Andrzej Kukla - cp's OEIS frontend

A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 48, 53, 54, 56, 59, 60, 61, 64, 66, 67, 71, 72, 73, 78, 79, 80, 83, 84, 88, 89, 90, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 120, 126, 127, 128

Offset: 1

Andrzej Kukla, Nov 11 2024

Equivalently, positive integers m such that every number k <= d is a sum of distinct proper divisors of m, where d is the largest proper divisor of m (follows from Corollary 2.11 in the Kukla and Miska paper).

Rao and Peng (2013) proved that a number is quasi practical if and only if it is prime or practical (also Theorem 2.9 in Kukla/Miska paper).

Andrzej Kukla, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Union of A000040 and A005153.

Cf. A002093, A103288, A033630, A119348, A174533, A174973, A083207.

QuasiPracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]] || PrimeQ[n]]; Select[Range[200], QuasiPracticalQ] (* Created based on code by T. D. Noe, Apr 02 2010 *)

A347363 Number of self-avoiding knight's paths from the lower left corner to the lower right corner of a 3 X n chessboard.

Original entry on oeis.org

1, 0, 2, 8, 32, 156, 871, 5292, 28702, 154162, 845532, 4662014, 25579463, 140098348, 767973001, 4212065280, 23097682805, 126643657272, 694390484065, 3807499106946, 20877386149018, 114474503105178, 627683328355315, 3441701959286326, 18871492466212538

Offset: 1

Andrzej Kukla, Aug 29 2021

If we enumerate the squares in the 3 X n board like this:
------------------------------------
| 1 | 4 | 7 | 10 | 13 | ... | 3n-2 |
------------------------------------
| 2 | 5 | 8 | 11 | 14 | ... | 3n-1 |
------------------------------------
| 3 | 6 | 9 | 12 | 15 | ... |  3n  |
------------------------------------
then a(n) is the number of self-avoiding knight's paths on such a board from square 3 to square 3n.

			For n = 4 we have exactly 8 self-avoiding paths starting at square 3 and ending at square 12:
  3,  4,  9, 10,  5, 12;
  3,  4,  9,  2,  7, 12;
  3,  8,  1,  6,  7, 12;
  3,  4, 11,  6,  7, 12;
  3,  8,  1,  6, 11,  4,  9,  2,  7, 12;
  3,  4, 11,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6, 11,  4,  9, 10,  5, 12;

Cf. A118067, A169696, A212715.

a(8)-a(15) from Pontus von Brömssen, Aug 30 2021

Terms a(16) and beyond from Andrew Howroyd, Nov 19 2021

A345406 Integers k such that k = d1^(c) + d2^(c) + ... + dc^(c), where d^(c) denotes the rising factorial of d, c is the length of k and di is the i-th digit of k in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 90, 744, 840

Offset: 1

Andrzej Kukla, Jun 18 2021

The rising factorial d^(c) is defined as d*(d+1)*(d+2)*...*(d+c-1).

			7^(3) + 4^(3) + 4^(3) = 7*8*9 + 4*5*6 + 4*5*6 = 504 + 120 + 120 = 744, therefore 744 is in the list.

Cf. A014080 (factorions), A265609 (rising factorials), A345405.

q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[(d + nd - 1)!/(d - 1)!, {d, dig}] == n]; Select[Range[0, 1000], q] (* Amiram Eldar, Jun 18 2021 *)

A345405 Integers k such that k = (d1)_c + (d2)_c + ... + (dc)_c, where (d)_c denotes the descending factorial of d, c is the length of k and di is the i-th digit of k in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 86, 15960

Offset: 1

Andrzej Kukla, Jun 18 2021

The descending factorial (d)_c is defined as d*(d-1)*(d-2)*...*(d-c+1).

			(8)_2 + (6)_2 = 8*7 + 6*5 = 56 + 30 = 86, therefore 86 is in the list.

Cf. A014080 (factorions), A068424 (descending factorials), A345406.

q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[d!/(d - nd)!, {d, dig}] == n]; Select[Range[0, 16000], q] (* Amiram Eldar, Jun 18 2021 *)

A345404 a(n) is the smallest positive integer k such that |tan(k) - round(tan(k))| is smaller than 10^(-n), but greater than 10^(-n-1).

