Ivan Stoykov - cp's OEIS frontend

A358348 Numbers k such that k == k^k (mod 9).

1, 4, 7, 9, 10, 13, 16, 17, 18, 19, 22, 25, 27, 28, 31, 34, 35, 36, 37, 40, 43, 45, 46, 49, 52, 53, 54, 55, 58, 61, 63, 64, 67, 70, 71, 72, 73, 76, 79, 81, 82, 85, 88, 89, 90, 91, 94, 97, 99, 100, 103, 106, 107, 108, 109, 112, 115, 117, 118, 121, 124, 125, 126

Offset: 1

Ivan Stoykov, Nov 11 2022

Each multiple of 9 is in the sequence. Additionally, the squares are also present.

			4 is a term since 4^4 = 256 == 4 (mod 9).

M. Fujiwara and Y. Ogawa, Introduction to Truly Beautiful Mathematics. Tokyo: Chikuma Shobo, 2005.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A007953, A010888, A082576, A189510.

A358348 := proc(n)
    2*(n+1)-op(modp(n,9)+1,[2,3,2,1,1,2,1,0,1]) ;
end proc:
seq(A358348(n),n=1..50) ; # R. J. Mathar, Mar 29 2023

Select[Range[130], MemberQ[{0, 1, 4, 7, 9, 10, 13, 16, 17}, Mod[#, 18]] &] (* Amiram Eldar, Nov 12 2022 *)

isok(k) = k == Mod(k,9)^k; \\ Michel Marcus, Nov 22 2022

def A358348(n):
    return ((0, 1, 4, 7, 9, 10, 13, 16, 17)[m := n % 9]
         + (n - m << 1))  # Chai Wah Wu, Feb 09 2023

G.f.: x*(x+1)*(x^7+3*x^5+x^3+x^2+2*x+1)/((1-x)^2*(1+x^3+x^6)*(1+x+x^2)). - Alois P. Heinz, Feb 08 2023

a(n) = 2*(n+1) - b(n) where b(n>=0) = 2,3,2,1,1,2,1,0,1,2,3,2,... has period 9. - Kevin Ryde, Mar 26 2023

A333448 Smallest positive divisibility coefficient of A045572(n).

Original entry on oeis.org

1, 1, 5, 1, 10, 4, 12, 2, 19, 7, 19, 3, 28, 10, 26, 4, 37, 13, 33, 5, 46, 16, 40, 6, 55, 19, 47, 7, 64, 22, 54, 8, 73, 25, 61, 9, 82, 28, 68, 10, 91, 31, 75, 11, 100, 34, 82, 12, 109, 37, 89, 13, 118, 40, 96, 14, 127, 43, 103, 15, 136, 46, 110, 16, 145, 49, 117

Offset: 1

Ivan Stoykov, Mar 21 2020

nonn

The sequence was generated in an attempt to create a universal divisibility test. Namely, taking the last digit of the number inspected, multiplying it by a number (the "divisibility coefficient"), and adding it to the inspected number without the last digit. Then, if the result is divisible by the number we are checking, so is our original number. This test works only for numbers coprime to 10, hence the sequence is based on A045572. The sequence lists the smallest positive divisibility coefficients of the members of A045572.

a(n) may equivalently be defined as the multiplicative inverse of 10 modulo A045572(n). - Ely Golden, Mar 27 2024

			For example, let us check whether 21 is divisible by 7. First, we take off the last digit, 1. Since 7 is the third member of A045572, its divisibility coefficient is the third member of this sequence, namely 5. Then we multiply 5 times 1 to obtain 5, and we add it to the original number without the last digit, in our case, 2. We get 7, and since it is clearly divisible by 7, so is 21.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A045572.

