Sebastian Karlsson - cp's OEIS frontend

A377918 a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

1, 11, 77, 566, 4197, 31148, 231193, 1716043, 12737453, 94544693, 701765055, 5208903636, 38663477066, 286982552081, 2130149470506, 15811193864583, 117359769764941, 871111674250772, 6465891595866732, 47993564275737877, 356235822660837879, 2644187054283807954, 19626676300599636003

Offset: 1

Sebastian Karlsson and N. J. A. Sloane, Nov 30 2024

These are the points in the graph of A342042 where the separate paths come together.

The first differences are in A377917, which is the more fundamental sequence. To get this sequence from A377917, add an initial zero, take partial sums, and add 1 to each term.

Ray Chandler, Table of n, a(n) for n = 1..1149
Index entries for linear recurrences with constant coefficients, signature (6,10,5,-5,-9,-5,-1).

Cf. A342042, A377912, A377917.

A377918 := proc(n) local S; option remember;
S:=[1, 11, 77, 566, 4197, 31148, 231193, 1716043];
if n <= 8 then S[n] else
6*A377918(n-1)+10*A377918(n-2)+5*A377918(n-3)-5*A377918(n-4)-9*A377918(n-5)-5*A377918(n-6)-A377918(n-7); fi;
end;
[seq(A377918(i),i=1..20)];

LinearRecurrence[{6, 10, 5, -5, -9, -5, -1}, {1, 11, 77, 566, 4197, 31148, 231193, 1716043}, 25] (* Paolo Xausa, Dec 02 2024  *)

G.f. = (x^7+6*x^6+15*x^5+19*x^4+11*x^3-x^2-5*x-1)/((1-x)*(x^6+6*x^5+15*x^4+20*x^3+15*x^2+5*x-1)) (From g.f. for A377917).
Recurrence: See Maple code.
The smallest root of the denominator of the g.f. is 0.134724138401519... whose reciprocal is (say) c1 = 7.422574840... Then a(n) is asymptotically c2*c1^n, for n >= 0, where c2 = 1.3824387... This is an excellent approximation. It gives a(22) = 0.1962667617*10^20, compared with a(22) = 19626676300599636003.
This also enables us to give a formula for the lower envelope of A342042 - see that entry for details.

More terms added based on A377917. - N. J. A. Sloane, Dec 01 2024

A377917 Number of n-digit terms in A377912.

Original entry on oeis.org

10, 66, 489, 3631, 26951, 200045, 1484850, 11021410, 81807240, 607220362, 4507138581, 33454573430, 248319075015, 1843166918425, 13681044394077, 101548575900358, 753751904485831, 5594779921615960, 41527672679871145, 308242258385100002, 2287951231622970075, 16982489246315828049

Offset: 1

Sebastian Karlsson and N. J. A. Sloane, Nov 30 2024

Also number of n-digit terms in A342042.
a(1149) has 1001 digits. - Michael S. Branicky, Nov 30 2024
The terms of A377912 as decimal digit strings are a regular language so can be counted using the transitions in a state machine matching those strings. - Kevin Ryde, Dec 01 2024

			The 66 two-digit terms in A377912 are
  10,11,12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,
  38,39,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,67,68,69,70,71,72,73,74,
  75,76,77,78,79,89,90,91,92,93,94,95,96,97,98,99.
There is an obvious division into 5 blocks of size 10 and blocks of sizes 7, 5, 3, and 1.

Michael S. Branicky, Table of n, a(n) for n = 1..1148
Michael S. Branicky, Python program for A377917
Index entries for linear recurrences with constant coefficients, signature (5,15,20,15,6,1).

Cf. A342042, A377912.

First differences of A377918.

