A308070
Numbers k such that k is the substring identical to the most significant digits of its base-7 representation.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 102616333034, 102620103253, 103055445560, 206154633166, 206154633200, 212216263215, 212220033434, 315315450515, 321340554340, 424436332033, 424440102253, 430461435550, 430501403606, 533560623156, 533600556144
Offset: 1
102616333034_10 = 10261633303415_7, which also begins with '102616333034'.
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isok(n) = my(vb=digits(n, 7), vd=digits(n)); vd == vector(#vd, k, vb[k]); \\ Michel Marcus, May 17 2019
A308071
Numbers k such that k is the substring identical to the most significant digits of its base-8 representation.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 13663722656306465044, 13664030703642511032, 13707143454447524553, 13710552565235121771, 14336552505653142213, 14340267205650777016, 14340267205651006010, 15013125376232575510, 15013125401147630620, 10142740706726414575022612720
Offset: 1
13663722656306465044_10 = 1366372265630646504424_8, which also begins with '13663722656306465044'.
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isok(n) = my(vb=digits(n, 8), vd=digits(n)); vd == vector(#vd, k, vb[k]); \\ Michel Marcus, May 17 2019
A308072
Numbers k such that k is the substring identical to the most significant digits of its base-9 representation.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 1066338786883726756382, 1066338787045067082685, 1066338805156287287066, 1066338805156287287067, 1066338805156287300050, 1066338805156301216066, 1066341132816741837214, 1070235882243468707818, 1070235882243470056412, 1070236076684636644063
Offset: 1
1066338786883726756382_10 = 10663387868837267563825_9, which also begins with '1066338786883726756382'.
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isok(n) = my(vb=digits(n, 9), vd=digits(n)); vd == vector(#vd, k, vb[k]); \\ Michel Marcus, May 17 2019
A359183
a(n) is the smallest number such that when written in all bases from base 2 to base n its leading digit equals the base - 1.
Original entry on oeis.org
1, 2, 54, 13122, 15258789062500
Offset: 2
a(2) = 1 as 1 = 1_2, which has 1 = 2 - 1 as its leading digit.
a(3) = 2 as 2 = 10_2 = 2_3, which have 1 = 2 - 1 and 2 = 3 - 1 as their leading digits.
a(4) = 54 as 54 = 110110_2 = 2000_3 = 312_4, which have 1 = 2 - 1, 2 = 3 - 1 and 3 = 4 - 1 as their leading digits.
a(5) = 13122 as 13122 = 11001101000010_2 = 200000000_3 = 3031002_4 = 404442_5, which have 1 = 2 - 1, 2 = 3 - 1, 3 = 4 - 1 and 4 = 5 - 1 as their leading digits.
a(6) = 15258789062500 as 15258789062500 = 110000010110110101100111010011101100100_2 = 2000000201121020121212112011_3 = 3132002312230322131210_4 = 4000000000000000000_5 = 52241442501204004_6, which have 1 = 2 - 1, 2 = 3 - 1, 3 = 4 - 1, 4 = 5 - 1 and 5 = 6 - 1 as their leading digits.
a(7) = 81582795696655426358720748526459181157825502882872103403434619627581986794626\
90448473536034793921827874140100908746255557234586263455831973302268738547817\
2585724832003163984432734404608 (Too large to include in the DATA section)
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from math import floor, log
def a(n):
arr = []
p = 0
while True:
for m in range(1, n):
for b in range(2, max(3, n)):
k = m*b**p
if k in arr:
continue
arr.append(k)
q = 1
for b in range(3, n+1):
if floor(k/b**floor(log(k)/log(b))) != b-1:
q = 0
break
if q:
return k
p += 1
# Christoph B. Kassir, Feb 10 2023
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