Simon R Blow - cp's OEIS frontend

A384468 a(0) = 1; for n >= 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = 2*a(n-1) + n.

1, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 19, 52, 26, 13, 43, 104, 52, 26, 13, 48, 24, 12, 6, 3, 33, 94, 47, 124, 62, 31, 95, 224, 112, 56, 28, 14, 7, 54, 27, 96, 48, 24, 12, 6, 3, 54, 27, 104, 52, 26, 13, 80, 40, 20, 10, 5, 69, 198, 99, 260, 130, 65, 195, 456, 228, 114, 57, 184

Offset: 0

Simon R Blow, May 30 2025

The sequence behaves similarly to the Collatz sequence but introduces a linearly increasing term n when updating odd values.

a(n) = 1 for n = 0, 5, 9, 60843, 19628571, and 772944372 and no other values up to 10^14. - David Radcliffe, Jun 18 2025

			a(2) = 2*3 + 2 = 8 (since a(1) is odd).
a(3) = 8/2 = 4 (since a(2) is even).

Cf. A000079, A006370.

a[0] = 1; a[n_] := a[n] = If[EvenQ[a[n-1]], a[n-1]/2, 2*a[n-1] + n]; Array[a, 100, 0] (* Amiram Eldar, May 30 2025 *)

def generate_sequence(n_terms):
    seq = [1]  # a(0) = 1
    for n in range(1, n_terms):
        prev = seq[-1]
        if prev % 2 == 0:
            seq.append(prev // 2)
        else:
            seq.append(2 * prev + n)
    return seq
seq = generate_sequence(1000)
print(seq)

a(0) = 1.
For n >= 1:
a(n) = a(n-1)/2 if a(n-1) is even;
a(n) = 2*a(n-1) + n if a(n-1) is odd.

A378625 The first subsequence starting at the first digit after the decimal point in the decimal expansion of Pi that is divisible by n.

Original entry on oeis.org

1, 14, 141, 141592, 1415, 141592653589793238, 14, 141592, 141592653, 14159265358979323846264338327950, 141592, 1415926535897932384626433832795028, 14159265358979, 14, 14159265, 1415926535897932384, 141592653589793, 141592653589793238, 141592653589793238462

Offset: 1

Simon R Blow, Dec 02 2024

For every positive integer n there exists a contiguous subsequence of the decimal expansion of Pi that is divisible by n (conjectured).

			a(3) = 141 is the first integer in the sequence that is divisible by 3.
a(6) = 141592653589793238 is the first integer in the sequence that is divisibe by 6.

Cf. A000796, A088143.

Cf. A014777 (if start can move).

from mpmath import mp
mp.dps = 1000000
pi_digits = str(mp.pi)[2:]
def first_divisible(n):
    current_number = 0
    for digit in pi_digits:
        current_number = current_number * 10 + int(digit)
        if current_number % n == 0:
            return current_number
    return None
results = [first_divisible(n) for n in range(1, 21)]
print(results)

a(n) = n * A088143(n). - Alois P. Heinz, Dec 05 2024

A377421 Numbers whose binary reversal is prime and unequal to the original number.

Original entry on oeis.org

6, 10, 11, 12, 13, 14, 20, 22, 23, 24, 25, 26, 28, 29, 34, 37, 40, 41, 43, 44, 46, 47, 48, 50, 52, 53, 55, 56, 58, 61, 62, 67, 68, 71, 74, 77, 80, 82, 83, 86, 88, 91, 92, 94, 96, 97, 100, 101, 104, 106, 110, 112, 113, 115, 116, 121, 122, 124, 131, 134, 136, 142

Offset: 1

Simon R Blow, Oct 27 2024

Contains A080790 and p*2^i for all primes p in A074832 union A080790 and i > 0. - Michael S. Branicky, Oct 29 2024

			6 = 110_2 is a term since reversed it is 011_2 = 3 which is prime.
7 = 111_2 is not a term since base 2 palindromic numbers are not included.

James C. McMahon, Table of n, a(n) for n = 1..10000

Supersequence of A080790.

Cf. A030101, A006995, A074832, A204232.