LinearRecurrence[{5, 15, 20, 15, 6, 1}, {10, 66, 489, 3631, 26951, 200045, 1484850}, 25] (* Paolo Xausa, Dec 01 2024 *)

From Kevin Ryde, Dec 01 2024: (Start)
a(n) = 5*a(n-1) + 15*a(n-2) + 20*a(n-3) + 15*a(n-4) + 6*a(n-5) + a(n-6) for n>=8.
G.f.: -1 + x + (1+x)^4 / (1 - 5*x - 15*x^2 - 20*x^3 - 15*x^4 - 6*x^5 - x^6). (End)

a(6) and beyond from Michael S. Branicky, Nov 30 2024

A353776 a(n) = Sum_{d|n} (n/d mod d).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 3, 1, 4, 1, 7, 1, 4, 6, 3, 1, 7, 1, 8, 5, 4, 1, 14, 1, 4, 4, 10, 1, 14, 1, 7, 6, 4, 8, 11, 1, 4, 5, 17, 1, 16, 1, 10, 13, 4, 1, 19, 1, 9, 6, 8, 1, 16, 7, 17, 5, 4, 1, 32, 1, 4, 13, 7, 9, 19, 1, 8, 6, 23, 1, 27, 1, 4, 10, 10, 12, 16, 1, 23

Offset: 1

Sebastian Karlsson, May 07 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A055155, A057670, A082909.

import Math.NumberTheory.ArithmeticFunctions
a n = sum $ map (\d -> n `quot` d `rem` d) $ divisorsList n

a[n_] := DivisorSum[n, Mod[n/#, #] &]; Array[a, 100] (* Amiram Eldar, May 07 2022 *)

A353776(n) = sumdiv(n,d,((n/d)%d)); \\ Antti Karttunen, May 08 2022

A349423 Index of first occurrence of n in A348179, or -1 if n never appears.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1020, 11, 1023, 31, 1024, 51, 1063, 71, 1028, 91, 20, 21012, 12, 220035, 24, 20025, 26, 20074, 28, 220095, 230, 13, 300025, 33, 10413, 53, 300026, 73, 300085, 93, 40, 1041, 42, 3000461, 14, 5041, 46, 700451, 48, 9041, 520, 15

Offset: 0

Sebastian Karlsson, Nov 17 2021

			a(10) = 1020, because 1020 is the first number k such that A348179(k) = 10.
a(1111111110) = -1 (found by _Kevin Ryde_).

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 453 terms from Sebastian Karlsson)

Cf. A336668, A348179, A349422.

def A348179(n):
    s, l = str(n), len(str(n))
    return int("".join(s[(i + int(s[i])) % l] for i in range(l)))
terms = {}
for n in range(0, 3000462): # 3000462 to get all terms of the data-section without -1s.
    if (k := A348179(n)) not in terms: terms[k] = n
for n in range(0, 52):
    try: print(terms[n], end=", ")
    except: print(-1, end=", ") # These -1s are conjectured.

A349422 Replace each decimal digit d of n with the digit that is d steps to the left of d. Interpret the digits of n as a cycle: one step to the left from the first digit is considered to be the last.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 22, 31, 44, 51, 66, 71, 88, 91, 20, 22, 22, 22, 24, 22, 26, 22, 28, 22, 0, 13, 22, 33, 44, 53, 66, 73, 88, 93, 40, 44, 42, 44, 44, 44, 46, 44, 48, 44, 0, 15, 22, 35, 44, 55, 66, 75, 88, 95, 60, 66, 62, 66, 64, 66, 66, 66, 68, 66, 0, 17, 22, 37, 44, 57, 66, 77, 88, 97, 80, 88, 82, 88, 84, 88, 86, 88, 88, 88, 0, 19, 22, 39, 44, 59, 66, 79, 88, 99, 0, 100

Offset: 0

Sebastian Karlsson, Nov 17 2021

First differs from A348179 at a(101).

			a(3210) = 2020, because:
Moving 3 steps to the left from 3 gives: 3 -> 0 -> 1 -> 2.
Moving 2 steps to the left from 2 gives: 2 -> 3 -> 0.
Moving 1 step to the left from 1 gives: 1 -> 2.
Moving 0 steps to left from 0 gives: 0.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000

Cf. A336668 (fixed points), A348179 (to the right).

