Freddy Barrera - cp's OEIS frontend

A371862 Positive integers that can be written as the product of two or more other integers, none of which uses any of the digits in the number itself.

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 49, 52, 54, 56, 57, 58, 60, 63, 64, 66, 68, 69, 70, 72, 76, 78, 80, 81, 84, 86, 87, 88, 90, 96, 98, 99, 100, 102, 104, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120

Offset: 1

Bernardo Recamán and Freddy Barrera, Apr 09 2024

Infinite since 10^k = 2^k * 5^k and 10^k - 1 = 3^2 * (10^k - 1)/9 are terms for k > 0 and are the smallest (k+1)-digit and largest k-digit terms, resp. All repdigits consisting of 4's, 6's, 8's, or 9's are also terms. - Michael S. Branicky, Apr 09 2024
No number ending in 5 is a term. - Jon E. Schoenfield, Apr 09 2024
Terms are composite. If 9 consecutive positive integers are terms then they are between two consecutive primes at least 14 apart. - David A. Corneth, Apr 10 2024
All products of x (repdigits consisting of 3's or 6's) and 10x + 7 are terms. - Ivan N. Ianakiev, Apr 11 2024

			60 is a term because it can be expressed as 4 * 15, avoiding its own digits 6 and 0. 50 isn't because there is no way of expressing 50 avoiding both 5 and 0.
112 is a term since 112 = 4*4*7 and is the first term requiring a product with three factors.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Bernardo Recamán Santos, Melissa's Numbers, Puzzling Stack Exchange, Mar 13 2024.

Cf. A001055, A002808, A162247.

from sympy import divisors, isprime
from functools import cache
@cache
def ok(n, avoid=tuple()):
    if avoid == tuple(): avoid = set(str(n))
    else: avoid = set(avoid)
    if n%10 == 5 or len(avoid) == 10 or isprime(n): return False
    for d in divisors(n)[1:-1]:
        if set(str(d)) & avoid == set():
            if set(str(n//d)) & avoid == set(): return True
            if ok(n//d, tuple(sorted(avoid))): return True
    return False
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Apr 10 2024

A357474 Squarely correct numbers.

Original entry on oeis.org

1, 4, 9, 11, 14, 16, 19, 25, 36, 41, 44, 49, 64, 81, 91, 94, 99, 100, 111, 114, 116, 119, 121, 125, 136, 141, 144, 149, 161, 164, 169, 181, 191, 194, 196, 199, 225, 251, 254, 256, 259, 289, 324, 361, 364, 369, 400, 411, 414, 416, 419, 425, 436, 441, 444, 449, 464

Offset: 1

Freddy Barrera, Sep 29 2022

A positive integer is a squarely correct number if its base-10 representation consists entirely of one or more adjacent blocks of digits that are positive perfect squares with no leading zeros. (See George Berzsenyi link below.)

			14401 is not a term, but 14411 is.

George Berzsenyi, Consecutively and Squarely Correct (P439), Math Horizons, Vol. 30, Issue 1, The Playground, August 2022.

Cf. A000290, A036435, A018851.

is(n)=if(n<11, issquare(n), n%10, my(d=digits(n)); for(i=1,#d, if(issquare(n%10^i) && d[#d+1-i] && is(n\10^i), return(1))); 0, my(k=valuation(n,10)); k%2==0 && is(n/10^k)) \\ Charles R Greathouse IV, Oct 04 2022

from sympy.ntheory.primetest import is_square
def ok(n):
    if is_square(n): return True
    s = str(n)
    return any(s[i]!="0" and ok(int(s[:i])) and ok(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1, 465) if ok(k)]) # Michael S. Branicky, Oct 03 2022

5n/3 < a(n) << n^k for n > 1, where k = log 10/log 3 = 2.0959.... - Charles R Greathouse IV, Oct 03 2022

A348300 a(n) is the largest number that is the digit sum of the square of an n-digit number.

Original entry on oeis.org

13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297, 310, 331, 343, 360, 378, 396

Offset: 1

Bernardo Recamán and Freddy Barrera, Oct 10 2021

18*n-a(n) appears to be nondecreasing. - Chai Wah Wu, Nov 18 2021
According to new data 18*n-a(n) sometimes decreases. - David A. Corneth, Feb 21 2024
a(n) is the digit sum of the square of the last n-digit integer in A067179. - Zhao Hui Du, Mar 04 2024
a(n) appears to be approximately equal to 16.5*n. - Zhining Yang, Mar 12 2024
a(n) modulo 9 is either 0, 1, 4 or 7. - Chai Wah Wu, Apr 04 2024

			a(3) = 46 because 46 is the largest digital sum encountered among the squares (that of 937) of all 3-digit numbers. Such maximal digital sum can be achieved by more than one square (squares of 836 and 883 also have digital sum 46). Largest of these is A348303.