Array[# - (# Mod[PowerMod[#, 3, 10], 10] - 1)/10 &[1/2*(5*# + Mod[3*# + 2, 4] - 4)] &, 67] (* Michael De Vlieger, Oct 05 2020 *)

lista(nn) = {for (n=1, nn, if (gcd(n,10) == 1, my(m=n % 10, k=n\10, x); if (m == 1, x = 9*k+1); if (m == 3, x = 3*k+1); if (m == 7, x = 7*k+5); if (m == 9, x = k+1); print1(x, ", ");););} \\ Michel Marcus, May 04 2020

def a(n):
    u = 10*((n-1) // 4) + [1, 3, 7, 9][(n-1) % 4]
    return pow(10, -1, u) + (u == 1)
print(*(a(i) for i in range(1,101)), sep=", ")
# Ely Golden, Mar 27 2024

The sequence can be defined piecewise: 9m+1 for numbers of the form 10m+1; 3m+1 for numbers of the form 10m+3; 7m+5 for numbers of the form 10m+7 and m+1 for numbers of the form 10m+9.
From Lorenzo Sauras Altuzarra, Sep 29 2020: (Start)
a(n) = 1/10 - (1 - 2*(floor((n + 1)/4) + n))*(1 - (1 + (floor(16*9^n/205) mod 9))/10).
a(n) = b(n) - (((b(n) mod 10)^3 mod 10)*b(n) - 1)/10, where b(n) = A045572(n). (End)

More terms from Michel Marcus, May 04 2020

A306583 Positive integers that cannot be represented as a sum or difference of factorials of distinct integers.

Original entry on oeis.org

11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 107, 108, 109, 131, 132, 133, 155, 156, 157

Offset: 1

Ivan Stoykov, Feb 25 2019

nonn

It can be proved that any number in the gap between n! + (n-1)! + (n-2)! + ... + 2! + 1! + 0! and (n+1)! - (n! + (n-1)! + (n-2)! + ... + 2! + 1! + 0!) is in this sequence.

0! and 1! are treated as distinct. - Bernard Schott, Feb 25 2019

			10 can be represented as 10 = 0! + 1! + 2! + 3!, so it is not a term.
11 cannot be represented as a sum or a difference of factorials, so it is a term.

Cf. A000142 and A007489.

Cf. A059589 (Sums of factorials of distinct integers with 0! and 1! treated as distinct), A059590 (Sums of factorials of distinct integers with 0! and 1! treated as identical), A005165 (Alternating factorials).

Complement[Range[160], Total[# Range[0, 5]!] & /@ (IntegerDigits[ Range[3^6 - 1], 3, 6] - 1)] (* Giovanni Resta, Feb 27 2019 *)

More terms from Giovanni Resta, Feb 27 2019

A322406 a(n) = n + n*n^n.

Original entry on oeis.org

2, 10, 84, 1028, 15630, 279942, 5764808, 134217736, 3486784410, 100000000010, 3138428376732, 106993205379084, 3937376385699302, 155568095557812238, 6568408355712890640, 295147905179352825872, 14063084452067724991026, 708235345355337676357650, 37589973457545958193355620

Offset: 1

Ivan Stoykov, Dec 07 2018

All terms are produced by successively applying the three basic operations: exponentiation, multiplication and addition.

			a(3) = 3 + 3*3^3 = 3 + 3*27 = 8 + 81 = 84.

Muniru A Asiru, Table of n, a(n) for n = 1..380
R. L. Goodstein, Transfinite Ordinals in Recursive Number Theory, Journal of Symbolic Logic, Vol. 12, No. 4 (Dec. 1947), pp. 123-129.

Cf. A066068, A055897.

Equals 2 * A108398.

List([1..20],n->n+n*n^n); # Muniru A Asiru, Dec 07 2018

[n+n*n^n$n=1..20]; # Muniru A Asiru, Dec 07 2018

a[n_]:=n+n*n^n; Array[a, 20] (* Stefano Spezia, Dec 07 2018 *)

a(n) = n+n*n^n \\ Felix Fröhlich, Dec 07 2018

E.g.f.: exp(x) * x - LambertW(-x)/(1 + LambertW(-x))^3. - Vaclav Kotesovec, Dec 20 2018

a(12)-a(19) from Stefano Spezia, Dec 07 2018

A321970 Numbers k such that 7^k ends with k.