Cf. A139337, A346576, A347323.

import Data.Char (digitToInt)
a n = read [s !! mod (i - digitToInt (s !! i)) l | i <- [0..l-1]] :: Integer
    where s = show n; l = length s

a(n) = { my (d=digits(n)); fromdigits(vector(#d, k, d[1+(k-1-d[k])%#d])) } \\ Rémy Sigrist, Nov 17 2021

def a(n):
    s, l = str(n), len(str(n))
    return int("".join(s[(i - int(s[i])) % l] for i in range(l)))

A348179 Replace each decimal digit d of n with the digit that is d steps to the right of d. Interpret the digits of n as a cycle: one step to the right from the last digit is considered to be the first.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 22, 31, 44, 51, 66, 71, 88, 91, 20, 22, 22, 22, 24, 22, 26, 22, 28, 22, 0, 13, 22, 33, 44, 53, 66, 73, 88, 93, 40, 44, 42, 44, 44, 44, 46, 44, 48, 44, 0, 15, 22, 35, 44, 55, 66, 75, 88, 95, 60, 66, 62, 66, 64, 66, 66, 66, 68, 66, 0, 17, 22, 37, 44, 57, 66, 77, 88, 97, 80, 88, 82, 88, 84, 88, 86, 88, 88, 88, 0, 19, 22, 39, 44, 59, 66, 79, 88, 99, 0, 1

Offset: 0

Sebastian Karlsson, Oct 05 2021

First differs from A349422 at a(101). - Sebastian Karlsson, Dec 31 2021

			a(102345) = 004124 = 4124. For example, 4 gets replaced by 2 because moving 4 steps to the right gives: 4 -> 5 -> 1 -> 0 -> 2. Note that from 5 we went to the first digit of the number.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000

Cf. A336668 (fixed points), A349422 (to the left), A349423 (index of first appearance of n).

Cf. A106649, A139337, A322131, A339023, A341767, A341953, A346576, A347323.

import Data.Char (digitToInt)
a n = let s = show n; l = length s in
  read [s !! (mod (i + digitToInt (s !! i)) l) | i <- [0..l-1]] :: Integer

Table[FromDigits@Table[v[[If[(p=Mod[k+v[[k]],t])==0,t,p]]],{k,t=Length[v=IntegerDigits[n]]}],{n,0,67}] (* Giorgos Kalogeropoulos, Oct 08 2021 *)

f(k, d) = d[(k+d[k]-1)%#d + 1];
a(n) = my(d=digits(n), dd=vector(#d, k, f(k, d))); fromdigits(dd); \\ Michel Marcus, Oct 07 2021

def a(n):
    s, l = str(n), len(str(n))
    return int("".join(s[(i + int(s[i])) % l] for i in range(l)))

a(68)-a(101) from Sebastian Karlsson, Dec 31 2021

A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -2, -3, 1, -1, 1, -3, -4, 1, 1, -2, 3, -4, -5, 1, -1, 4, -3, 6, -5, -6, 1, 0, -2, 9, -4, 10, -6, -7, 1, 0, 0, -3, 16, -5, 15, -7, -8, 1, 1, 1, -1, -4, 25, -6, 21, -8, -9, 1, -1, 4, 3, -4, -5, 36, -7, 28, -9, -10, 1, 0, -2, 9, 6, -10, -6

