cp's OEIS Frontend

Since 8 <= A357973(n) <= 26 the sequence is finite and begins with offset 8.

A074458 is a strictly decreasing function A063720(x) < x whenever x >= 10 and all single digit x reach a fixed point A063720(x) = x with x in {4, 5}.

This sequence is unbounded and the first occurrence of a(n) = k is at n = A338255(k + 2) for any k >= 3.

a(A216261(n)) = n and a(m) <> n for m < A216261(n). - Reinhard Zumkeller, Mar 15 2013

If we mark with * resp. ' the graphical representations which use more resp. less segments, we have the following variants:

A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),

_____________ this: A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*).

Sequences A234691, A234692 and variants make precise which segments are lit in each digit. These are related through the Hamming weight function A000120, e.g., A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). - M. F. Hasler, Jun 17 2020

Except for 1 and 3 every positive integer occurs; A143616 and A143617 give record values and where they occur. - Reinhard Zumkeller, Aug 27 2008

The difference between this sequence and A006942 lies in the representation chosen for the digit 7,

| | |

| (here), vs. | in A006942.

If we mark with ' the "sans serif" graphical representation which uses one segment less and with * the "heavier" version, we have the following variants:

A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),

___________________ A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*) = this.

Sequences A234691, A234692 and variants make precise which segments are lit in each digit. They are related through the Hamming weight A000120, see formula. The sequence could be extended to negative arguments with a(-n) = a(n)+1. - M. F. Hasler, Jun 17 2020

If we mark with * resp. ' the glyph variants (graphical representations) which use more resp. less segments, we have the following variants:

A063720 (this: 6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'), _________________________ A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*). Sequences A234691, A234692 and variants make precise which segments are lit in each digit. These are related through the Hamming weight function A000120, e.g., A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). - M. F. Hasler, Jun 17 2020

Another version of A006942. Here the digit "6" is represented with six segments (the same as in A006942) but the digit "9" is represented with five segments instead of six segments. - Omar E. Pol, Sep 30 2016

If we mark with * resp. ' the graphical representations which use one more resp. one less segment, we have the following variants:

A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),

___________________ A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*).

Sequences A234691 and A234692 make precise which segments are lit in each digit. They are related through the Hamming weight function A000120, e.g., A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). - M. F. Hasler, Jun 17 2020

The bits 0-6 are assigned to the segments as follows: top (bit 0), upper left, upper right, middle, lower left, lower right, bottom (bit 6). Other conventions are common in engineering (as well for the segment-to-bit correspondence as for the glyphs), see sequence A234692 (the so-called ABCDEF coding), Wikipedia and the Example for a(7).

For n > 9, the decimal representation of n is encoded using a group of 7 bits for each digit, e.g., a(10) = a(1)*2^7 + a(0).

The sequence could have been extended differently to indices > 9, for example, by coding 7-segment representations of upper- and/or lowercase letters, as used for hexadecimal and base-64 representation.

The Hamming weight A000120 of the terms of this sequence yields the count of lit segments, A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). For that sequence, 4 other variants are in the OEIS, depending on the number of segments used to represent digits 6, 7 and 9: A063720 (6', 7', 9'), A277116 (7', 9'), A074458 (9') and A006942 (7'), where x' means that the "sans serif" variant (one segment less than here) is used for digit x. - M. F. Hasler, Jun 17 2020

The bits 0-6 are assigned to the segments according to the "abcdefg" scheme (top, upper right, lower right, bottom, lower left, upper left, center), cf. section "Displaying letters" of the Wikipedia page (3rd column of the table). Other conventions are common in engineering (as well for the segment-to-bit correspondence as for the glyphs), see sequence A234691, the Wikipedia page and the comment after the Example for a(7).

For n > 9, each of the digits of the base-10 representation is coded in a separate group of 7 bits, for example, a(10) = a(1)*2^7 + a(0) = 831.

Alternatively, for n >= 10 one could define a(n) to represent a 7-segment variant of the characters A-Z and/or a-z, as in hexadecimal or base-64 encoding. In that case, one could also use a few more bits for additional segments, e.g., four half-diagonals to represent K, M, N, R, V, X, Z correctly and S distinctly from 5. But as mentioned on the Wikipedia page, a possible ambiguity of representations of alphabetic characters is not always an obstacle to common use, since whole words are usually readable nonetheless.

The Hamming weight A000120 of the terms of this sequence yields the count of lit segments, A010371(n) = A000120(a(n)) = A000120(A234691(n)). For that sequence, 5 other variants are in the OEIS, depending on the number of segments used to represent digits 6, 7 and 9: A063720 (6', 7', 9'), A277116 (7', 9'), A074458 (9') and A006942 (7'), where x' means that the "sans serif" variant (one segment less than here) is used for digit x. - M. F. Hasler, Jun 17 2020

Essentially the same as A038619 and A143617. One could argue that a(3) should rather be -1 (prior to adding "positive" in the definition), which does use 3 segments on typical 7-segment displays, and is smaller than 7. Also, most pocket calculators and the Unicode standard (cf. links) use 4 rather than 3 segments to represent a '7' (as in A074458 and A010371, rather than A063720, A277116 or A006942), in which case a(3) is undefined if negative numbers are not allowed. No digit '9' will ever occur here, whether it would be represented with 6 or only 5 segments. However, digit '6' does occur, as the second smallest digit using 6 segments as does '0', which cannot occur as leading digit. If '6' is represented with 5 segments, any prefix 68 would be replaced with 80. - M. F. Hasler and Kevin Ryde, Jun 17 2020

The nonnegative integers are displayed as in A006942, where a 7 is depicted by 3 segments.

Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = f(6) = 3 and f(s) = 1 elsewhere, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see first formula).

The same sequence is obtained when 7 and 9 are depicted respectively by 4 and 5 segments (A074458). - Stefano Spezia, Apr 11 2021