A067251 -id:A067251 - cp's OEIS frontend

A076641 Integers read backwards, but with repetitions omitted.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 21, 31, 41, 51, 61, 71, 81, 91, 12, 22, 32, 42, 52, 62, 72, 82, 92, 13, 23, 33, 43, 53, 63, 73, 83, 93, 14, 24, 34, 44, 54, 64, 74, 84, 94, 15, 25, 35, 45, 55, 65, 75, 85, 95, 16, 26, 36, 46, 56, 66, 76, 86, 96, 17, 27, 37, 47, 57, 67, 77, 87, 97

Offset: 0

Francois Jooste (phukraut(AT)hotmail.com), Oct 23 2002

Although this is a list, it seems most natural here to have offset 0. - N. J. A. Sloane, Dec 09 2020

			10 backwards is 01=1, but 1 already appears, so omit it.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A004086.

a076641 = a004086 . a067251  -- Reinhard Zumkeller, Aug 12 2013

s := (L::list)->sum(L[i]*10^(nops(L)-i),i=1..nops(L)); d := (X::posint)->convert(X,base,10); SD := K->[seq(s(d(k)),k=1..K)];

a(0) = 0 added by N. J. A. Sloane, Dec 09 2020

A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

Original entry on oeis.org

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 127, 133, 163, 193, 197, 199, 203, 206, 209, 214, 218, 233, 234, 236, 247, 254, 258, 263, 266, 274, 293, 294, 296, 298, 299, 309, 367, 399, 406, 409, 417, 428, 436, 466, 468, 487, 496, 499, 509, 537, 599, 609, 638, 657, 678, 699, 709, 799, 809, 899

Offset: 1

Eric Angelini and M. F. Hasler, Mar 22 2012

This is a restricted or simplified version of the definition of modest numbers A054986.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Angelini, Not Modest, Mar 22 2012
E. Angelini, Not modest [Cached copy, with permission]

Cf. A054986, A178158.

A subsequence of A067251, disjoint with A034709.

a210582 n = a210582_list !! (n-1)
a210582_list = filter (\x -> mod x (a010879 x) == a000030 x) a067251_list
-- Reinhard Zumkeller, Mar 26 2012

[ n: n in [1..1002] | not IsZero(d[1]) and n mod d[1] eq d[#d] where d is Intseq(n) ];  // Bruno Berselli, Mar 26 2012

is_nm( x )=x%10 && x%(x%10)==x\10^(#Str(x)-1)
for(n=1,999,is_nm(n)&print1(n","))

a(n) mod A010879(a(n)) = A000030(a(n)). [Reinhard Zumkeller, Mar 26 2011]

Edited by M. F. Hasler, Jan 14 2014

A255357 Natural numbers n, other than multiples of 10, such that n, n^2 and n^3 lack the digit 1 in their decimal expansion.

Original entry on oeis.org

2, 3, 7, 62, 63, 65, 66, 67, 74, 76, 77, 78, 84, 86, 87, 92, 93, 94, 95, 202, 207, 274, 275, 282, 284, 287, 288, 292, 295, 298, 305, 307, 452, 453, 457, 587, 588, 592, 594, 607, 624, 632, 635, 636, 637, 638, 653, 664, 665, 666, 667, 668, 675, 686, 688, 695, 697, 698, 702, 703, 705, 707

Offset: 1

Zak Seidov, Feb 23 2015

			Numbers {2, 3, 7, 62, 63}, their squares {4, 9, 49, 3844, 3969} and cubes {8, 27, 343, 238328, 250047} all are "one-less".

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000027, A052383, A067251, A255398, A255430, A255431.

filter:= proc(n) local L,i;
  for i from 1 to 3 do
    L:= convert(n^i,base,10);
    if member(1,L) then return false fi;
  od;
  true
end proc:
select(filter, [seq(seq(10*i+j, j=2..8),i=0..100)]); # Robert Israel, Apr 03 2024

Select[Range[800],NumberDigit[#,0]!=0&&FreeQ[Flatten[ IntegerDigits/@ {#,#^2,#^3}],1]&] (* Harvey P. Dale, Sep 24 2021 *)

Definition modified by Harvey P. Dale, Sep 24 2021

A379982 Nonmultiples of 10 that are divisible by the square of the sum of the squares of their digits.