Select[Range[142],PrimeQ[r=FromDigits[Reverse[IntegerDigits[#,2]],2]]&&r!=#&] (* James C. McMahon, Nov 18 2024 *)

from sympy import isprime
def ok(n): return (b:=bin(n)[2:]) != (br:=b[::-1]) and isprime(int(br, 2))
print([k for k in range(1, 143) if ok(k)]) # Michael S. Branicky, Oct 28 2024

# alternate program constructing terms directly from primes
from sympy import primerange
def auptobits(maxbits):
    alst = []
    for p in primerange(3, 1<Michael S. Branicky, Oct 29 2024

a = A204232 - A006995 (as sets). - Michael S. Branicky, Oct 29 2024

A365762 Greatest common divisor of n and the product of its digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 20, 1, 2, 1, 8, 5, 2, 1, 4, 1, 30, 1, 2, 3, 2, 5, 18, 1, 2, 3, 40, 1, 2, 1, 4, 5, 2, 1, 16, 1, 50, 1, 2, 1, 2, 5, 2, 1, 2, 1, 60, 1, 2, 9, 8, 5, 6, 1, 4, 3, 70, 1, 2, 1, 2, 5, 2, 7, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 90, 1, 2, 3, 2, 5, 6, 1, 2, 9, 100

Offset: 1

Simon R Blow, Sep 18 2023

This sequence will contain all numbers whose prime factors are exclusively <10 (7-smooth numbers, A002473).

			a(11)=1 as 1*1=1; 11 and 1 share 1 as a gcd.
a(15)=5 as 1*5=5; 15 and 5 share 5 as a gcd.
a(10)=10 as 1*0=0; 10 and 0 share 10 as a gcd.

Cf. A066750, A007954, A002473, A214587.

a[n_]:=GCD[n,Product[Part[IntegerDigits[n],i],{i,IntegerLength[n]}]]; Array[a,100] (* Stefano Spezia, Sep 20 2023 *)

a(n) = gcd(n, vecprod(digits(n))); \\ Michel Marcus, Sep 20 2023

from math import gcd, prod
def a(n): return gcd(n, prod(map(int, str(n))))
print([a(n) for n in range(1, 101)])

A364786 We exclude powers of 10 and numbers of the form 11...111 in which the number of 1's is a power of 10. Then a(n) is the smallest number (not excluded) whose trajectory under iteration of "x -> sum of n-th powers of digits of x" reaches 1.

Original entry on oeis.org

19, 7, 112, 11123, 1111222, 111111245666689, 1111133333333335, 1111122333333333333333333346677777777888, 22222222222222222226666668888888, 233444445555555555555555555555555555555555555555555577, 1222222222233333333333333444444444455555555555555556666666666666666666666677778888889

Offset: 1

Simon R Blow, Aug 07 2023

For n!=2, it appears that the first step in the trajectory is always to a power of 10, so that the task would be to find the shortest and lexicographically smallest partition of a power of 10 into parts 1^n,...,9^n.

			a(1) = 19 since 1^1 + 9^1 = 10 and 1^1 + 0^1 = 1.
a(3) = 112 since 1^3 + 1^3 + 2^3 = 10 and 1^3 + 0^3 = 1.

Cf. A007770, A035497, A046519.

a(6), a(8), and a(9) corrected by, and a(10) and a(11) from Jon E. Schoenfield, Aug 10 2023

Definition clarified by N. J. A. Sloane, Sep 15 2023

A364479 Happy palindromic primes.

Original entry on oeis.org

7, 313, 383, 11311, 15451, 30103, 30803, 35053, 36263, 71317, 74047, 94349, 94649, 95959, 98689, 1221221, 1257521, 1262621, 1281821, 1311131, 1444441, 1551551, 1594951, 1597951, 1658561, 1703071, 1737371, 1764671, 1829281, 1924291, 1957591, 1970791, 1981891, 1988891, 3001003

Offset: 1

Simon R Blow, Jul 26 2023

All terms in the sequence are prime numbers that read the same backward and forward. Each term is a happy number, meaning it converges to 1 under the process of repeatedly summing the squares of its digits. The sequence is a subset of both the palindromic prime numbers (A002385) and the happy numbers (A007770).

