Kalle Siukola - cp's OEIS frontend

A379938 Numbers k such that the k-th prime is a power of two reversed.

1, 9, 18, 142, 575, 23652, 3633466, 10846595429, 802467018101, 2289255503212477

Offset: 1

Kalle Siukola, Jan 06 2025

			The 9th prime is 23, 23 reversed is 32, and 32 = 2^5, so 9 is a term.

Cf. A000040, A000079, A004087, A004094, A057708, A102385.

import sympy
for (k, p) in enumerate(sympy.primerange(10**8)):
    rev = int(str(p)[::-1])
    # is rev a power of two (or zero)?
    if rev & (rev - 1) == 0:
        print(k + 1, end=",")
print()

A000040(a(n)) = A102385(n).

a(n) = A000720(A102385(n)). - Michel Marcus, Jan 07 2025

a(8)-a(10) from Amiram Eldar, Jan 07 2025

A376898 Positive numbers k such that all the digits in the octal expansion of k^3 are distinct.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 14, 15, 22, 30, 37, 41, 49, 61, 74, 98, 122

Offset: 1

Kalle Siukola, Oct 08 2024

There are no terms >= 2^8 because 2^24-1 is the largest eight-digit octal number.

			11 is a term because 11^3 = 1331 = 2463_8 in octal and no octal digit occurs more than once.

Cf. A007094, A129525, A376897.

Select[Range[2^8],Length[IntegerDigits[#^3,8]]==Length[Union[IntegerDigits[#^3,8]]]&] (* James C. McMahon, Oct 16 2024 *)

for k in range(1, 2**8):
    octal = format(k**3, "o")
    if len(octal) == len(set(octal)): print(k, end=",")

A376897 Positive numbers k such that all the digits in the octal expansion of k^2 are distinct.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 14, 15, 18, 20, 21, 28, 30, 37, 39, 43, 44, 45, 53, 55, 63, 78, 84, 103, 110, 113, 117, 127, 149, 155, 156, 161, 162, 172, 173, 174, 175, 179, 220, 236, 242, 270, 286, 293, 299, 301, 340, 343, 350, 356, 361, 395, 407, 412, 425, 439, 461, 475, 499, 674, 819, 1001, 1211, 1230, 1244, 1323, 1764, 2450, 2751, 3213

Offset: 1

Kalle Siukola, Oct 08 2024

There are no terms >= 2^12 because 2^24-1 is the largest eight-digit octal number.

			110 is in the sequence because 110^2 = 12100 = 27504_8 and no octal digit occurs more than once.

Cf. A007094, A119509, A376898.

Select[Range[2^12], DuplicateFreeQ[IntegerDigits[#^2, 8]] &] (* Michael De Vlieger, Oct 12 2024 *)

for k in range(1, 2**12):
    octal = format(k**2, "o")
    if len(octal) == len(set(octal)): print(k, end=",")

A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

Original entry on oeis.org

4, 6, 8, 9, 22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 537, 737

Offset: 1

Kalle Siukola, Oct 25 2023

There are no terms greater than 999 because the only three-digit prime whose substrings are all primes is 373 (see A085823) and prepending or appending any prime digit to it would create a different three-digit substring.

			237 is included because it is composite and 2, 3, 7, 23 and 37 are all primes.
4 is included because it is composite and has no proper substrings.

Subsequence of A002808.
Cf. A000040.
Cf. A061371, A062115, A202262.
Cf. A033274, A068669, A085823, A279366.

from itertools import combinations
from sympy import isprime
for n in range(2, 1000):
    if not isprime(n):
        properSubstrings = set(
            int(str(n)[start:end]) for (start, end)
            in combinations(range(len(str(n)) + 1), 2)
        ) - set((n,))
        if all(isprime(s) for s in properSubstrings):
            print(n, end=', ')

A307342 Products of four primes, except fourth powers of primes.

Original entry on oeis.org

24, 36, 40, 54, 56, 60, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 390

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) and more than one distinct prime factor.

Numbers n such that bigomega(n) = 4 and omega(n) > 1.

Setwise difference of A014613 and A030514.
Union of A046386, A065036, A085986 and A085987.
Cf. A307682.