Original entry on oeis.org

11, 22, 1120, 355, 14817, 286602, 5117932, 144316263, 167004362, 8984683957

Offset: 1

Andrzej Kukla, Jun 18 2021

In other words, a(n) is the smallest positive integer k such that the distance between tan(k) and nearest integer to tan(k) is smaller than 10^(-n), but greater than 10^(-n-1).

			For n=3, a(n)=1120, because 1120 is the smallest positive integer such that |tan(1120) - round(tan(1120))| = 0.0008709... < 10^(-3) and 0.0008709... > 10^(-4).

Cf. A000209, A345328.

n := 1: for i from 2 to 10^10 do if 10^(-n - 1) < abs(evalf(tan(i)) - floor(evalf(tan(i)) + 1/2)) and abs(evalf(tan(i)) - floor(evalf(tan(i)) + 1/2)) < 10^(-n) then print(i); n := n + 1; i := 1; end if; end do;

Transpose[Table[Catch[Table[Table[{i, j};
      If[10^(-i - 1) < Abs[Tan[j] - Round[Tan[j]]] &&
        Abs[Tan[j] - Round[Tan[j]]] < 10^(-i),
       Throw[{i, j}]], {i}], {j, 10^(i+1)}]], {i, 10}]][[-1]] (* Bence Bernáth, Jul 07 2021 *)

a(n) = my(m); default(realprecision, 2*n); for(k=1, oo, if(10^-n > (m=abs(tan(k)-round(tan(k)))) && m > 10^(-n-1), return(k))); \\ Jinyuan Wang, Jun 18 2021

A345328 a(n) is the smallest integer k>1 such that |log(k)-round(log(k))| is smaller than 10^(-n).

Original entry on oeis.org

3, 20, 1096, 2981, 59874, 442413, 8886110, 65659969, 178482301, 3584912846, 26489122130, 195729609429, 3931334297144, 78962960182680, 214643579785916, 4311231547115195, 31855931757113756, 86593400423993747, 12851600114359308275, 34934271057485095348

Offset: 1

Andrzej Kukla, Jun 14 2021

nonn

In other words, a(n) is the smallest integer k>1 such that the distance between log(k) and nearest integer to log(k) is smaller than 10^(-n).

			For n=4 a(n)=2981, because 2981 is the smallest integer greater than 1 such that |log(2981)-round(2981)| = 0.00001409... < 10^(-4).

Cf. A000193, A000227, A079663, A080021.

n := 1: for i from 2 to 10^10 do if abs(evalf(log(i)) - floor(log(i) + 1/2)) < 10^(-n) then print(i); n := n + 1 fi end do;

\\ suitable precision needed.
a(n)={my(epsilon=1.0/10^n); for(k=1, oo, my(t=floor(exp(k))); if(k-log(t)Andrew Howroyd, Jun 14 2021

Terms a(10) and beyond from Andrew Howroyd, Jun 14 2021

A343187 Decimal expansion of Sum_{k>=1} 1/af(k), where af is the alternating factorial.

Original entry on oeis.org

2, 2, 6, 4, 4, 0, 5, 5, 1, 7, 9, 3, 2, 5, 3, 1, 7, 0, 6, 2, 9, 3, 4, 5, 7, 9, 7, 0, 3, 3, 6, 2, 9, 5, 3, 8, 4, 3, 7, 7, 0, 7, 9, 1, 2, 7, 4, 3, 7, 4, 4, 8, 9, 0, 3, 0, 4, 9, 6, 6, 7, 1, 0, 6, 1, 9, 8, 7, 0, 9, 1, 4, 2, 5, 9, 8, 7, 8, 7, 6, 8, 1, 2, 7, 2, 4, 7, 9, 3, 0, 4, 0, 7, 7, 0, 9, 0, 2, 8, 9, 8, 2, 7, 9, 9

Offset: 1

Andrzej Kukla, Apr 07 2021

			2.2644055179325317... = 1/1 + 1/1 + 1/5 + 1/19 + 1/101 +....

