Mohammed Yaseen - cp's OEIS frontend

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

1, 3, 8, 10, 18, 24, 30, 5, 9, 15, 28, 40, 72, 84, 90, 120, 7, 21, 26, 56, 70, 78, 108, 126, 168, 210, 34, 45, 52, 102, 140, 156, 252, 360, 420, 11, 33, 88, 110, 198, 264, 330, 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840, 58, 98, 174, 294

Offset: 1

Mohammed Yaseen, Nov 16 2024

			Triangle begins:
  n=1: 1, 3, 8, 10, 18, 24, 30;
  n=2: 5, 9, 15, 28, 40, 72, 84, 90, 120;
  n=3: 7, 21, 26, 56, 70, 78, 108, 126, 168, 210;
  n=4: 34, 45, 52, 102, 140, 156, 252, 360, 420;
  n=5: 11, 33, 88, 110, 198, 264, 330;
  n=6: 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840;
  n=7: 58, 98, 174, 294;
  ...

Cf. A000005 (tau), A000010 (phi).
Cf. A020488 (row 1), A062516 (row 2), A063469 (row 3), A063470 (row 4).
Cf. A112954 (row lengths), A175667 (1st column), A112955 (right column), A020491 (ordered terms).

A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 2, 1, 5, 16, 27, 8, 1, 6, 25, 32, 27, 1, 1, 7, 36, 125, 128, 81, 4, 1, 8, 49, 27, 625, 128, 243, 4, 1, 9, 64, 343, 216, 3125, 512, 243, 16, 1, 10, 81, 256, 2401, 81, 3125, 1024, 729, 1, 1, 11, 100, 729, 2048, 16807, 972, 15625, 4096, 2187, 4, 1

Offset: 0

Mohammed Yaseen, Aug 03 2024

			Array begins:
+-----+-----------------------------------------------+
| n\k |   2    3     4    5      6     7       8  ... |
+-----+-----------------------------------------------+
|  0  |   1    2     3    4      5     6       7  ... |
|  1  |   1    4     9   16     25    36      49  ... |
|  2  |   1    2    27   32    125    27     343  ... |
|  3  |   1    8    27  128    625   216    2401  ... |
|  4  |   1    1    81  128   3125    81   16807  ... |
|  5  |   1    4   243  512   3125   972  117649  ... |
|  6  |   1    4   243 1024  15625   486  823543  ... |
|  7  |   1   16   729 4096  78125 11664  823543  ... |
|  8  |   1    1  2187 2048 390625  2187 5764801  ... |
| ... | ...  ...   ...  ...    ...   ...     ...  ... |
+-----+-----------------------------------------------+

Cf. A374895 (numerators).

Cf. A008292, A121985, A277542.

T(n,k) = denominator(polylog(-n, 1/k));
matrix(7,7,n, k, T(n-1,k+1)) \\ Michel Marcus, Aug 04 2024

T(n,k) = denominator(polylog(-n, 1/k)).
T(n,k) = denominator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
T(0,k) = k-1.
T(1,k) = (k-1)^2.
T(2,k) = A277542(k-1).
T(n,2) = 1.
T(n,n) = A121985(n).

A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 3, 26, 1, 5, 20, 33, 150, 1, 6, 15, 44, 15, 1082, 1, 7, 42, 115, 380, 273, 9366, 1, 8, 7, 366, 285, 4108, 1491, 94586, 1, 9, 72, 91, 4074, 3535, 17780, 38001, 1091670, 1, 10, 45, 776, 70, 11334, 26355, 269348, 17295, 14174522, 1, 11, 110, 531, 10440, 2149, 189714, 458555, 4663060, 566733, 204495126

