A359183 - OEIS

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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.

1, 2, 54, 13122, 15258789062500 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`2,2`
	COMMENTS	`Each term can be represented in some base < n as a number < n multiplied by the base to some power. The terms given in the data section are a(2) = 1, a(3) = 2, a(4) = 54 = 23^3, a(5) = 13122 = 23^8, a(6) = 15258789062500 = 45^18, a(7) = 8158...4608 (186 digits) = 34^308. The other known terms (too large to write in the data section) are a(8) = 9532...8658 (3448 digits) = 23^7226, a(9) = a(10) = 9123...2500 (10344 digits) = 45^14798.` `Assuming a(11) exists, it is greater than 10^22500.`
	LINKS	`Table of n, a(n) for n=2..6.`
	EXAMPLE	`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)`
	PROG	`(Python)` `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 = mbp` `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`
	CROSSREFS	`Cf. A347053, A004053, A258107, A181929, A307254, A307255.` `Sequence in context: A306266 A117681 A221603 * A089180 A034013 A340211` `Adjacent sequences: A359180 A359181 A359182 * A359184 A359185 A359186`
	KEYWORD	`nonn,base`
	AUTHOR	`Scott R. Shannon, Dec 18 2022`
	STATUS	`approved`

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Last modified March 24 21:37 EDT 2023. Contains 361511 sequences. (Running on oeis4.)