Offset: 1

Sebastian Karlsson, Aug 20 2021

			  n\k| 1    2    3    4    5    6    7    8    9   10   11    12 ...
  ---+--------------------------------------------------------------
   1 | 1   -1   -1    0   -1    1   -1    0    0    1   -1     0 ...
   2 | 1   -2   -2    1   -2    4   -2    0    1    4   -2    -2 ...
   3 | 1   -3   -3    3   -3    9   -3   -1    3    9   -3    -9 ...
   4 | 1   -4   -4    6   -4   16   -4   -4    6   16   -4   -24 ...
   5 | 1   -5   -5   10   -5   25   -5  -10   10   25   -5   -50 ...
   6 | 1   -6   -6   15   -6   36   -6  -20   15   36   -6   -90 ...
   7 | 1   -7   -7   21   -7   49   -7  -35   21   49   -7  -147 ...
   8 | 1   -8   -8   28   -8   64   -8  -56   28   64   -8  -224 ...
   9 | 1   -9   -9   36   -9   81   -9  -84   36   81   -9  -324 ...
  10 | 1  -10  -10   45  -10  100  -10 -120   45  100  -10  -450 ...
  11 | 1  -11  -11   55  -11  121  -11 -165   55  121  -11  -605 ...
  12 | 1  -12  -12   66  -12  144  -12 -220   66  144  -12  -792 ...
  13 | 1  -13  -13   78  -13  169  -13 -286   78  169  -13 -1014 ...
  14 | 1  -14  -14   91  -14  196  -14 -364   91  196  -14 -1274 ...
  15 | 1  -15  -15  105  -15  225  -15 -455  105  225  -15 -1575 ...
  ...

Sebastian Karlsson, Antidiagonals n = 1..140, flattened

Row n=1..10 give A008683, A007427, A007428, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
Main diagonal gives A341837.
Cf. A077592, A163767,

T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2021 *)

T(n, k) = my(f=factor(k)); for (k=1, #f~, f[k,1] = binomial(n, f[k,2])*(-1)^f[k,2]; f[k,2]=1); factorback(f); \\ Michel Marcus, Aug 21 2021

from sympy import binomial, primefactors as pf, multiplicity as mult
from math import prod
def T(n, k):
    return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))

If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
Dirichlet g.f. of the n-th row: 1/zeta^n(s).
T(n, p) = -n.
T(n, n) = A341837(n).

A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 18, 25, 30, 36, 30, 54, 36, 54, 72, 66, 48, 90, 54, 108, 108, 90, 66, 150, 108, 108, 134, 162, 84, 216, 90, 168, 180, 144, 216, 270, 108, 162, 216, 300, 120, 324, 126, 270, 360, 198, 138, 396, 234, 324, 288, 324, 156, 402, 360, 450, 324

Offset: 1

Sebastian Karlsson, Aug 17 2021

Dirichlet convolution of A000010 with A029935.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A000010, A029935, A166633.

f[p_, e_] := (1/2)*(p - 1)*p^(e - 3)*(e^2*(p - 1)^2 + 3*e*(p^2 - 1) + 2*(p^2 + p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 17 2021 *)

from sympy import divisors as div, totient as phi
def D(f, g, n):
    return sum(f(d)*g(n//d) for d in div(n))
def phi_o_phi(n):
    return D(phi, phi, n)
def a(n):
    return D(phi, phi_o_phi, n)

Dirichlet g.f.: zeta(s - 1)^3 / zeta(s)^3.
Multiplicative with a(p^e) = (1/2)*(p-1)*p^(e-3)*(e^2*(p-1)^2 + 3*e*(p^2-1) + 2*(p^2 + p + 1)).
Sum_{k=1..n} a(k) ~ 27*n^2/Pi^10 * (2*Pi^4*log(n)^2 - 2*Pi^4*log(n)*(1 + 6*log(2) - 72*(1/12 - zeta'(-1)) + 6*log(Pi)) + Pi^4*(1 + 6*gamma*(2*gamma - 1) - 12*sg1) + 864*zeta'(2)^2 - 36*Pi^2*((6*gamma - 1)*zeta'(2) + zeta''(2))), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 24 2022

A341953 Replace each digit d in the decimal representation of n with the digital root of d^n.

Original entry on oeis.org

1, 4, 9, 4, 2, 9, 7, 1, 9, 10, 11, 11, 19, 17, 18, 19, 14, 11, 19, 40, 81, 77, 59, 11, 25, 49, 81, 71, 59, 90, 91, 94, 99, 94, 92, 99, 97, 91, 99, 40, 71, 11, 49, 77, 18, 49, 74, 11, 49, 70, 81, 47, 29, 11, 55, 79, 81, 41, 29, 90, 91, 94, 99, 94, 92, 99, 97

Offset: 1

Sebastian Karlsson, Feb 24 2021

The digits 0, 1, 3, 6 and 9 will always be replaced by the same digits: 0 -> 0, 1 -> 1, 3 -> 9, 6 -> 9 and 9 -> 9.

			a(14) = 17, since 1^14 = 1 and 4^14 = 268435456. 2 + 6 + 8 + 4 + 3 + 5 + 4 + 5 + 6 = 43 and 4 + 3 = 7. Thus, the digital root of 268435456 is 7. This means that for 14, "1" gets replaced by "1" and "4" gets replaced by "7".

