A068407 -id:A068407 - cp's OEIS frontend

A072495 Automorphic numbers: numbers k such that k^21 ends with k. Also m-morphic numbers for any m such that (m-1)/10 is an even integer not divisible by 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99, 101, 107, 125, 143

Offset: 1

Benoit Cloitre, Oct 19 2002

Definition: k is an m-morphic number if k^m ends with k. For this sequence m can be 21, 41, 61, ...

3-morphic numbers = 7-morphic numbers, see A033819; 5-morphic numbers = 13-morphic numbers, see A068407.

Cf. A003226, A033819, A068407, A068408, A072496.

isok(n, m=21)={n == 0 || (n^m)%(10^(1+logint(n,10))) == n}

Missing terms inserted by Sean A. Irvine, Oct 05 2024

A072496 k-morphic numbers for any k such that (k-1)/10 is an odd integer not divisible by 5.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 11, 16, 19, 21, 24, 25, 29, 31, 36, 39, 41, 44, 49, 51, 56, 59, 61, 64, 69, 71, 75, 76, 79, 81, 84, 89, 91, 96, 99, 101, 125, 149, 151, 176, 199, 201, 224, 249, 251, 299, 301, 349, 351, 375, 376, 399, 401, 424, 449, 451, 499, 501, 549, 551

Offset: 1

Benoit Cloitre, Oct 19 2002

nonn

Definition: n is a k-morphic number if n^k ends with n.

Cf. A072495.

3-morphic numbers = 7-morphic numbers, see A033819; 5-morphic numbers = 13-morphic numbers, see A068407.

k=11; for(n=1,10000,if((n^k)%(10^ceil(log(n)/log(10)))==n, print1(n,","))); \\ starting with 4, 5, ...

def automorphic(maxdigits, pow, base=10) :
    morphs = [[0]]
    for i in range(maxdigits):
        T=[d*base^i+x for x in morphs[-1] for d in range(base)]
        morphs.append([x for x in T if x^pow % base^(i+1) == x])
    res = list(set(sum(morphs, []))); res.sort()
    return res
# (call with pow=11 for this sequence), Eric M. Schmidt, Jul 30 2013

Sequence corrected by Eric M. Schmidt, Jul 30 2013

A227071 Let s(m) = the set of k > 0 such that k^m ends with k. Then a(n) = least m such that s(m) = s(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 3, 2, 9, 2, 11, 2, 5, 2, 3, 6, 17, 2, 3, 2, 21, 2, 3, 2, 9, 26, 3, 2, 5, 2, 11, 2, 33, 2, 3, 6, 5, 2, 3, 2, 41, 2, 3, 2, 5, 6, 3, 2, 17, 2, 51, 2, 5, 2, 3, 6, 9, 2, 3, 2, 21, 2, 3, 2, 65, 6, 3, 2, 5, 2, 11, 2, 9, 2, 3, 26, 5, 2, 3, 2, 81, 2

Offset: 1

T. D. Noe, Jul 29 2013

See A227070 for more details and for the numbers n such that n = a(n).

The entries in the b-file have been tentatively obtained by comparing the terms < 10^30 in the sets s(n). - Giovanni Resta, Jul 30 2013

Giovanni Resta, Table of n, a(n) for n = 1..10000 (warning: contains tentative terms)

Cf. A003226 (n=2), A033819 (n=3), A068407 (n=5), A068408 (n=6).
Cf. A072496 (n=11), A072495 (n=21), A076650 (n=26).
Cf. A227070 (n such that n = a(n)).

ts = {{}}; t2 = {1}; te = {1}; Do[s = Select[Range[0, 10^7], PowerMod[#, n, 10^IntegerLength[#]] == # &]; If[MemberQ[ts, s], AppendTo[t2, te[[Position[ts, s, 1, 1][[1, 1]]]]], AppendTo[ts, s]; AppendTo[te, n]; AppendTo[t2, n]], {n, 2, 82}]; t2

Conjecture: a(n+1) = A132741(n) + 1. - Eric M. Schmidt, Jul 30 2013

Mathematica program and some entries corrected by Giovanni Resta, Jul 30 2013

A224904 Primes p such that the decimal expansion of p^5 ends in p.