Offset: 0

Mohammed Yaseen, Jul 22 2024

			Array begins:
+-----+--------------------------------------------------------------+
| n\k |       2     3       4       5        6      7         8  ... |
+-----+--------------------------------------------------------------+
|  0  |       1     1       1       1        1      1         1  ... |
|  1  |       2     3       4       5        6      7         8  ... |
|  2  |       6     3      20      15       42      7        72  ... |
|  3  |      26    33      44     115      366     91       776  ... |
|  4  |     150    15     380     285     4074     70     10440  ... |
|  5  |    1082   273    4108    3535    11334   2149    174728  ... |
|  6  |    9366  1491   17780   26355   189714   3311   3525192  ... |
|  7  |   94586 38001  269348  458555  3706518 285929  11870648  ... |
|  8  | 1091670 17295 4663060 1139685 82749954 220430 319735800  ... |
| ... |     ...   ...     ...     ...      ...    ...       ...  ... |
+-----+--------------------------------------------------------------+

Cf. A374896 (denominators).

Cf. A000629, A008292, A121376, A276805.

T(n,k) = numerator(polylog(-n, 1/k));
matrix(7,7,n,k,T(n-1, k+1)) \\ Michel Marcus, Aug 04 2024

T(n,k) = numerator(polylog(-n, 1/k)).
T(n,k) = numerator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
T(0,k) = 1.
T(1,k) = k.
T(2,k) = A276805(k-1).
T(n,2) = A000629(n).
T(n,n) = A121376(n).

A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.

Original entry on oeis.org

3, -5, -8, 25, 22, 30, -119, -122, -114, -144, 721, 718, 726, 696, 840, -5039, -5042, -5034, -5064, -4920, -5760, 40321, 40318, 40326, 40296, 40440, 39600, 45360, -362879, -362882, -362874, -362904, -362760, -363600, -357840, -403200, 3628801, 3628798, 3628806, 3628776, 3628920, 3628080, 3633840, 3588480, 3991680

Offset: 2

Mohammed Yaseen, Jun 24 2024

			Triangle begins:
      3;
     -5,    -8;
     25,    22,    30;
   -119,  -122,  -114,  -144;
    721,   718,   726,   696,   840;
  -5039, -5042, -5034, -5064, -4920, -5760;
  ...

Cf. A000142, A153805, A373966.

Unsigned diagonals: A001048, A213167.

T[n_,k_]:= (-1)^n*n! + (-1)^(k+1)*k!; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* Stefano Spezia, Jun 24 2024 *)

Integral_{1..e} (log(x)^k - log(x)^n) dx = A373966(n,k)*e + T(n,k).

A373966 Triangle read by rows: T(n,k) = (-1)^(n+1) * A000166(n) + (-1)^(k) * A000166(k) for n >= 2 and 1 <= k <= n-1.

Original entry on oeis.org

-1, 2, 3, -9, -8, -11, 44, 45, 42, 53, -265, -264, -267, -256, -309, 1854, 1855, 1852, 1863, 1810, 2119, -14833, -14832, -14835, -14824, -14877, -14568, -16687, 133496, 133497, 133494, 133505, 133452, 133761, 131642, 148329, -1334961, -1334960, -1334963, -1334952, -1335005, -1334696, -1336815, -1320128, -1468457

Offset: 2

Mohammed Yaseen, Jun 24 2024

			Triangle begins:
    -1;
     2,    3;
    -9,   -8,  -11;
    44,   45,   42,   53;
  -265, -264, -267, -256, -309;
  1854, 1855, 1852, 1863, 1810, 2119;
  ...

Cf. A153805, A373967.
Unsigned columns: A000166, A000240.
Unsigned diagonals: A000255, A018934.

T[n_,k_]:= (-1)^(n+1)*Subfactorial[n] + (-1)^k*Subfactorial[k]; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* Stefano Spezia, Jun 24 2024 *)

Integral_{1..e} (log(x)^k - log(x)^n) dx = T(n,k)*e + A373967(n,k).

A374170 a(n) is the least nonsquare k such that sigma_n(k) divides sigma_2n(k).

