] | IsPrime(p) and (p-1)/Modorder(2, p) le 2];

isok(p) = p%2 && isprime(p) && (p-1)/znorder(Mod(2, p))<=2;

from itertools import islice
from sympy import nextprime, n_order
def A363286_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue,3)-1
    while (p:=nextprime(p)):
        if n_order(2,p)<<1 >= p-1:
            yield p
A363286_list = list(islice(A363286_gen(),30)) # Chai Wah Wu, Jul 17 2023

a(n) ~ (3/2)*n*log((3/2)*n).

A361900 Numbers k such that 3153479820268467961^22^k + 1 is prime.

Original entry on oeis.org

600, 810, 1074, 7974, 22290, 43086

Offset: 1

Arkadiusz Wesolowski, Mar 28 2023

Let p be a prime number of the form 3*153479820268467961^2*2^k + 1 with k > 0, then the multiplicative order of 2 modulo p is not of the form 2^(m+1), m >= 0. Hence, p does not divide any Fermat number F(m) = 2^(2^m) + 1.

Cf. A000215, A229852, A351332, A361898, A361899.

Select[Range[2, 10^4, 2], PrimeQ[3*153479820268467961^2*2^# + 1] &]

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User: Arkadiusz Wesolowski

A383267 Decimal expansion of (4/11)^(1/3).

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A375459 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squared primes, or 0 if no such magic square exists.

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A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

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Author

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Comments

Examples

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Programs

Magma

PARI

PARI

A368340 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.

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Author

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PARI

Formula

A368339 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A364413 Odd numbers m such that for every k >= 1, m*2^k + 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

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Author

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Links

Crossrefs

Formula

A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

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Author

Keywords

Links

Crossrefs

Formula

A364186 Primes p such that p divides 2^((p-1)/x) - 1, where x is the smallest odd prime factor of p - 1.

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Author

Keywords

References

Crossrefs

Programs

Magma

PARI

PARI

A363286 Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.

Views

A368340 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.

A368339 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

A361900 Numbers k such that 3153479820268467961^22^k + 1 is prime.