Original entry on oeis.org

1, 2023, 4332, 30324, 31311, 43011, 52022, 52215, 71824, 101376, 110201, 116964, 120213, 120472, 120612, 131072, 141312, 145152, 202312, 230202, 233928, 244634, 298374, 305252, 320305, 327184, 409374, 506056, 511104, 519168, 565152, 615627, 652118, 667815, 680625

Offset: 1

Amiram Eldar, Jan 07 2025

			2023 is a term since 2023 is not divisible by 10 and it is divisible by (2^2 + 0^2 + 2^2 + 3^2)^2 = 289.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pradip Kumar Pal and Kaushik Gopalan, Second Order Harshad Number, International Journal of Mathematical Education, Vol. 13, No. 1 (2023), pp. 25-26.

Intersection of A067251 and A379980.

Select[Range[10^6], ! Divisible[#, 10] && Divisible[#, (Plus @@ (IntegerDigits[#]^2))^2] &]

isok(k) = k % 10 && !(k % vecsum(apply(x -> x^2, digits(k)))^2);

def ok(n): return n%10 and n%sum(di**2 for di in map(int, str(n)))**2 == 0
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 10 2025

A103951 Procedure "Remove every 10th term!" executed 10 times.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 23, 25, 27, 29, 32, 35, 38, 42, 46, 49, 51, 54, 56, 59, 62, 65, 68, 72, 75, 76, 79, 83, 84, 87, 92, 93, 96, 102, 103, 105, 106, 113, 114, 116, 117, 125, 126, 128, 129, 137, 138, 139, 142, 143, 152, 153, 154, 157, 158, 162, 168, 169, 171

Offset: 1

Zak Seidov, Feb 22 2005

nonn

Procedure A067251, "Every 10th number has been omitted", executed 10 times.
0. Start with the sequence of natural numbers:
s0=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
1. Remove each 10th term:
s1=1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,...
2. Remove each 10th term:
s2=1,2,3,4,5,6,7,8,9,12,13,14,15,16,17,18,19,21,23,24,...
After 10 executions we have:
s10=1,2,3,4,5,6,7,8,9,21,23,25,27,29,32,35,38,42,46,49,...
This algorithm may be generalized to
"Remove each k-th term k times!". We have then
k=2: s2=1,5,9,13,17,21,25,29,33,37,41,45,... a(n)=4n-3 (n=1,2,...)
k=3: s3=1,2,7,10,14,16,20,23,28,29,34,37,41,43,47,50,55, (no simple rule here and further?)
k=4: s4=1,2,3, 9,11,14,18,22,23,29,30,33,38,39,43,46,50,51
k=5, s5=1,2,3, 4,11,13,16,19,23,26,28,32,34,39,41,42,48,51,52
k=6, s6=1,2,3, 4, 5,13,15,17,20,23,27,31,32,37,38,44,45,47,52,53
k=7, s7=1,2,3, 4, 5, 6,15,17,19,22,25,29,33,34,38,39,44,45,51,52,54
k=8, s8=1,2,3, 4, 5, 6, 7,17,19,21,23,26,29,33,37,39,42,44,47,50,53
k=9, s9=1,2,3, 4, 5, 6, 7, 8,19,21,23,25,28,31,34,38,42,43,47,48,52,
Is there any general rule for a_k(n)?

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A067251.

ra=Range[1000];k=10;Do[ra=Drop[ra, {k, Length[ra], k}], {i, k}];ra
Nest[Flatten[Most/@Partition[#,10]]&,Range[300],10] (* Harvey P. Dale, Mar 17 2015 *)

A255431 Numbers n such that integers n through n+6 and their squares all lack the digit 1 in their decimal expansion.

Original entry on oeis.org

22, 62, 82, 472, 522, 662, 822, 832, 932, 972, 2362, 2382, 2522, 2562, 2632, 2972, 4522, 4832, 4862, 4882, 4932, 4972, 5022, 5062, 5082, 5322, 5472, 6062, 6382, 6632, 6662, 6782, 6972, 7022, 7382, 7432, 7472, 7632, 8322, 8332, 8362, 8672, 8882

Offset: 1

Zak Seidov, Feb 23 2015

Integers n through n+6 are terms in A255430.

