A370362
Numbers k such that any two consecutive decimal digits of k^2 differ by 1 after arranging the digits in decreasing order.
Original entry on oeis.org
0, 1, 2, 3, 18, 24, 66, 74, 152, 179, 3678, 3698, 4175, 4616, 5904, 5968, 6596, 7532, 8082, 8559, 9024, 10128, 10278, 11826, 12363, 12543, 12582, 13278, 13434, 13545, 13698, 14442, 14676, 14766, 15681, 15963, 16854, 17529, 17778, 18072, 19023, 19377, 19569, 19629
Offset: 1
18^2 = 324 consists of the consecutive digits 2, 3 and 4;
24^2 = 576 consists of the consecutive digits 5, 6 and 7;
66^2 = 4356 consists of the consecutive digits 3, 4, 5 and 6;
74^2 = 5476 consists of the consecutive digits 4, 5, 6 and 7.
The actual squares are given by
A370610.
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isconsecutive(m, {b=10})=my(v=vecsort(digits(m, b))); for(i=2, #v, if(v[i]!=1+v[i-1], return(0))); 1 \\ isconsecutive(k, b) == 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) = isconsecutive(n^2)
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from math import isqrt
from sympy.ntheory import digits
def afull(): return([i for i in range(isqrt(10**10)+1) if len(d:=sorted(str(i*i))) == ord(d[-1])-ord(d[0])+1 == len(set(d))])
print(afull()) # Michael S. Branicky, Feb 23 2024
A370371
Largest m such that any two consecutive digits of the base-n expansion of m^2 differ by 1 after arranging the digits in decreasing order.
Original entry on oeis.org
1, 1, 15, 2, 195, 867, 3213, 18858, 99066, 528905, 2950717, 294699, 105011842, 659854601, 4285181505, 1578809181, 198009443151, 1404390324525, 10225782424031, 3635290739033, 583655347579584, 4564790605900107, 36485812146621733, 297764406866494254, 2479167155959358950
Offset: 2
Base 4: 15^2 = 225 = 3201_4;
Base 6: 195^2 = 38025 = 452013_6;
Base 7: 867^2 = 751689 = 6250341_7;
Base 8: 3213^2 = 10323369 = 47302651_8;
Base 9: 18858^2 = 355624164 = 823146570_9;
Base 10: 99066^2 = 9814072356;
Base 11: 528905^2 = 279740499025 = A8701245369_11;
Base 12: 2950717^2 = 8706730814089 = B8750A649321_12;
Base 13: 294699^2 = 86847500601 = 8260975314_13.
The actual squares are given by
A370611.
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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) = forstep(m=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)), 0, -1, if(isconsecutive(m^2,n), return(m)))
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from math import isqrt
from sympy import multiplicity
from sympy.ntheory import digits
def a(n):
ub = isqrt(sum(i*n**i for i in range(n)))
if n%2 == 1 and multiplicity(2, n-1)%2 == 0:
ub = isqrt(sum(i*n**(i-2) for i in range(2, n)))
return(next(i for i in range(ub, -1, -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, 13)]) # Michael S. Branicky, Feb 23 2024
A370610
Squares such that any two consecutive decimal digits differ by 1 after arranging the digits in decreasing order.
Original entry on oeis.org
0, 1, 4, 9, 324, 576, 4356, 5476, 23104, 32041, 13527684, 13675204, 17430625, 21307456, 34857216, 35617024, 43507216, 56731024, 65318724, 73256481, 81432576, 102576384, 105637284, 139854276, 152843769, 157326849, 158306724, 176305284, 180472356, 183467025, 187635204
Offset: 1
The square roots are given by
A370362.
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isconsecutive(m, {b=10})=my(v=vecsort(digits(m, b))); for(i=2, #v, if(v[i]!=1+v[i-1], return(0))); 1 \\ isconsecutive(k, b) == 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) = issquare(n) && isconsecutive(n)
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from math import isqrt
from sympy.ntheory import digits
def afull(): return([i*i for i in range(isqrt(10**10)+1) if len(d:=sorted(str(i*i))) == ord(d[-1])-ord(d[0])+1 == len(set(d))])
print(afull()) # Michael S. Branicky, Feb 23 2024
A370611
Largest square such that any two consecutive digits of its base-n expansion differ by 1 after arranging the digits in decreasing order.
Original entry on oeis.org
1, 1, 225, 4, 38025, 751689, 10323369, 355624164, 9814072356, 279740499025, 8706730814089, 86847500601, 11027486960232964, 435408094460869201, 18362780530794065025, 2492638430009890761, 39207739576969100808801, 1972312183619434816475625, 104566626183621314286288961, 13215338757299095309775089
Offset: 2
See the Example section of A370371.
The square roots are given by
A370371.
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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) = forstep(m=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)), 0, -1, if(isconsecutive(m^2, n), return(m^2)))
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