A345397 -id:A345397 - cp's OEIS frontend

A345497 Numbers that are the sum of eight squares in ten or more ways.

70, 71, 73, 74, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			71 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 8^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 4^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
   = 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 4^2 + 4^2
so 71 is a term.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A025427, A025429, A345487, A345496, A345540, A346803.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 8):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 10])
    for x in range(len(rets)):
        print(rets[x])

def A345397(n): return (70, 71, 73, 74, 77, 78, 79, 80, 82, 83)[n-1] if n<11 else n+74 # Chai Wah Wu, May 09 2024

From Chai Wah Wu, May 09 2024: (Start)
All integers >= 85 are terms. Proof: since 594 can be written as the sum of 3 positive squares in 10 ways (see A025427) and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 628 can be written as a sum of 8 positive squares in 10 or more ways. Integers from 85 to 627 are terms by inspection.
a(n) = 2*a(n-1) - a(n-2) for n > 12.
G.f.: x*(-x^11 + x^10 - x^9 + x^8 - 2*x^5 + 2*x^4 - x^3 + x^2 - 69*x + 70)/(x - 1)^2. (End)

A353139 Digitally balanced numbers (A031443) whose squares are also digitally balanced.

Original entry on oeis.org

212, 781, 794, 806, 817, 838, 841, 844, 865, 2962, 3101, 3130, 3171, 3178, 3185, 3213, 3219, 3226, 3269, 3274, 3335, 3353, 3354, 3356, 3370, 3378, 3490, 3496, 3521, 3528, 3595, 3597, 3606, 3610, 3626, 3651, 3672, 3718, 3777, 11797, 11798, 11850, 11938, 12049

Offset: 1

Alex Ratushnyak, Apr 26 2022

Numbers x such that both x and x^2 are terms of A031443, that is, have the same number of 0's as 1's in their binary representations.

Cf. A031443, A345397.

balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[12000], balQ[#] && balQ[#^2] &] (* Amiram Eldar, Apr 26 2022 *)

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
def A031443gen(): yield from (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1)))
def agen():
    for k in A031443gen():
        if isbalanced(k**2):
            yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Apr 26 2022

A345396 a(n) is the least prime p such that {p, p^2, ..., p^n} are all digitally balanced numbers in base 2 (A031443).

Original entry on oeis.org

2, 12049, 52673, 937253, 1000099, 3844790441, 62911443401, 1052000152157

Offset: 1

Amiram Eldar, Jun 17 2021

a(9) > 2.5 * 10^12, if it exists.

			a(1) = 2 since 2 is digitally balanced: its binary representation, 10, has the same number of 0's and 1's.
a(2) = 12049 since both 12049 and 12049^2 are digitally balanced: the binary representation of 12049, 10111100010001, has 7 0's and 7 1's, and the binary representation of 12049^2, 1000101001110011111100100001, has 14 0's and 14 1's.

Subsequence of A031443, A066196 and A345395.

Cf. A345397.

balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; f[p_] := Module[{e = 0, r = p}, While[balQ[r], r *= p; e++]; e]; mx = 5; s = Table[0, {mx}]; c = 0; p = 2; While[c < mx, k = f[p]; Do[If[s[[i]] == 0, s[[i]] = p; c++], {i, 1, k}]; p = NextPrime[p]]; s

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
def A066196gen():
    yield from filter(isprime, (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1))))
def a(n):
    for p in A066196gen():
        if all(isbalanced(p**i) for i in range(2, n+1)):
            return p
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, May 15 2022

A353140 Digitally balanced numbers (A031443) whose squares and cubes are also digitally balanced.

Original entry on oeis.org

3274, 13453, 13492, 13706, 14726, 15113, 15498, 15528, 52049, 52251, 52330, 52673, 52778, 53478, 53684, 53775, 53972, 54295, 54411, 54598, 54601, 55057, 55449, 55462, 55505, 55512, 55689, 56333, 58066, 58260, 58446, 58453, 58470, 58918, 59266, 59722, 59786

Offset: 1

Alex Ratushnyak, Apr 26 2022

Numbers x such that x, x^2 and x^3 are terms of A031443, that is, have the same number of 0's as 1's in their binary representations.

Cf. A031443, A345397, A353139.

balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[60000], balQ[#] && balQ[#^2] && balQ[#^3] &] (* Amiram Eldar, Apr 26 2022 *)

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
def A031443gen(): yield from (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1)))
def agen():
    for k in A031443gen():
        if isbalanced(k**2) and isbalanced(k**3):
            yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Apr 26 2022

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A345497 Numbers that are the sum of eight squares in ten or more ways.

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A353139 Digitally balanced numbers (A031443) whose squares are also digitally balanced.

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A345396 a(n) is the least prime p such that {p, p^2, ..., p^n} are all digitally balanced numbers in base 2 (A031443).

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Author

Keywords

Comments

Examples

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Python

A353140 Digitally balanced numbers (A031443) whose squares and cubes are also digitally balanced.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python