A062273 -id:A062273 - cp's OEIS frontend

A077578 Terms of A062273 which are multiples of their indices.

1, 456, 12345, 90123456, 678901234567890, 1234567890123456, 45678901234567890123456, 901234567890123456789012345678901234567890123456

Offset: 1

Amarnath Murthy, Nov 12 2002

Amiram Eldar, Table of n, a(n) for n = 1..11

Cf. A062273, A077577, A077579.

a(n) = A062273(A077579(n)). - Amiram Eldar, May 26 2024

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 11 2003

Offset corrected by Amiram Eldar, May 26 2024

A077577 a(n) = floor(A062273(n)/n).

Original entry on oeis.org

1, 11, 152, 1972, 2469, 113150, 335096, 11265432, 87668038, 678901234, 6171829405, 65751028806, 693257359915, 1675484929453, 45260082304526, 77160493132716, 4641249092228830, 25377167352537716, 123456784275503573

Offset: 1

Amarnath Murthy, Nov 12 2002

			a(6) = Floor[678901/6]= 113150.

Cf. A062273, A077578.

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 11 2003

A077579 Numbers k that divide A062273(k).

Original entry on oeis.org

1, 3, 5, 8, 15, 16, 23, 48, 96, 155, 208, 1956, 4245, 6816, 6907, 28515, 55568, 57765, 203376, 437336, 746448, 963408, 1247088, 18661696, 32187455, 113280465, 238564335, 368111296, 1491581031, 1689287568, 1831503936, 1838097728, 2426280688

Offset: 1

Amarnath Murthy, Nov 12 2002

From David A. Corneth, May 25 2024: (Start)
a(n) cannot be equal to 2, 4, 6, 10, 12, 14, 18 (mod 20).
We can ease verification of terms in this sequence by rewriting A062273(k). A062273(k) can be written as blocks of 10 digits followed by some remaining digits like this:
A062273(k) = (B * Sum_{i = 0..(floor(k/10)-1)} (10^10)^i)*10^(k mod 10) + floor(B / (10 ^ (9 - (k-1) mod 10))) = (B * Sum_{i = 0..(floor(k/10)-1)} (10^10)^i)*10^(k mod 10) + floor(B / (10 ^ (9 - (k-1) mod 10))) where B is a block of 10 digits starting with (((k-1)*k)/2 + 1) mod 10 and each successive digit is one more than the previous except after 9 there is 0.
Using modular arithmetic and a simplification for geometric sums we can compute A062273(k) mod k and see if the result is 0.
Also terms are 11670334251, 12057978588, 15925928169. Any missing term must have gcd(t, floor(10^10/9)) > 1. (End)

			From _David A. Corneth_, May 25 2024: (Start)
48 is in the sequence as 48 divides A062273(48). We may write A062273(48) as (9012345678 * Sum_{i = 0..(floor(48)-1)} (10^10)^i)*10^8 + 90123456
= 9012345678 * ((10^10)^4 - 1) / (10^10 - 1) * 10^8 + 90123456
= 1001371742 * ((10^10)^4 - 1) / ((10^10 - 1)/9) * 10^8 + 90123456 and using modular arithmetic (note that (10^10 - 1)/9 is coprime to 48 so we can compute the inverse of the modular class of (10^10 - 1)/9  mod 48. (End)

David A. Corneth, PARI program

Cf. A062273, A077577, A077578.

PARI
```
\\ See PARI link
```

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 11 2003
Offset corrected and a(12)-a(16) added by Amiram Eldar, May 21 2024
a(17)-a(20) from Michael S. Branicky, May 22 2024
a(21)-a(23) from David A. Corneth and Michael S. Branicky, May 24 2024
a(24)-a(31) from David A. Corneth, May 25 2024

A077576 a(1) = 1; for n > 1, a(n) = largest prime divisor of A062273(n).

Original entry on oeis.org

1, 23, 19, 263, 823, 678901, 50993, 1279, 211, 150866941, 353594393, 757819, 9012345678901, 213408443, 11904282563, 435503, 21339439879, 7403356241, 13212859242013, 27961, 205126079, 5201061865265117273, 54843374833199, 1384232185401581113277, 12792129877

Offset: 1

Amarnath Murthy, Nov 11 2002

Daniel Suteu, Table of n, a(n) for n = 1..131

Cf. A006530, A062273.

f(n) = sum(i=1, n, ((n*(n-1)/2+i) % 10)*10^(n-i)); \\ A062273
a(n) = my(x=f(n)); if (x==1, 1, vecmax(factor(x)[,1])); \\ Michel Marcus, May 26 2022

a(n) = A006530(A062273(n)). - Michel Marcus, May 26 2022

More terms from Sascha Kurz, Jan 12 2003

a(24)-a(25) from Daniel Suteu, May 25 2022

A059043 Numbers in which each digit is the (immediate) successor of the previous one (if it exists) and 0 is considered the successor of 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 23, 34, 45, 56, 67, 78, 89, 90, 123, 234, 345, 456, 567, 678, 789, 890, 901, 1234, 2345, 3456, 4567, 5678, 6789, 7890, 8901, 9012, 12345, 23456, 34567, 45678, 56789, 67890, 78901, 89012, 90123, 123456, 234567, 345678, 456789