Robert Israel, Table of n, a(n) for n = 1..10000
Sebastian Karlsson, Generalized and plotted in arbitrary bases

Cf. A010888, A106649, A139337, A322131, A339023, A341767.

digroot[n_] := If[n == 0, 0, Mod[n - 1, 9] + 1]; a[n_] := FromDigits[digroot /@ (IntegerDigits[n]^n)]; Array[a, 100] (* Amiram Eldar, Feb 24 2021 *)

r(n) = if(n, (n-1)%9+1) \\ A010888
a(n) = fromdigits(apply(x->r(x^n), digits(n))); \\ Michel Marcus, Mar 21 2021

def D(d, n):
    return 0 if d == 0 else (pow(d, n, 9)-1)%9 + 1
def a(n):
    return int(''.join(str(D(int(d), n)) for d in str(n)))

A341767 Replace each digit d in the decimal representation of n with the digital root of n^d.

Original entry on oeis.org

1, 4, 9, 4, 2, 9, 7, 1, 9, 11, 22, 39, 41, 54, 69, 71, 88, 99, 11, 41, 93, 77, 78, 99, 44, 11, 99, 11, 48, 91, 14, 87, 99, 17, 88, 99, 11, 84, 99, 41, 45, 99, 71, 11, 99, 11, 72, 99, 41, 21, 96, 44, 88, 99, 11, 51, 99, 77, 28, 91, 17, 11, 99, 11, 15, 99, 14

Offset: 1

Sebastian Karlsson, Feb 19 2021

If n == 1 (mod 9), then every digit will be replaced by "1". If n == 0 (mod 9), then all nonzero digits will be replaced by "9".

The corresponding n of values a(n)= 1, a(n)= 11, a(n)= 111,... creates a subsequence of A236653. - Davide Rotondo, Mar 04 2024

			a(26) = 11, since 26^2 = 676 and 26^6 = 308915776. 6 + 7 + 6 = 19, 1 + 9 = 10 and 1 + 0 = 1, so the digital root of 676 is 1. 3 + 0 + 8 + 9 + 1 + 5 + 7 + 7 + 6 = 46, 4 + 6 = 10 and 1 + 0 = 1, so the digital root of 308915776 is 1. Thus, for 26, both "2" and "6" will be replaced by "1".

Sebastian Karlsson, Table of n, a(n) for n = 1..10000
Sebastian Karlsson, Generalized and plotted in arbitrary bases

Cf. A010888, A106649, A139337, A322131, A339023.

a[n_] := FromDigits[Mod[n^IntegerDigits[n] - 1, 9] + 1]; Array[a, 100] (* Amiram Eldar, Feb 19 2021 *)

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = my(d=digits(n)); fromdigits(vector(#d, k, dr(n^d[k]))); \\ Michel Marcus, Feb 19 2021

def a(n):
    return int(''.join(str((pow(n, int(d), 9)-1)%9 + 1) for d in str(n)))

a(10*n) = 10*a(n) + 1.

cp's OEIS Frontend

User: Sebastian Karlsson

A377918 a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

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A377917 Number of n-digit terms in A377912.

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A353776 a(n) = Sum_{d|n} (n/d mod d).

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A349423 Index of first occurrence of n in A348179, or -1 if n never appears.

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A349422 Replace each decimal digit d of n with the digit that is d steps to the left of d. Interpret the digits of n as a cycle: one step to the left from the first digit is considered to be the last.

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A348179 Replace each decimal digit d of n with the digit that is d steps to the right of d. Interpret the digits of n as a cycle: one step to the right from the last digit is considered to be the first.

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A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

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