All terms are congruent to 2 mod 10.

			22,23,24,25,26,27,28 and their squares 484,529,576,625,676,729,784 are "one-less".

Cf. A000027, A052383, A067251, A255398, A255430.

A337836 a(n) is the smallest base of the form 8 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

Original entry on oeis.org

8, 18, 68, 2318, 7318, 1068, 32318, 501068, 7532318, 3626068, 23157318, 120813568, 3538782318, 1097376068, 110960657318, 49925501068, 1880980188568, 355101282318, 53760863001068, 15613890344818, 587818480188568, 2495167113001068

Offset: 1

Marco Ripà, Sep 24 2020

Let n >= 1. For any t == 8 (mod 10), if 5^n divides (t^2 + 1) and 5^(n + 1) does not divide (t^2 + 1), then V(t) = n (where V(t) represents the convergence speed of t). In particular, the aforementioned property holds for any a(n), since a(n) belongs to the residue class 8 modulo 10 for any n. Moreover, 5^n always divides (a(n) + A340345(n)).
From Marco Ripà, Dec 31 2021: (Start)
In general, any tetration base m = A067251(n) which is congruent to {2,8}(mod 10) is characterized by a convergence speed equal to the 5-adic valuation of m^2 + 1. Similarly, if m is congruent to 4(mod 10), then the convergence speed of m is given by m + 1, whereas if m belongs to the congruence class 6 modulo 10, then its convergence speed is m - 1. Lastly, for any m congruent to 5 modulo 10, the congruence speed exceeds by 1 the 2-adic valuation of m^2 - 1
Moreover, assuming m > 1, m^m is not congruent to m^m^m if and only if m belongs to the congruence class 18 modulo 20 or 2 modulo 20, whereas if m = A067251(n) is not coprime to 10 and is not equal to 5, then the number of new stable digits from m^m^m to m^m^m^m is always equal to the convergence speed of m. The aforementioned statement, in general, is untrue if m is coprime to 10 (see "Number of stable digits of any integer tetration" in the Links section).
(End)

			For n = 3, a(3) = 68 is characterized by a convergence speed of 3, and it is the smallest base such that V(a) = 3. Moreover, 5^3 has to divide a(3) (i.e., a(3)^2+1 = 4625 = 5^3*37 is a multiple of 5^3).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà, Number of stable digits of any tetration, ResearchGate, December 2021.

Cf. A317905, A337392, A337833, A340345, A349425.

a(n) = g(n) + u(n), where g(n) = (-2^5^n (mod 10^n)) (mod 2*5^n) and where u(n) = [0 iff g(n) <> g(n + 1); 2*5^n iff g(n) = g(n + 1)].

a(n) = 5-adic valuation of a(n)^2 + 1. - Marco Ripà, Dec 31 2021

A340345 a(n) is the smallest base of the form 2 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

Original entry on oeis.org

2, 32, 432, 182, 5182, 30182, 123932, 1061432, 280182, 15905182, 74498932, 367467682, 1344030182, 23316686432, 11109655182, 255250280182, 1170777623932, 7274293248932, 22533082311432, 175120972936432, 365855836217682, 7041576051061432