Original entry on oeis.org

2, 3, 5, 7, 43, 193, 251, 307, 443, 499, 557, 751, 1249, 1693, 3307, 4999, 5443, 5807, 7057, 7499, 20807, 22943, 31249, 49999, 52057, 54193, 56249, 79193, 97943, 281249, 672943, 4218751, 4999999, 5422943, 8281249, 8704193, 17077057, 74218751, 407922943

Offset: 1

K. D. Bajpai, Jul 25 2013

Subsequence of A068407. - Giovanni Resta, Jul 25 2013

			193 is a prime and 193^5=267785184193 ends in 193, hence 193 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A068407.

with(numtheory):with(StringTools):KD := proc() local a,b,d,e,f; a:= ithprime(n);b:= a^5;d:=length(a);e:=floor(b/(10^d))*10^d;f:=b-e;if a=f then RETURN (a) fi:end:seq(KD(),n=1..500000);

d[n_] := Block[{x}, Select[x /. List@ ToRules@ Reduce[x^5 == x, {x}, Modulus -> 10^n], # > 10^(n-1) && PrimeQ@# &]]; Union @@ d /@ Range@ 9 (* Giovanni Resta, Jul 25 2013 *)

is(n)=isprime(n) && Mod(n,10^#digits(n))^5==n \\ Charles R Greathouse IV, Jul 26 2013

a(35)-a(39) from Giovanni Resta, Jul 25 2013

A227070 Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 17, 21, 26, 33, 41, 51, 65, 81, 101, 126, 129, 161, 201, 251, 257, 321, 401, 501, 513, 626, 641, 801, 1001, 1025, 1251, 1281, 1601, 2001, 2049, 2501, 2561, 3126, 3201, 4001, 4097, 5001, 5121, 6251, 6401, 8001, 8193, 10001

Offset: 1

T. D. Noe, Jul 29 2013

These numbers might be called automorphic powers because the sets s(n) are called automorphic numbers. It appears that all numbers of the form 1 + 5^i are here. In fact, these appear to produce the only even numbers here. The set s(4) equals s(2). The set s(7) equals s(3). The set s(9) does not differ from s(5) until k = 10443. The set s(17) does not differ from s(9) until k = 108307. The sequence also has 126, 201, 251, 501, and 626, but there may be missing numbers.

Entries a(17)-a(49) have been tentatively obtained by comparing the terms < 10^30 in the sets s(n), for 2 <= n <= 10001. - Giovanni Resta, Jul 30 2013

Cf. A003226 (n=2), A033819 (n=3), A068407 (n=5), A068408 (n=6).
Cf. A072496 (n=11), A072495 (n=21), A076650 (n=26).
Cf. A227071.

ts = {}; t = {}; Do[s = Select[Range[11000000], PowerMod[#, n, 10^IntegerLength[#]] == # &]; If[! MemberQ[ts, s], Print[n]; AppendTo[ts, s]; AppendTo[t, n]], {n, 2, 101}]; t = Join[{1}, t]

Conjecture: a(n+1) = A003592(n) + 1. - Eric M. Schmidt, Jul 30 2013

a(17)-a(49) from Giovanni Resta, Jul 30 2013

A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).

Original entry on oeis.org

2, 2, 5, 307, 807, 72943, 795807, 1295807, 16295807, 166295807, 16666295807, 31666295807, 81666295807, 8581666295807, 26581907922943, 503581666295807, 2003581666295807, 90476581907922943, 140476581907922943, 6847003581666295807, 61847003581666295807, 911847003581666295807

Offset: 1

Marco Ripà, Jan 05 2025

The present sequence is a subsequence of A068407.
Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases whose congruence speed does not stabilize in less than b + 1 iterations, for any chosen positive integer b.
As a nontrivial example, the congruence speed of m := 45115161423787862411847003581666295807 becomes stable at height 41, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(45115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.

			a(5) = 807 since the congruence speed of 807 is 0 at height 1, 4 at heights 2, 3, 4, and 5, finally matching the value of the constant congruence speed of 807 at height 6 (and it is the smallest integer whose congruence speed stabilizes at height 6 or above).

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, Volume 26, 2020, Number 3, Pages 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Wikipedia, Tetration.