Offset: 1

Thomas Schulze (jazariel(AT)tiscalenet.it), Feb 12 2001

Numbers that are contiguous substrings of those in A062273. - Michael S. Branicky, May 26 2022

Michael S. Branicky, Table of n, a(n) for n = 1..9001

Cf. A161760, A006055, A062273.

def ok(n):
    s = str(n)
    return s == "".join(str((int(s[0])+i)%10) for i in range(len(s)))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 26 2022

from itertools import count, islice
def agen(): # generator of terms
    yield 0
    for d in count(1):
        for s0 in range(1, 10):
            yield int("".join(str((s0+i)%10) for i in range(d)))
print(list(islice(agen(), 50))) # Michael S. Branicky, May 26 2022

More terms from Brian DiCesare (bdicesar(AT)ashland.edu), Oct 11 2004

a(1) = 0 inserted by Michael S. Branicky, May 26 2022

A061501 a(1) = 1, a(n+1) = (a(n) + n) mod 10.

Original entry on oeis.org

1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1, 1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1, 1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1, 1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1, 1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1, 1, 2, 4, 7, 1

Offset: 1

Amarnath Murthy, May 06 2001

First row of array shown below.
a(n) = most significant digit of A062273(n).
Period 20: repeat [1, 2, 4, 7, 1, 6, 2, 9, 7, 6, 6, 7, 9, 2, 6, 1, 7, 4, 2, 1]. - Peter M. Chema, Feb 12 2017

			1 2 4 7 1 6 2 9 7 6 6 ...
3 5 8 2 7 3 0 8 7 7 ...
6 9 3 8 4 1 9 8 8 ...
0 4 9 5 2 0 9 9 ...
5 0 6 3 1 0 0 ...
1 7 4 2 1 1 ...

Cf. A061352, A061353.
Cf. A062273, A077577, A077578.
Cf. A000124.

a = {1}; Do[AppendTo[a, Mod[a[[n - 1]] + n - 1, 10]], {n, 2, 120}]; a (* Michael De Vlieger, Feb 13 2017 *)

a(n)=if(n==1,return(1));(a(n-1)+n-1)%10
for(n=1,50,print1(a(n),", ")) \\ Derek Orr, Feb 26 2017

a(n) = A008954(n-1) + 1.
a(n) = A000124(n) mod 10. - Peter M. Chema, Feb 11 2017
From Chai Wah Wu, Jan 09 2020: (Start)
a(n) = a(n-5) - a(n-10) + a(n-15) for n > 15.
G.f.: x*(-x^14 - 2*x^13 - 4*x^12 - 7*x^11 - x^10 - 5*x^9 - 5*x^7 - 5*x^5 - x^4 - 7*x^3 - 4*x^2 - 2*x - 1)/(x^15 - x^10 + x^5 - 1). (End)

Better description and more terms from Larry Reeves (larryr(AT)acm.org), May 08 2001

A100814 Digits 9 to 0 are written in order with increasing number of digits for each member of the sequence. Leading zeros are counted, but are not written down.

Original entry on oeis.org

9, 87, 654, 3210, 98765, 432109, 8765432, 10987654, 321098765, 4321098765, 43210987654, 321098765432, 1098765432109, 87654321098765, 432109876543210, 9876543210987654, 32109876543210987, 654321098765432109

Offset: 1

Parthasarathy Nambi, Jan 05 2005

This is similar to A062273 and A007923

			The first number in the sequence is 9.
The second number in the sequence is 87.
The third number in the sequence is 654.

C. Ashbacher, "Some problems concerning the Smarandache deconstructive sequence", Journal of Recreational Mathematics, vol. 29(2), 82-84 (1998)
Russell Euler and Jawad Sadek, "Some divisibility patterns in the Smarandache deconstructive sequence", Journal of Recreational Mathematics, vol. 31(1), 12-14 (2002-2003)

Cf. A007923, A062273.

With[{c=PadRight[{},250,Range[9,0,-1]]},Table[FromDigits[Take[c,{(n(n+1))/2+1,((n+1)(n+2))/2}]],{n,0,20}]] (* Harvey P. Dale, Jan 17 2017 *)

More terms from Robert G. Wilson v and Lior Manor, Jan 14 2005

cp's OEIS Frontend

A077578 Terms of A062273 which are multiples of their indices.

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A077577 a(n) = floor(A062273(n)/n).

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A077579 Numbers k that divide A062273(k).

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A077576 a(1) = 1; for n > 1, a(n) = largest prime divisor of A062273(n).

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A059043 Numbers in which each digit is the (immediate) successor of the previous one (if it exists) and 0 is considered the successor of 9.

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Python

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A061501 a(1) = 1, a(n+1) = (a(n) + n) mod 10.

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A100814 Digits 9 to 0 are written in order with increasing number of digits for each member of the sequence. Leading zeros are counted, but are not written down.

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