Original entry on oeis.org

20, 20, 6050, 7203

Offset: 1

Mohammed Yaseen, Jun 30 2024

a(1) = A227771(1); a(2) = A046871(5).
a(5) > 10^9 if it exists.
a(6) = 17328, a(7) = 50, a(13) = 761378.

Cf. A020487, A227771, A046871.

a(n) = my(k=1, f=factor(k)); while (issquare(k) || (sigma(f, 2*n) % sigma(f, n)), f=factor(k++)); k; \\ Michel Marcus, Jun 30 2024

A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

Original entry on oeis.org

4, 64, 4, 7168, 4, 606528, 4, 64, 4, 4194304, 4

Offset: 2

Mohammed Yaseen, Mar 20 2023

a(13) <= 31525197391593472. - David A. Corneth, Mar 20 2023
From Thomas Scheuerle, Mar 22 2023: (Start)
a(17) <= 15211807202738752817960438464512 and a(19) <= 2^190*11.
Conjecture: a(n) is of the form 2^b*p1^c*p2^d*...*pk^j with b > 0 and A020639(n) divides b*(c+1)*(d+1)*...*(j+1). (p1, p2, ..., pk are distinct odd prime numbers). (End)

Cf. A003601, A020486, A046839, A046840.
Cf. A000005, A000203, A001157, A001158.
Cf. A020639, A010709 (bisection).

a[n_] := Module[{k = 1, d}, While[Divisible[DivisorSigma[1, k], (d = DivisorSigma[0, k])] || !Divisible[DivisorSigma[n, k], d], k++]; k]; Array[a, 11, 2] (* Amiram Eldar, Mar 20 2023 *)

isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f,n) % nd);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, Mar 20 2023

a(2*m) = 4 for m >= 1.
a(6*m-3) = 64 for m >= 1.
From Thomas Scheuerle, Mar 22 2023: (Start)
a(m) <= a(A020639(m)) if a(A020639(m)) <> -1.
Conjecture: For primes q > p, a(q) > a(p). If true, we could replace "<=" with "=" in the above formula. (End)

A359800 a(n) is the least m such that the concatenation of n^2 and m is a square.

Original entry on oeis.org

6, 9, 61, 9, 6, 1, 284, 516, 225, 489, 104, 4, 744, 249, 625, 3201, 444, 9, 201, 689, 4201, 416, 984, 4801, 681, 5201, 316, 996, 5801, 601, 6201, 144, 936, 6801, 449, 7201, 7401, 804, 7801, 225, 8201, 8401, 6, 8801, 9001, 9201, 9401, 324, 9801, 19344, 769, 38025

Offset: 1

Mohammed Yaseen, Jan 13 2023

The only one-digit terms are 1, 4, 6 and 9. Proof: Squares mod 10 are 0, 1, 4, 5, 6 and 9. Concatenation of a square and 0 is 10 times that square, which is not a square. So 0 is ruled out. Squares with last digit 5 have second last digit 2. Since no square ends in 2, 5 is also ruled out.
From Thomas Scheuerle, Jan 14 2023: (Start)
The only term with two digits is a(3) = 61.
Some terms with an odd number of digits appear infinitely often, for example, 516 for n = 8, 1352, 632958674, ... .
If a term has an even number of digits and is of the form 1+2*k*10^(d+1) with 10^d <= 2*k < 10^(d+1), then it appears only once at k = n in this sequence. Are terms with an even number of digits possible which are not of this form? (End)

			For n=3, 61 is the least number m such that the concatenation of 3^2 and m is a square: 961 = 31^2. So a(3) = 61.
For n=7, 284 is the least number m such that the concatenation of 7^2 and m is a square: 49284 = 222^2. So a(7) = 284.