Cf. A005165 (alternating factorial).

evalf(sum(1/sum((-1)^(k - i)*i!, i = 1 .. k), k = 1 .. infinity));

f(n) = sum(k=0, n-1, (-1)^k*(n-k)!); \\ A005165
suminf(n=1, 1/f(n)) \\ Michel Marcus, Apr 07 2021

A342157 Base-10 numbers m such that k*m = d||d||...||d (here || appears k-1 times), where k is the length of m, d is any m's digit and || represents concatenation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 148, 185, 148148

Offset: 1

Andrzej Kukla, Mar 02 2021

All numbers satisfying such conditions must be smaller than 10^9, because if we take any 10-digit number m, 10m is an 11-digit number while d||d||...||d is a 10-digit number.

148 and 148148 are the only numbers in the sequence for which d is not necessarily the last digit (for 148 we take d=4, which is the second digit of 148 and for 148148 we take d=8, which is the last, but also the third digit).

			m=148 is in the sequence, because if we multiply 148 by k=3 (length of 148) we obtain 444, which is d||d||d for d=4 (second digit of 148)

A340163 For n>=1, smallest integer k such that for all m>=k: m^(1/n)+(m+1)^(1/n) >= (2^n*m+2^(n-1)-1)^(1/n).

Original entry on oeis.org

0, 0, 1, 2, 3, 7, 14, 28, 57, 115, 233, 469, 945, 1902, 3823, 7680, 15420, 30948, 62087, 124518, 249661, 500457, 1002986, 2009771, 4026532

Offset: 1

Andrzej Kukla, Dec 30 2020

For k>1, a(n) <= ceiling(2^(k-3)). This sequence refers to a conjecture, which is a generalization of a Question 723. (iii) from "Collected Papers", Srinivasa Ramanujan.

			For n=6, a(6)=7, because for all m<7: m^(1/n)+(m+1)^(1/n) < (2^n*m+2^(n-1)-1)^(1/n) and for all m>=7: m^(1/n)+(m+1)^(1/n) >= (2^n*m+2^(n-1)-1)^(1/n).

Srinivasa Ramanujan, Collected Papers, Question 723 in p. 332, Providence RI: AMS / Chelsea (2000).

B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), page 14 (see Q723, JIMS VII).
MathOverflow, Generalization of a problem, involving radicals and the floor function, proposed by Ramanujan to the Journal of the Indian Mathematical Society

A335961 Alternating factorions: Numbers m such that m = S_af(m) = af(d_1)+af(d_2)+...+af(d_k) where d_1 d_2 ... d_n is the decimal expansion of m and af(m) = m!-(m-1)!+(m-2)!+...1! (alternating factorial) with af(0) = 0 (base 10).

Original entry on oeis.org

0, 1, 620, 621, 643

Offset: 1

Andrzej Kukla, Jul 01 2020

Largest k such that S_af(k) > k is 1599999. That's why there are only five numbers such that S_af(m) = m. Proved by computer calculations.

If m has eight or more digits then S_af(m) < m. Proved directly.

			For m = 620, S_af(620) = af(6)+af(2)+af(0) = 619+1+0 = 620.

Cf. A005165 (alternating factorial), A014080 (factorions).

af[0] = 0; af[n_] := af[n] = n! - af[n - 1]; Select[Range[1000], Total[af /@ IntegerDigits[#]] == # &] (* Amiram Eldar, Jul 02 2020 *)

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User: Andrzej Kukla

A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

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A347363 Number of self-avoiding knight's paths from the lower left corner to the lower right corner of a 3 X n chessboard.

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A345406 Integers k such that k = d1^(c) + d2^(c) + ... + dc^(c), where d^(c) denotes the rising factorial of d, c is the length of k and di is the i-th digit of k in base 10.

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A345405 Integers k such that k = (d1)_c + (d2)_c + ... + (dc)_c, where (d)_c denotes the descending factorial of d, c is the length of k and di is the i-th digit of k in base 10.

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A345404 a(n) is the smallest positive integer k such that |tan(k) - round(tan(k))| is smaller than 10^(-n), but greater than 10^(-n-1).

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A345328 a(n) is the smallest integer k>1 such that |log(k)-round(log(k))| is smaller than 10^(-n).

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A343187 Decimal expansion of Sum_{k>=1} 1/af(k), where af is the alternating factorial.

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A342157 Base-10 numbers m such that k*m = d||d||...||d (here || appears k-1 times), where k is the length of m, d is any m's digit and || represents concatenation.

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A340163 For n>=1, smallest integer k such that for all m>=k: m^(1/n)+(m+1)^(1/n) >= (2^n*m+2^(n-1)-1)^(1/n).

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