A230422 -id:A230422 - cp's OEIS frontend

A258010 First differences of A230422.

7, 6, 2, 2, 4, 11, 2, 2, 4, 4, 6, 2, 4, 4, 10, 4, 7, 5, 9, 10, 10, 2, 2, 4, 4, 6, 2, 4, 4, 10, 4, 7, 5, 9, 10, 9, 2, 6, 2, 4, 4, 10, 4, 7, 5, 9, 10, 9, 2, 8, 9, 4, 7, 5, 9, 10, 9, 2, 8, 2, 6, 2, 9, 9, 10, 9, 2, 8, 8, 16, 8, 9, 2, 8, 8, 29, 2, 8, 8, 9, 6, 13, 10, 2, 2, 4, 4, 6, 2, 4, 4, 10, 4, 7, 5, 9, 10, 9, 2, 6, 2, 4, 4, 10, 4, 7, 5, 9

Offset: 1

Antti Karttunen, Jun 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17516

Cf. A230422, A258008.

(define (A258010 n) (- (A230422 (+ 1 n)) (A230422 n)))

a(n) = A230422(n+1) - A230422(n).

A230410 After a(0)=0, a(n) = A230415(A219666(n),A219666(n-1)).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 3, 1, 3, 4, 2, 2, 2, 4, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 3, 1, 3, 4, 2, 1, 3, 3, 3, 2, 4, 1, 3, 1, 3, 3, 3, 1, 3, 4, 2, 1, 2, 2, 2, 2, 3, 2, 2, 4, 3, 1, 3, 4, 2, 1, 2, 2, 2, 2, 3, 2, 1, 3, 2, 5, 2

Offset: 0

Antti Karttunen, Nov 10 2013

After zero, a(n) = number of positions where digits in the factorial base representations of successive nodes A219666(n-1) and A219666(n) in the infinite trunk of the factorial beanstalk differ from each other.

			a(8) = 1, because A219666(8)=23, whose factorial base representation (A007623(23)) is '321', and A219666(7)=17, whose factorial base representation (A007623(17)) is '221', and they differ just in one digit position.
a(9) = 3, because A219666(9)=25, '...01001' in factorial base, which differs from '...0321' in three digit positions.
Note that A226061(4)=8 (A226061(n) tells the position of (n!)-1 in A219666), and 1+2+3 = 6 happens to be both a triangular number (A000217) and a factorial number (A000142).
The next time 1 occurs in this sequence because of this coincidence is at x=A226061(16) (whose value is currently not known), as at that point A219666(x) = 16!-1 = 20922789887999, whose factorial base representation is (15,14,13,12,11,10,9,8,7,6,5,4,3,2,1), and A000217(15) = 120 = A000142(5), which means that A219666(x-1) = A219651(20922789887999) = 20922789887879, whose factorial base representation is (15,14,13,12,11,10,9,8,7,6,4,4,3,2,1), which differs only in one position from the previous.
Of course 1's occur in this sequence for other reasons as well.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A230415, A230406, A231717, A231719, A232094. A230422 gives the positions of ones.

nn = 1200; m = 1; While[m! < nn, m++]; m; f[n_] := IntegerDigits[n, MixedRadix[Reverse@ Range[2, m]]]; Join[{0}, Function[w, Count[Subtract @@ Map[PadLeft[#, Max@ Map[Length, w]] &, w], k_ /; k != 0]]@ Map[f@ # &, {#1, #2}] & @@@ Partition[#, 2, 1] &@ TakeWhile[Reverse@ NestWhileList[# - Total@ f@ # &, nn, # > 0 &], # <= 500 &]] (* Michael De Vlieger, Jun 27 2016, Version 10 *)

(define (A230410 n) (if (zero? n) n (A230415bi (A219666 n) (A219666 (- n 1))))) ;; Where bi-variate function A230415bi has been given in A230415.

a(0)=0, and for n>=1, a(n) = A230415(A219666(n),A219666(n-1)).

For all n, a(A226061(n+1)) = A232094(n).

A231718 Positions of ones in A231717.

Original entry on oeis.org

1, 8, 14, 18, 22, 33, 37, 41, 45, 53, 57, 61, 71, 75, 87, 116, 120, 124, 128, 136, 140, 144, 154, 158, 170, 208, 212, 216, 226, 230, 242, 289, 293, 305, 362, 544, 548, 552, 556, 564, 568, 572, 582, 586, 598, 636, 640, 644, 654, 658, 670, 717, 721, 733, 790, 1021

Offset: 1

Antti Karttunen, Nov 12 2013

nonn

This sequence gives all n at which positions the successive terms A219666(n-1) & A219666(n) in the infinite trunk of the factorial beanstalk differ only in one digit position in their factorial base representations (A007623) and the difference of those digits is exactly one.

			14 is included, because A219666(13) = 40 = '1220' in factorial base representation, while A219666(14) = 46 = '1320' in factorial base, and they differ only by the third least significant digits, and 3-2 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..11905

Subset of A230422.

For all n, A230406(a(n)) = A000142(A231719(a(n))).

cp's OEIS Frontend

A258010 First differences of A230422.

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A230410 After a(0)=0, a(n) = A230415(A219666(n),A219666(n-1)).

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A231718 Positions of ones in A231717.

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