Cf. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387.

As long as ceiling(log_10(a(n))) < n, for any n > 3, the least significant floor(log_10(a(n))) digits of a(n) (from right to left) are given by the first floor(log_10(a(n))) entries of A290372(n), or A290373(n), or A290374(n), or A290375(n) (i.e., all but the first digit of each a(n) are described by ({5^2^k}_oo + {2^5^k}_oo) := ...17196359523418092077057, ({5^2^k}_oo - {2^5^k}_oo) := ...37588152996418333704193, (- {5^2^k}_oo + {2^5^k}_oo) := ...2411847003581666295807, and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943).

A380031 Smallest integer of d digits, greater than 1 and not ending in 0, whose constant congruence speed is not yet constant at height d + 2.

Original entry on oeis.org

5, 807, 81666295807, 81907922943, 161423787862411847003581666295807, 115161423787862411847003581666295807, 45115161423787862411847003581666295807, 44317662666830362972182803640476581907922943, 776138023544317662666830362972182803640476581907922943

Offset: 1

Marco Ripà, Jan 10 2025

The present sequence is a subsequence of A068407, but it is not a subsequence of A379906 (e.g., a(4) is not a term of A379906).
Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases of d digits whose congruence speed does not stabilize in less than d + 3 iterations (e.g., the congruence speed of 807, a 3-digit number, becomes constant only at height).
As a nontrivial example, the congruence speed of a(10) := 712222747129609220545115161423787862411847003581666295807 (a 57-digit number whose constant congruence speed is also 57) becomes stable at height 60, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(712222747129609220545115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.
The smallest integer of d digits whose constant congruence speed is not yet constant at height d + 3 is 435525708925199660525680385844696084258785712222747129609220545115161423787862411847003581666295807 (a 99-digit number whose congruence speed stabilizes at height 104 to its constant value of 101).
For any n >= 2, terms of this sequence derive from one digit 5 that appears in any of the two 10-adic solutions (- {5^2^k}_oo + {2^5^k}_oo) :=  ...2411847003581666295807 and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943 of the fundamental 10-adic equation y^5 = y (see "The congruence speed formula" in Links). The only other candidate terms can arise from the remaining two symmetric 10-adic solutions ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 and ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193 of y^5 = y as particular patterns of 0s and 5 may occur in the corresponding (neverending) strings (e.g., '50...0').
Consequently, if n > 1 is given, a(n) is always congruent modulo 50 to 7 or 3.

			a(2) = 807 since the corresponding 10-adic solution of y^5 = y is ...61423787862411847003581666295807 where the rightmost digit 5 appears to the left side of a(2) itself, while no smaller numbers with the same feature are achievable by cutting the 10-adic integer ...30362972182803640476581907922943 (also one of the 15 solutions of the fundamental 10-adic equation y^5 = y) in correspondence of its rightmost digit 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, Volume 26, 2020, Number 3, Pages 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Wikipedia, Tetration.

Cf. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387, A379906.

For any n > 1, a(n) corresponds to a cut on the right side of a digit 5 that appears inside one of the two strings (- {5^2^k}_oo - {2^5^k}_oo) := ...96579486665776138023544317662666830362972182803640476581907922943 and (- {5^2^k}_oo + {2^5^k}_oo) := ...84258785712222747129609220545115161423787862411847003581666295807, or even to a cut on the right side of a 5 belonging to rare digit-patterns consisting of juxtaposed 5 and trailing 0's occurring inside ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 or ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193.

cp's OEIS Frontend

A072495 Automorphic numbers: numbers k such that k^21 ends with k. Also m-morphic numbers for any m such that (m-1)/10 is an even integer not divisible by 10.

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A072496 k-morphic numbers for any k such that (k-1)/10 is an odd integer not divisible by 5.

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A227071 Let s(m) = the set of k > 0 such that k^m ends with k. Then a(n) = least m such that s(m) = s(n).

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A224904 Primes p such that the decimal expansion of p^5 ends in p.

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Maple

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A227070 Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

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A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).

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A380031 Smallest integer of d digits, greater than 1 and not ending in 0, whose constant congruence speed is not yet constant at height d + 2.

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