Cf. A004159, A348303, A370522.

Cf. A371728.

Array[Max@ Map[Total@ IntegerDigits[#^2] &, Range[10^(# - 1), 10^# - 1]] &, 8] (* Michael De Vlieger, Oct 12 2021 *)

def A348300(n): return max(sum(int(d) for d in str(m**2)) for m in range(10**(n-1),10**n)) # Chai Wah Wu, Jun 26 2024

def A348300(n):
    return max(sum((k^2).digits()) for k in (10^(n-1)..10^n-1))

a(n) = Max_{k=10^(n-1)..10^n-1} A004159(k).

a(11) from Chai Wah Wu, Nov 18 2021
a(12)-a(13) from Martin Ehrenstein, Nov 20 2021
a(14)-a(24) from Zhao Hui Du, Feb 23 2024
Name edited by Jon E. Schoenfield, Mar 10 2024

A348303 a(n) is the largest n-digit number whose square has a digital sum equal to A348300(n).

Original entry on oeis.org

7, 83, 937, 9417, 94863, 987917, 9893887, 99483667, 994927133, 9486778167, 99497231067, 999949483667, 9892825177313

Offset: 1

Bernardo Recamán and Freddy Barrera, Oct 10 2021

Cf. A004159, A348300.

Array[#1 + Position[#2, Max[#2]][[-1, -1]] - 1 & @@ {#1, Map[Total@ IntegerDigits[#^2] &, Range[#1, #2]]} & @@ {10^(# - 1), 10^# - 1} &, 8] (* Michael De Vlieger, Oct 12 2021 *)

a(11) from Chai Wah Wu, Nov 18 2021

a(12)-a(13) from Martin Ehrenstein, Nov 20 2021

A340486 Number of unlabeled graphs on n vertices whose independence number is equal to clique number.

Original entry on oeis.org

1, 0, 2, 3, 8, 54, 380, 3462, 83606, 4822240

Offset: 1

Freddy Barrera, Jan 09 2021

			a(3) = 2 because, of the four graphs of order three, only the path graph and its complement have equal independence and clique number, 2 in this case.

Eric Weisstein's World of Mathematics, Clique number.
Eric Weisstein's World of Mathematics, Independence Number.

A306841 Number of rectangles of integer sides whose area or perimeter is n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 4, 2, 4, 1, 6, 1, 5, 2, 7, 1, 7, 1, 8, 2, 7, 1, 10, 2, 8, 2, 10, 1, 11, 1, 11, 2, 10, 2, 14, 1, 11, 2, 14, 1, 14, 1, 14, 3, 13, 1, 17, 2, 15, 2, 16, 1, 17, 2, 18, 2, 16, 1, 21, 1, 17, 3, 20, 2, 20, 1, 20, 2, 21, 1, 24, 1, 20, 3, 22, 2, 23

Offset: 1

Freddy Barrera, Mar 12 2019

nonn

			a(4) = 3 because there are two rectangles of integer sides of area 4 (2 X 2 and 1 X 4) and one rectangle of integer sides of perimeter 4 (1 X 1).

MathOverflow, Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N

Cf. A038548 (area n), A004526 (perimeter 2n).

a(n) = ceiling(d(n)/2) + floor(n/4) if n is even, a(n) = ceiling(d(n)/2) otherwise, where d(n) is the number of divisors of n.

A306488 Number of ways of expressing n as a + b + c, with a, b, and c positive integers, gcd(a, b) = 1, but gcd(a, c) and gcd(b, c) both greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 4, 0, 1, 0, 9, 0, 7, 1, 4, 1, 15, 0, 13, 1, 4, 2, 16, 0, 24, 4, 10, 1, 29, 0, 32, 4, 5, 3, 41, 0, 38, 2, 17, 6, 54, 1, 43, 6, 26, 10, 70, 0, 65, 9, 20, 11, 68, 1, 86, 14, 35, 2, 99, 1, 99, 15, 18, 16, 104, 1, 125, 10, 53, 19, 134, 0, 114, 21, 58

Offset: 0

Freddy Barrera, Feb 18 2019

nonn

			a(11) = 1 because of the ten partitions of 11 into three parts, only 6 + 3 + 2 satisfies the conditions. But a(210) = 0, because 210 does not have any partition that satisfies the conditions.