There are no happy palindromic primes with an even number of digits: every palindromic number with an even number of digits is divisible by 11, so 11 itself is the only palindromic prime, and it is not a happy number.

			313 is a term as it is palindromic (can be reversed), is a prime and is happy: 3^2 + 1^2 + 3^2 = 19, 1^2 + 9^2 = 82, 8^2 + 2^2 = 68, 6^2 + 8^2 = 100, 1^2 + 0^2 + 0^2 = 1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Happy Number.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Intersection of A002385 and A007770.

happyQ[n_] := NestWhile[Plus @@ (IntegerDigits[#]^2) &, n, UnsameQ, All] == 1; Select[Prime[Range[220000]], PalindromeQ[#] && happyQ[#] &] (* Amiram Eldar, Jul 28 2023 *)

def is_prime(num):
    return num > 1 and all(num % i != 0 for i in range(2, int(num ** 0.5) + 1))
def is_palindrome(num):
    return str(num) == str(num)[::-1]
def is_happy(num):
    while num != 1 and num != 4:
        num = sum(int(digit) ** 2 for digit in str(num))
    return num == 1
happy_palindromic_primes = [num for num in range(1, 10000000) if is_prime(num) and is_palindrome(num) and is_happy(num)]
print(happy_palindromic_primes)

from itertools import islice
from sympy import isprime
def A364479_gen(): # generator of terms
    n = 1
    while True:
        for z in (1,3,5,7,9):
            for y in range(z*n, (z+1)*n):
                k, m = y//10, 0
                while k >= 10:
                    k, r = divmod(k, 10)
                    m = 10*m + r
                if isprime(a:=y*n + 10*m + k):
                    b = a
                    while b not in {1,37,58,89,145,42,20,4,16}:
                        b = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(b))
                    if b == 1:
                        yield a
        n *= 10
A364479_list = list(islice(A364479_gen(),20)) # Chai Wah Wu, Aug 02 2023

A362584 Integers k > 1 such that k >= the square of the sum of their prime factors (A074373(k)).

Original entry on oeis.org

243, 256, 270, 288, 300, 320, 324, 336, 360, 375, 378, 384, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 495, 500, 504, 512, 525, 528, 540, 550, 560, 567, 576, 585, 588, 594, 600, 616, 624, 625, 630, 640, 648, 650, 660, 672, 675, 686, 693, 700, 702, 704

Offset: 1

Simon R Blow, Jun 23 2023

nonn

			243 >= A001414(243)^2 = (3+3+3+3+3=15)^2 = 225 so 243 is a term.
800 >= A001414(800)^2 = (2+2+2+2+2+5+5=20)^2 = 400 so 800 is a term.

Simon R Blow, Table of n, a(n) for n = 1..5000

Cf. A001414, A074373.

q:= n-> n>=add(i[1]*i[2], i=ifactors(n)[2])^2:
select(q, [$2..800])[];  # Alois P. Heinz, Jun 23 2023

Select[Range[2, 700], # >= (Plus @@ Times @@@ FactorInteger[#])^2 &] (* Amiram Eldar, Jun 24 2023 *)

isok(n) = ((p=factor(n))[, 1]~*p[, 2])^2 <= n \\ Thomas Scheuerle, Jun 23 2023

cp's OEIS Frontend

User: Simon R Blow

A384468 a(0) = 1; for n >= 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = 2*a(n-1) + n.

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A378625 The first subsequence starting at the first digit after the decimal point in the decimal expansion of Pi that is divisible by n.

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A377421 Numbers whose binary reversal is prime and unequal to the original number.

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A365762 Greatest common divisor of n and the product of its digits.

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A364786 We exclude powers of 10 and numbers of the form 11...111 in which the number of 1's is a power of 10. Then a(n) is the smallest number (not excluded) whose trajectory under iteration of "x -> sum of n-th powers of digits of x" reaches 1.

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A364479 Happy palindromic primes.

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A362584 Integers k > 1 such that k >= the square of the sum of their prime factors (A074373(k)).

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Maple

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