Original entry on oeis.org

3, 43, 343, 2343, 72343, 172343, 5172343, 65172343, 565172343, 1565172343, 11565172343, 511565172343, 5511565172343, 65511565172343, 265511565172343, 1265511565172343, 31265511565172343, 331265511565172343, 3331265511565172343, 43331265511565172343

Offset: 1

Ivan Stoykov, Nov 26 2018

Leftmost digit of a(n) is A133617(n-1) for n <= 30. - Alois P. Heinz, Nov 26 2018

			7^3 = 343, and it ends with 3, so 3 is a term.

J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x == x (mod b^n), J. Int. Seq. 12 (2009) #09.8.8.

Cf. A133617.

Sequence A064541 is similar, but uses the smallest single-digit prime as a base, unlike this one, which uses the largest single-digit prime as a base.

a[1] = 3; a[n_] := a[n] = For[ida = IntegerDigits[a[n-1]]; k = 1, True, k++, idk = IntegerDigits[k]; pm = PowerMod[7, an = FromDigits[Join[idk, ida]], 10^IntegerLength[an]]; If[pm == an, Return[an]]]; Array[a, 20] (* after Jean-François Alcover in A064541 *)

A321771 Numbers whose digit product equals the number of their digits.

Original entry on oeis.org

1, 12, 21, 113, 131, 311, 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111, 11115, 11151, 11511, 15111, 51111, 111116, 111123, 111132, 111161, 111213, 111231, 111312, 111321, 111611, 112113, 112131, 112311, 113112, 113121, 113211, 116111, 121113

Offset: 1

Ivan Stoykov, Nov 21 2018

Idea is similar to A061384, which uses addition instead of multiplication.

			12 has two digits, and their product is also 2, as 1*2=2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061384.
Cf. A007954, A055642. Subsequence of A007602.
Subsequence of A052382 (zeroless numbers).

Select[Range[1000000], Length[IntegerDigits[#]] == Times @@ IntegerDigits[#] &] (* Amiram Eldar, Nov 21 2018 *)

isok(n) = my(d=digits(n)); vecprod(d) == #d; \\ Michel Marcus, Nov 22 2018

More terms from Amiram Eldar, Nov 21 2018

A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 28, 29, 30, 42, 46, 50, 52, 99, 134, 148, 165, 205, 245, 249, 315, 390, 441, 461, 525, 560, 763, 962, 1596, 1666, 1716, 1847, 1854, 1860, 3515, 4501, 5179, 6850, 7345, 7867, 8940, 9491, 9523, 15688, 19988, 23574, 23956

Offset: 1

Ivan Stoykov, Jun 17 2018

nonn

			For k=6, A055228(6) = ceiling(sqrt(6!)) = 27, and 27^2-27 = 702, which is a multiple of 6.

Hazewinkel, Michiel, ed. (2001) [1994], Gamma Function, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Chai Wah Wu, Table of n, a(n) for n = 1..89 (all terms up to 10^6, n = 1..70 from Jon E. Schoenfield)

Cf. A000188, A055228.

Select[Range[4600],Divisible[Ceiling[Sqrt[#!]]^2-Ceiling[Sqrt[#!]],#]&] (* Harvey P. Dale, Mar 02 2023 *)

default(realprecision,10^5); for(n=1,10^4, if( Mod( ceil(sqrt(n!)) - ceil(sqrt(n!))^2 , n) == 0, print1(n,", "))); \\ Joerg Arndt, Jun 17 2018

Terms > 99 from Joerg Arndt, Jun 17 2018

cp's OEIS Frontend

User: Ivan Stoykov

A358348 Numbers k such that k == k^k (mod 9).

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A333448 Smallest positive divisibility coefficient of A045572(n).

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A306583 Positive integers that cannot be represented as a sum or difference of factorials of distinct integers.

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A322406 a(n) = n + n*n^n.

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A321970 Numbers k such that 7^k ends with k.

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A321771 Numbers whose digit product equals the number of their digits.

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Author

Keywords

Comments

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Mathematica

PARI

Extensions

A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).

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