F. Barrera, B. Recamán and S. Wagon, Problem 12044, Amer. Math. Monthly 125 (2018), p. 466.

Freddy Barrera, Table of n, a(n) for n = 0..1000

a[n_] := Length@ Select[ IntegerPartitions[ n, {3}], (t = Sort[GCD @@@ Subsets[#, {2}]]; t[[1]] == 1 && t[[2]] > 1 && t[[3]] > 1) &]; a /@ Range[0, 87] (* Giovanni Resta, Feb 20 2019 *)

def a(n):
    if n < 3: return 0
    r = 0
    t = [False, True, True]
    for p in Partitions(n, length=3, min_part=2, max_slope=-1):
        s = sorted(gcd(a, b) > 1 for a, b in Subsets(p, 2))
        r += int(s == t)
    return r
[a(n) for n in (0..100)]

A261573 A variation of Recamán's sequence A005132: Define a(0) = 0, and for n > 0, a(n) = a(n-1) - (n+2) if positive and not already in the sequence, otherwise a(n) = a(n-1) + (n+2).

Original entry on oeis.org

0, 3, 7, 2, 8, 1, 9, 18, 28, 17, 5, 18, 4, 19, 35, 52, 34, 15, 35, 14, 36, 13, 37, 12, 38, 11, 39, 10, 40, 71, 103, 70, 104, 69, 33, 70, 32, 71, 31, 72, 30, 73, 29, 74, 120, 167, 119, 168, 118, 67, 119, 66, 120, 65, 121, 64, 6, 65, 125, 186, 124, 61, 125, 60, 126, 59, 127, 58, 128, 57

Offset: 0

Freddy Barrera, Aug 24 2015

nonn

As in Recamán's sequence, terms are repeated, the first being 18 = a(7) = a(11).
More generally, for k >= 0, a_k(0) = 0, and for n > 0, a_k(n) = a_k(n-1) - (n+k) if positive and not already in the sequence, otherwise a_k(n) = a_k(n-1) + (n+k).
For k = 0, this is Recamán's sequence A005132.

Freddy Barrera, Table of n, a(n) for n = 0..10000
Freddy Barrera, Line plot of the first 1000 terms.
Freddy Barrera, Scatter plot of the first 1000 terms.
Freddy Barrera, Line plot of the first 1000 terms when k = 0, 1, 2, ..., 9
Freddy Barrera, Scatter plot of the first 1000 terms when k = 0, 1, 2, ..., 9

Cf. A005132, A057167.

f[s_List] := Block[{a = s[[-1]], len = Length@ s}, Append[s, If[a > len + 1 && ! MemberQ[s, a - len - 2], a - len - 2, a + len + 2]]]; Nest[f, {0}, 70] (* Robert G. Wilson v, Sep 08 2015 *)

def sequence(n, k):
    """For n > 0 and k >= 0, generates the first n terms of the sequence"""
    A, a = {0}, 0
    yield a
    for n in range(1, n + 1):
        a = a - (n + k)
        if a > 0 and a not in A:
            A.add(a)
            yield a
        else:
            a = a + 2 * (n + k)
            A.add(a)
            yield a
# List of the first 1000 terms of the sequence with k = 2.
list(sequence(1000, 2))

cp's OEIS Frontend

User: Freddy Barrera

A371862 Positive integers that can be written as the product of two or more other integers, none of which uses any of the digits in the number itself.

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A357474 Squarely correct numbers.

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A348300 a(n) is the largest number that is the digit sum of the square of an n-digit number.

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A348303 a(n) is the largest n-digit number whose square has a digital sum equal to A348300(n).

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A340486 Number of unlabeled graphs on n vertices whose independence number is equal to clique number.

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A306841 Number of rectangles of integer sides whose area or perimeter is n.

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A306488 Number of ways of expressing n as a + b + c, with a, b, and c positive integers, gcd(a, b) = 1, but gcd(a, c) and gcd(b, c) both greater than 1.

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A261573 A variation of Recamán's sequence A005132: Define a(0) = 0, and for n > 0, a(n) = a(n-1) - (n+2) if positive and not already in the sequence, otherwise a(n) = a(n-1) + (n+2).

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