Cf. A000290, A071176, A075836, A084070, A221874, A246560.

a(n)={my(m=n^2, b=1); while(1, m*=10; my(r=(sqrtint(m+b-1)+1)^2-m); b*=10; if(rAndrew Howroyd, Jan 13 2023

from math import isqrt
def a(n):
    t, k = str(n*n), isqrt(10*n**2)
    while not (s:=str(k*k)).startswith(t) or s[len(t)]=="0": k += 1
    return int(s[len(t):])
print([a(n) for n in range(1, 53)]) # Michael S. Branicky, Jan 15 2023

from math import isqrt
from sympy.ntheory.primetest import is_square
def A359800(n):
    m = 10*n*n
    if is_square(m): return 0
    a = 1
    while (k:=(isqrt(a*(m+1)-1)+1)**2-m*a)>=10*a:
        a *= 10
    return k # Chai Wah Wu, Feb 15 2023

a(n) = A071176(n^2) = A071176(A000290(n)).
From Thomas Scheuerle, Jan 13 2023: (Start)
a(A084070(n)) = 1.
a(2*A084070(n)) = 4.
a(A221874(n)) = 6.
a(A075836(n)) = 9. (End)

A359224 Numbers whose decimal representation is the reverse of their base-7 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 23, 46, 2116, 15226, 32361

Offset: 1

Mohammed Yaseen, Dec 21 2022

			23 is a term since 23_7 = 32 which reversed is 23.
32361 is a term since 32361_7 = 163230 which reversed is 032361 and the leading 0 is disregarded.

Cf. A004086, A007093.

isok(n) = fromdigits(Vecrev(digits(n, 7))) == n

a(12) from Jon E. Schoenfield, Dec 21 2022

A358340 a(n) is the smallest n-digit number whose fourth power is zeroless.

Original entry on oeis.org

1, 11, 104, 1027, 10267, 102674, 1026708, 10266908, 102669076, 1026690113, 10266901031, 102669009704, 1026690096087, 10266900960914, 102669009608176, 1026690096080369, 10266900960803447, 102669009608034434, 1026690096080341627, 10266900960803409734, 102669009608034097731, 1026690096080340972491

Offset: 1

Mohammed Yaseen, Nov 10 2022

It has been proved that there exist infinitely many zeroless squares and cubes but there is apparently no proof for 4th powers, 5th powers, etc.

This sequence approaches the decimal expansion of 9000^(-1/4). Similar sequences of other small powers k seem to approach the decimal expansion of (9*10^(k-1))^(-1/k).

Michael S. Branicky, Table of n, a(n) for n = 1..69
Eric Weisstein's World of Mathematics, Zerofree

Cf. A052382, A052040, A052044.

Cf. A253643, A252484, A253644, A253647, A124648, A124649.

a(n) = my(x=10^(n-1)); while(! vecmin(digits(x^4)), x++); x; \\ Michel Marcus, Nov 10 2022

a(n) = { my(s = sqrtnint(10^(4*n - 3) \ 9, 4)); for(i = s, oo, c = i^4; if(vecmin(digits(c)) > 0, return(i) ) ) } \\ David A. Corneth, Nov 10 2022

from itertools import count
from sympy import integer_nthroot
def a(n):
    start = integer_nthroot(int("1"*(4*(n-1)+1)), 4)[0]
    return next(i for i in count(start) if "0" not in str(i**4))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Nov 10 2022

a(n) ~ 10^(n + 1/4) / sqrt(3).

More terms from David A. Corneth, Nov 10 2022

cp's OEIS Frontend

User: Mohammed Yaseen

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

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A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

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A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

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A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.

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A373966 Triangle read by rows: T(n,k) = (-1)^(n+1) * A000166(n) + (-1)^(k) * A000166(k) for n >= 2 and 1 <= k <= n-1.

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A374170 a(n) is the least nonsquare k such that sigma_n(k) divides sigma_2n(k).

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A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

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A359800 a(n) is the least m such that the concatenation of n^2 and m is a square.

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A359224 Numbers whose decimal representation is the reverse of their base-7 representation.

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