Offset: 1

Marco Ripà, Jan 04 2021

Let n >= 1. For any t == 2 (mod 10), if 5^n divides (t^2 + 1) and 5^(n + 1) does not divide (t^2 + 1), then V(t) = n (where V(t) represents the convergence speed of t). In particular, the aforementioned property holds for any a(n), since a(n) belongs to the residue class 2 modulo 10 for any n. Moreover, 5^n always divides (a(n) + A337836(n)).
From Marco Ripà, Dec 31 2021: (Start)
In general, any tetration base m = A067251(n) which is congruent to {2,8}(mod 10) is characterized by a convergence speed equal to the 5-adic valuation of m^2 + 1. Similarly, if m is congruent to 4(mod 10), then the convergence speed of m is given by m + 1, whereas if m belongs to the congruence class 6 modulo 10, then its convergence speed is m - 1. Lastly, for any m congruent to 5 modulo 10, the congruence speed exceeds by 1 the 2-adic valuation of m^2 - 1.
Moreover, assuming m > 1, m^m is not congruent to m^m^m if and only if m belongs to the congruence class 2 modulo 20 or 18 modulo 20, whereas if m = A067251(n) is not coprime to 10 and is not equal to 5, then the number of new stable digits from m^m^m to m^m^m^m is always equal to the convergence speed of m. The aforementioned statement, in general, is untrue if m is coprime to 10 (see "Number of stable digits of any integer tetration" in the Links section).
(End)

			For n = 4, a(4) = 182 is characterized by a convergence speed of 4, and it is the smallest base such that V(a) = 4. Moreover, 5 has to divide a(4)^2+1 exactly four times (i.e., a(4)^2+1 = 33125 = 5^4*53 is a multiple of 5^4 and is not divisible by 5^5).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà, Number of stable digits of any tetration, ResearchGate, December 2021.

Cf. A317905, A337392, A337833, A337836, A349425.

a(n) = g(n) + u(n), where g(n) = (2^5^n (mod 10^n)) (mod 2*5^n) and where u(n) = [0 iff g(n) <> g(n + 1); 2*5^n iff g(n) = g(n + 1)].

a(n) = 5-adic valuation of a(n)^2 + 1. - Marco Ripà, Dec 31 2021

A342851 Remove duplicates in the decimal digit-reversal of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82

Offset: 1

Michael De Vlieger, Mar 24 2021

Primitive terms in A004086.
Corresponds with A023804 for 1 <= n <= 73. The term 81 in this sequence is "100" in base 9, in which 2 digits are the same, therefore 81 does not appear in A023804.
0 plus integers that are not a multiple of 10. - Chai Wah Wu, Mar 25 2021
Differs "in substance" from A209931, because e.g. this sequence contains 214 and 214 is not in A209931 (because 107|214 and 107 contains a zero). - R. J. Mathar, Jul 29 2021
Differs from the finite sequence A023804. - R. J. Mathar, Jul 07 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding Structure in Sequences of Real Numbers via Graph Theory: a Problem List, arXiv:2012.04625 [math.CO], 2020.

Cf. A004086. Essentially the same as A067251.

Mathematica
```
Union@ IntegerReverse[Range[0, 100]]
```

A342851_list = [d for d in range(10**3) if d == 0 or d % 10] # Chai Wah Wu, Mar 25 2021

A342941 Numbers not ending with 0, that are not the quotient of a Zuckerman number divided by the product of its digits.

Original entry on oeis.org

15, 16, 24, 25, 26, 32, 35, 38, 39, 42, 43, 47, 54, 55, 58, 62, 65, 71, 73, 75, 78, 85, 87, 92, 95, 99, 105, 107, 108, 115, 116, 117, 119, 123, 125, 127, 131, 135, 137, 138, 139, 141, 142, 145, 146, 147, 155, 165, 175, 176, 178, 179, 181, 185, 189, 191, 193, 195, 197, 199

Offset: 1

Bernard Schott, Mar 30 2021

Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits (see link).

Multiples of 10 are never the quotient of a Zuckerman number divided by the product of its digits, but they are not present in this sequence (see A342593).

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Equals A342593 \ A008592.
Subsequence of A067251.
Cf. A007602, A056770, A288069.

cp's OEIS Frontend

A076641 Integers read backwards, but with repetitions omitted.

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A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

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A255357 Natural numbers n, other than multiples of 10, such that n, n^2 and n^3 lack the digit 1 in their decimal expansion.

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A379982 Nonmultiples of 10 that are divisible by the square of the sum of the squares of their digits.

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A103951 Procedure "Remove every 10th term!" executed 10 times.

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A255431 Numbers n such that integers n through n+6 and their squares all lack the digit 1 in their decimal expansion.

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A337836 a(n) is the smallest base of the form 8 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

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A340345 a(n) is the smallest base of the form 2 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

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