] = T[ - cp's OEIS frontend

A381233 Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

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

Offset: 0

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)

Cf. A196199, A381232.

Flatten[Table[(-1)^j*Floor[j/2], {k, 0, 10}, {j, 2*k + 1}]] (* Paolo Xausa, Mar 01 2025 *)

A381232 Count down from k to -k for k = 0, 1, 2, ... .

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)

Cf. A196199, A381233.

Flatten[Table[Range[k, -k, -1], {k, 0, 10}]] (* Paolo Xausa, Mar 01 2025 *)

from math import isqrt
def A381232(n): return (t:=isqrt(n))*(t+1)-n # Chai Wah Wu, Mar 01 2025

a(n) = -A196199(n) = floor(sqrt(n))*(floor(sqrt(n))+1)-n. - Chai Wah Wu, Mar 01 2025

A381104 a(n) is the number of prime factors with exponent 1 in the prime factorization of the n-th superabundant number.

Original entry on oeis.org

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

Offset: 1

Agustin T. Besteiro, Feb 14 2025

nonn

Alaoglu and Erdős proved that for all superabundant numbers, the exponents in their prime factorization are non-increasing. Moreover, there is always a sequence of prime factors with exponent 1 at the end of the factorization. The only exceptions for this sequence are 1, 4 and 36.

			For n=8 the 8th superabundant number is 48 = 2^4*3^1. Only one prime factor appears with exponent 1 so a(8) = 1.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017.

Cf. A004394, A056169.

a(n) = A056169(A004394(n)).

A365144 Numbers having each digit once and whose 4th power has each digit four times.

Original entry on oeis.org

5702631489, 7264103985, 7602314895, 7824061395, 8105793624, 8174035962, 8304269175, 8904623175, 8923670541, 9451360827, 9785261403, 9804753612, 9846032571

Offset: 1

T. D. Noe, Nov 09 2011

Currently same terms as A114260, but that sequence has more terms to follow. - Ray Chandler, Aug 23 2023

			5702631489 is a term since its 4th power 1057550783692741389295697108242363408641 contains four 5's, four 7's, four 0's and so on.

Cf. A050278 (pandigital numbers), A199630, A199631, A199633. Subsequence of A114260.

t = Select[Permutations[Range[0, 9]], #[[1]] > 0 &]; t2 = Select[t, Union[DigitCount[FromDigits[#]^4]] == {4} &]; FromDigits /@ t2 (* T. D. Noe, Nov 08 2011 *)

A363256 Number of length n strings on the alphabet {0,1,2,3} with digit sum at most 4.

Original entry on oeis.org

1, 4, 13, 32, 66, 121, 204, 323, 487, 706, 991, 1354, 1808, 2367, 3046, 3861, 4829, 5968, 7297, 8836, 10606, 12629, 14928, 17527, 20451, 23726, 27379, 31438, 35932, 40891, 46346, 52329, 58873, 66012, 73781, 82216, 91354, 101233, 111892, 123371, 135711

Offset: 0

Daniel T. Martin, May 23 2023

			For n=2, the 13 strings are all possible 2-character strings of '0', '1', '2' and '3' except the four strings '33', '32', '23'.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A227259 (the same for {0,1,2} with digit sum <= 4).
Cf. A105163 (the same for {0,1,2} with digit sum <= 3, shifted by 2).
Cf. A005718.

f[n_, r_, l_] := If[r < 0, 0, If[r==0, 1, If[l < 0, 0, If[l == 0, 1, Sum[f[n, r-j, l-1], {j, 0, n}]]]]]; Table[f[3, 4,x], {x, 0, 40}]

a(n) = (((n + 10)*n + 35)*n + 26)*n/24 + 1.
G.f.: -(x^4 - 3*x^3 + 3*x^2 - x + 1)/(x - 1)^5.
a(n) = 1 + A005718(n-1) for n>=1.

A361741 Starting positions of digit triples in the decimal expansion of Pi where the sum of the first 2 equals the third.

Original entry on oeis.org

1, 3, 10, 29, 61, 73, 83, 106, 117, 132, 177, 192, 195, 198, 241, 248, 251, 281, 309, 311, 333, 362, 381, 393, 432, 477, 486, 494, 504, 508, 525, 532, 536, 555, 602, 611, 628, 647, 662, 674, 689, 699, 710, 747, 755, 760, 771, 806, 853, 856, 887, 899, 927, 934, 966, 969, 989

Offset: 1

Aaron T Cowan, Mar 22 2023

The first digit of Pi, "3", is reckoned as position 1.