Select[Range@ 400, And[! PrimePowerQ@ #, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
Select[Range[400],PrimeOmega[#]==4&&PrimeNu[#]>1&] (* Harvey P. Dale, Aug 27 2021 *)

isok(n) = (bigomega(n)==4) && (omega(n) > 1); \\ Michel Marcus, Apr 03 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) > 1])

A307682 Products of four primes, two of which are distinct.

Original entry on oeis.org

24, 36, 40, 54, 56, 88, 100, 104, 135, 136, 152, 184, 189, 196, 225, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 441, 459, 472, 484, 488, 513, 536, 568, 584, 621, 632, 664, 676, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029

Offset: 1

Kalle Siukola, Apr 21 2019

nonn

Numbers with exactly four prime factors (counted with multiplicity) and exactly two distinct prime factors.
Numbers n such that bigomega(n) = 4 and omega(n) = 2.
Products of a prime and the cube of a different prime (pq^3) together with squares of squarefree semiprimes (p^2*q^2).

Union of A065036 and A085986.
Intersection of A007774 and A067801.
Intersection of A007774 and A195086.
Intersection of A014613 and A067801.
Intersection of A014613 and A195086.
Cf. A307342.

Select[Range@ 1050, And[PrimeNu@ # == 2, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)

isok(n) = (bigomega(n) == 4) && (omega(n) == 2); \\ Michel Marcus, Apr 22 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) == 2])

A307341 Products of four primes, not all distinct.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 414

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) but fewer than four distinct prime factors.

Numbers n such that bigomega(n) = 4 and omega(n) < 4.

Kalle Siukola, Table of n, a(n) for n = 1..10000

Setwise difference of A014613 and A046386.

Union of A030514, A065036, A085986 and A085987.

isok(n) = (bigomega(n) == 4) && (omega(n) < 4); \\ Michel Marcus, Apr 03 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) < 4])

A285508 Numbers with exactly three prime factors, not all distinct.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 343, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452

Offset: 1

Kalle Siukola, Apr 20 2017

Cubes of primes together with products of a prime and the square of a different prime.

Numbers k for which A001222(k) = 3, but A001221(k) < 3. - Antti Karttunen, Apr 20 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Kalle Siukola, Python program

Cf. A001221, A001222.
Setwise difference of A014612 and A007304.
Union of A030078 and A054753.

N:= 1000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/4,2)]): nP:= nops(P):
sort(select(`<=`,[seq(seq(P[i]*P[j]^2,i=1..nP),j=1..nP)],N)); # Robert Israel, Oct 20 2024

Select[Range[452], PrimeOmega[#] == 3 && PrimeNu[#] < 3 &] (* Giovanni Resta, Apr 20 2017 *)

isA285508(n) = ((omega(n) < 3) && (bigomega(n) == 3));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285508(n),write("b285508.txt", k, " ", n);k=k+1));
\\ Antti Karttunen, Apr 20 2017

from sympy import primefactors, primeomega
def omega(n): return len(primefactors(n))
def bigomega(n): return primeomega(n)
print([n for n in range(1, 501) if omega(n)<3 and bigomega(n) == 3]) # Indranil Ghosh, Apr 20 2017 and Kalle Siukola, Oct 25 2023

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A285508(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k**2))-(a<<1)+primepi(isqrt(x//k))-1 for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024

;; With my IntSeq-library.
(define A285508 (MATCHING-POS 1 1 (lambda (n) (and (= 3 (A001222 n)) (< (A001221 n) 3))))) ;; Antti Karttunen, Apr 20 2017

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User: Kalle Siukola

A379938 Numbers k such that the k-th prime is a power of two reversed.

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A376898 Positive numbers k such that all the digits in the octal expansion of k^3 are distinct.

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A376897 Positive numbers k such that all the digits in the octal expansion of k^2 are distinct.

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A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

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A307342 Products of four primes, except fourth powers of primes.

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A307682 Products of four primes, two of which are distinct.

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A307341 Products of four primes, not all distinct.

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A285508 Numbers with exactly three prime factors, not all distinct.

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