A370369 -id:A370369 - cp's OEIS frontend

A370370 Number of squares such that any two consecutive digits of their base-n expansions differ by 1 after arranging the digits in decreasing order.

2, 2, 6, 3, 10, 12, 14, 48, 160, 148, 226, 54, 1277, 2675, 6812, 2525

Offset: 2

Jianing Song, Feb 16 2024

			a(4) = 6 because there are 6 such squares in base 4: 0^2 = 0 = 0_4, 1^2 = 1 = 1_4, 2^2 = 4 = 10_4, 3^2 = 9 = 21_4, 6^2 = 36 = 210_4 and 15^2 = 225 = 3201_4.
a(6) = 10 because there are 10 such squares in base 6: 0^2 = 0 = 0_6, 1^2 = 1 = 1_6, 2^2 = 4 = 2_6, 9^2 = 81 = 213_6, 11^2 = 121 = 321_6, 21^2 = 441 = 2013_6, 50^2 = 2500 = 15324_6, 75^2 = 5625 = 42013_6, 85^2 = 7225 = 53241_6 and 195^2 = 38025 = 452013_6.
a(10) = 160 because there are 160 terms in A370362 (or A370610).

Cf. A215014, A370362, A370369, A370371, A370610, A370611.

Cf. A258103 (number of pandigital squares in base n).

isconsecutive(m,n)=my(v=vecsort(digits(m,n))); for(i=2, #v, if(v[i]!=1+v[i-1], return(0))); 1 \\ isconsecutive(k,n) == 1 if and only if any two consecutive digits of the base-n expansion of m differ by 1 after arranging the digits in decreasing order
a(n) = my(lim=sqrtint(if(n%2==1 && valuation(n-1, 2)%2==0, n^(n-1) - (n^(n-1)-1)/(n-1)^2, n^n - (n^n-n)/(n-1)^2)), count=0); for(m=0, lim, if(isconsecutive(m^2,n), count++)); count \\ See A258103 for the searching limit of m

# replace n**n with ub in A370371 for faster version
from math import isqrt
from sympy.ntheory import digits
def a(n): return(sum(1 for i in range(isqrt(n**n)+1) if len(d:=sorted(digits(i*i, n)[1:])) == d[-1]-d[0]+1 == len(set(d))))
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Feb 23 2024

a(15)-a(17) from Michael S. Branicky, Feb 23 2024

A175134 a(n) = number of pairs of permutations (p, q) of the same (arbitrary) size m such that f_m(f_{m-1}(...f_1(1)...)) = n, where f_i(x) = p_i*x + q_i.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 2, 4, 4, 4, 2, 4, 1, 3, 1, 2, 2, 1, 1, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 4, 8, 6, 8, 6, 8, 6, 16, 12, 14, 6, 12, 10, 16, 18, 14, 14, 10, 10, 10, 14, 12, 10, 10, 8, 4, 13, 15, 4, 13, 6, 8, 6, 8, 2, 11, 6, 5, 7, 2, 11, 7, 6, 6, 5, 3, 6, 6, 2, 5, 2, 4, 3, 4, 3, 3, 1, 3, 5, 5, 1, 2, 3, 2, 0, 3, 2, 2, 3, 3, 4, 4, 3, 4, 2, 0, 2, 3, 2, 0, 1, 3, 2, 3, 1, 1, 1, 0, 1, 4, 2, 2, 1, 4, 1, 2, 0, 2, 3, 6, 0, 1, 1, 0, 1, 3, 3, 2, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Leroy Quet, Feb 21 2010

nonn

For a fixed m, the smallest value of f_m(f_{m-1}(...f_1(1)...)) is achieved on the pair ((m,m-1,...,1), (1,2,...,m)) and is given by A370369(m) + m = A014144(m) + m!. Hence, to compute a(n) one can restrict m to those satisfying A014144(m) + m! <= n. - Max Alekseyev, Jan 29 2025

			For n = 13, suitable pairs of permutations are ((3,2,1),(1,2,3)) and ((3,2,1),(1,3,2)).

Cf. A014144, A175135, A370369.

Edited and terms a(14) onward added by Max Alekseyev, Jan 29 2025.

A175135 Numbers k such that there exist permutations p, q of some size m such that k = f_m(f_{m-1}(...f_1(1)...)), where f_i(x) = p_i*x + q_i.

Original entry on oeis.org

1, 2, 5, 6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 31, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 132, 133, 134, 135, 136, 137, 138, 139, 141, 142, 143, 145, 146, 148, 149, 150, 151, 153, 154, 157, 161, 162, 163, 165

Offset: 1

Leroy Quet, Feb 21 2010

nonn

Also, numbers k such that A175134(k) >= 1.

For a fixed m, the smallest value of f_m(f_{m-1}(...f_1(1)...)) is achieved on the pair ((m,m-1,...,1), (1,2,...,m)) and is given by A370369(m) + m = A014144(m) + m!. - Max Alekseyev, Jan 29 2025

Cf. A014144, A175134, A370369.

a(7) onward from Max Alekseyev, Dec 27 2024

Edited by Max Alekseyev, Jan 29 2025

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A370370 Number of squares such that any two consecutive digits of their base-n expansions differ by 1 after arranging the digits in decreasing order.

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A175134 a(n) = number of pairs of permutations (p, q) of the same (arbitrary) size m such that f_m(f_{m-1}(...f_1(1)...)) = n, where f_i(x) = p_i*x + q_i.

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A175135 Numbers k such that there exist permutations p, q of some size m such that k = f_m(f_{m-1}(...f_1(1)...)), where f_i(x) = p_i*x + q_i.

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