This pattern happens from the first digit of Pi, so it seems to be pretty basic.

			1 is the first term, since the first two digits 3 and 1 add up to 4.
3 is the second term, since 4 + 1 = 5.
10 is next, since 3 + 5 = 8.

Index entries for sequences related to the number Pi

Cf. A000796, A110883.

p=char(vpa(pi,1000));p(2)='3';
for i=2:strlength(p)-2
  if str2num(p(i))+str2num(p(i+1))==str2num(p(i+2)) fprintf('%i,',i-1)  end
end

Integers k such that A000796(k) + A000796(k+1) = A000796(k+2).
Equivalently, A110883(k) = A000796(k+2).
In a random model of this sequence (call it A(n)), A(n) ~ kn with probability 1, where k = 200/11. - Charles R Greathouse IV, Mar 28 2023

A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

Original entry on oeis.org

8, 13, 41, 53, 137, 173, 305, 397, 533, 877, 977, 1373, 1697, 1885, 2245, 2813, 3517, 3737, 4493, 5077, 5345, 6277, 6953, 7937, 9413, 10217, 10613, 11465, 12077, 12785, 16165, 17165, 18869, 19325, 22237, 22837, 24665, 26605, 27925, 29933, 32141, 32765, 36497, 37253, 38953, 39745

Offset: 1

Aaron T Cowan, Feb 20 2023

nonn

The value d is the square of the length of the diagonal of a trapezoid with a height and bases that are consecutive primes, respectively. The diagonal length is calculated using the Pythagorean theorem, but this distance is squared so that the value is an integer.

			        p(2)=3
        _ _ _ _
a(1):  |        \  d^2=2^2+(5-3)^2=8
p(1)=2 |_ _ _ _ _\
        p(3)=5
        p(3)=5
        _ _ _ _ _ _
a(2):  |           \    d^2=3^2 + (7-5)^2 = 9+4 = 13
p(2)=3 |            \
       |_ _ _ _ _ _ _\
        p(4)=7
a(3)= 5^2+(11-7)^2 = 25+16 = 41
a(7)= 17^2+(23-19)^2=305 = 5*61

Aaron T Cowan, Table of n, a(n) for n = 1..500

Cf. A001248, A076821.

Cf. A131019, A106171.

%shorter 1 line version
arrayfun(@(p) p^2+(nextprime(nextprime(p+1)+1)-nextprime(p+1))^2,[primes(10^6)])

Map[(#[[1]]^2 + (#[[3]] - #[[2]])^2) &, Partition[Prime[Range[50]], 3, 1]] (* Amiram Eldar, Feb 24 2023 *)

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2; \\ Michel Marcus, Feb 23 2023

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2.

a(n) = A001248(n) + A076821(n+1). - Michel Marcus, Feb 23 2023

A356718 T(n,k) is the total number of prime factors, counted with multiplicity, of k!*(n-k)!, for 0 <= k <= n. Triangle read by rows.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 5, 4, 3, 3, 4, 5, 7, 5, 5, 4, 5, 5, 7, 8, 7, 6, 6, 6, 6, 7, 8, 11, 8, 8, 7, 8, 7, 8, 8, 11, 13, 11, 9, 9, 9, 9, 9, 9, 11, 13, 15, 13, 12, 10, 11, 10, 11, 10, 12, 13, 15, 16, 15, 14, 13, 12, 12, 12

Offset: 0

Dario T. de Castro, Aug 24 2022

k!*(n-k)! is the denominator in binomial(n,k) = n!/(k!*(n-k)!) and all prime factors in the denominator cancel to leave an integer, so that T(n,k) = A022559(n) - A132896(n,k).

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7
  ---+--------------------------------------
   0 | 0
   1 | 0, 0;
   2 | 1, 0, 1;
   3 | 2, 1, 1, 2;
   4 | 4, 2, 2, 2, 4;
   5 | 5, 4, 3, 3, 4, 5;

Dario T. de Castro, Rows n = 0..140 of triangle, flattened

Cf. A007318, A001222, A022559, A132896, A303279.

T[n_,k_]:=PrimeOmega[Factorial[k]*Factorial[n-k]];
tab=Flatten[Table[T[n,k],{n,0,10},{k,0,n}]]

T(n,k) = bigomega(k!*(n-k)!), where 0 <= k <= n.

T(n,0) = T(n,n) = A022559(n).

A354139 a(n) is the least positive integer m such that (k+1)^n + (k+2)^n + ... + (k+m)^n == 0 (mod n) for every positive integer k.

Original entry on oeis.org

1, 4, 3, 8, 5, 36, 7, 16, 3, 20, 11, 72, 13, 28, 15, 32, 17, 108, 19, 200, 21, 44, 23, 144, 5, 52, 3, 56, 29, 180, 31, 64, 33, 68, 35, 216, 37, 76, 39, 400, 41, 1764, 43, 88, 15, 92, 47, 288, 7, 20, 51, 104, 53, 324, 55, 112, 57, 116, 59, 1800, 61, 124, 21, 128, 65, 396, 67, 136, 69, 140, 71

Offset: 1

Dimitrios T. Tambakos, May 18 2022

nonn

a(n) divides n * A007947(n).

			a(2) = 4 because, for every positive integer k, (k+1)^2 + (k+2)^2 + (k+3)^2 + (k+4)^2 == 0 (mod 2), and no smaller positive integer satisfies this condition.

sum[n_, r_] := Mod[Sum[k^r, {k, 1, n}], r];
rad[r_] := Product[i[[1]], {i, FactorInteger[r]}];
seq[r_] := Table[sum[n, r], {n, 1, r*rad[r]}];
A354139[r_] := Piecewise[   {    {rad[r], OddQ[r]},
    {2*r, EvenQ[r] && PrimePowerQ[r]},
    {Length[FindRepeat[seq[r]]], EvenQ[r] && Not[PrimePowerQ[r]]}
    }
   ];
Table[A354139[r], {r, 1, 20}] (* Improved by Dimitrios T. Tambakos, Feb 08 2023 *)

isok(k, n) = my(p=sum(i=1, k, Mod(i+x, n)^n)); if (p==0, return(1)); for (i=1, n, if (subst(p, x, i) != 0, return(0))); return(1);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, May 21 2022

a(2^t) = 2^(t+1) for integers t>0.
a(n) = A007947(n) for odd integers n.
Conjecture: a(n) = A007947(n) * A193267(n).

A348776 The numbers >= 2 with 3 repeated.

Original entry on oeis.org

2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 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

Offset: 1

N. J. A. Sloane, Nov 07 2021, following a suggestion from L. Guyot and T. Y. Lam

This sequence, 2, 3, 3, 4, 5, 6, 7, ..., gives the stable range of the polynomial rings Z, Z[x_1], Z[x_1, x_2], Z[x_1, x_2, x_3], ...

A note on terminology: "stable range" and "stable rank" are the same thing. In the English-speaking world, people have always used the term "stable range", which was what Bass had invented in the early '60s. When Russian workers wrote on this theme, of course they used a Russian translation of the term "stable range". When the term was translated back into English, it became "stable rank"! - T. Y. Lam, Nov 07 2021

T. Y. Lam, Excursions in Ring Theory, in preparation, 2021. See Section 24.

F. Grunewald, J. Mennicke, and L. Vaserstein, On the groups SL_2(Z[x]) and SL_2(k[x, y]), Israel J. Math., 86(1-3):157-193, 1994.
Luc Guyot, The stable rank of Z[x] is 3, arXiv:2111.02965 [math.AC], 2021-2025.
MathOverflow, Bass' stable range of Z[X].
L. N. Vaseršteĭn and Andrey Aleksandrovich Suslin, Serre's Problem on Projective Modules over Polynomial Rings, and Algebraic K-theory, Mathematics of the USSR-Izvestiya 10.5 (1976): 937 (Russian version).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

def A348776(n): return n+int(n<3) # Chai Wah Wu, Aug 09 2022

a(n) = n for n >= 3.
From Chai Wah Wu, Aug 09 2022: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 4.
G.f.: x*(x^3 - x^2 - x + 2)/(x - 1)^2. (End)
E.g.f.: x*(2*(1 + exp(x)) + x)/2. - Stefano Spezia, Apr 25 2025

cp's OEIS Frontend

User: ] = T[

A381233 Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A381232 Count down from k to -k for k = 0, 1, 2, ... .

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A381104 a(n) is the number of prime factors with exponent 1 in the prime factorization of the n-th superabundant number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A365144 Numbers having each digit once and whose 4th power has each digit four times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363256 Number of length n strings on the alphabet {0,1,2,3} with digit sum at most 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A361741 Starting positions of digit triples in the decimal expansion of Pi where the sum of the first 2 equals the third.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Formula

A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Mathematica

PARI

Formula

A356718 T(n,k) is the total number of prime factors, counted with multiplicity, of k!*(n-k)!, for 0 <= k <= n. Triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs