A008851 -id:A008851 - cp's OEIS frontend

A151971 Numbers n such that n^2 - n is divisible by 21.

0, 1, 7, 15, 21, 22, 28, 36, 42, 43, 49, 57, 63, 64, 70, 78, 84, 85, 91, 99, 105, 106, 112, 120, 126, 127, 133, 141, 147, 148, 154, 162, 168, 169, 175, 183, 189, 190, 196, 204, 210, 211, 217, 225, 231, 232, 238, 246, 252, 253, 259, 267, 273, 274, 280, 288, 294, 295, 301, 309

Offset: 1

N. J. A. Sloane, Aug 23 2009

Equivalently, numbers that are congruent to {0, 1, 7, 15} mod 21. - Bruno Berselli, Aug 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13:A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984.

Cf. A215202.

[n: n in [0..309] | IsZero((n^2-n) mod 21)]; // Bruno Berselli, Aug 06 2012

A151971:=n->(42*n+14*I^((n-1)*n)-3*I^(2*n)-3)/8-7: seq(A151971(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

Select[Range[0,400], Divisible[#^2-#,21]&] (* Harvey P. Dale, Jun 04 2012 *)

makelist((42*n+14*%i^((n-1)*n)-3*(-1)^n-3)/8-7, n, 1, 60); /* Bruno Berselli, Aug 06 2012 */

From Bruno Berselli, Aug 06 2012: (Start)
G.f.: x^2*(1+6*x+8*x^2+6*x^3)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = (42*n +14*i^((n-1)*n) -3*(-1)^n -3)/8 -7, where i=sqrt(-1). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Wesley Ivan Hurt, Jun 07 2016
E.g.f.: (24 + (21*x - 31)*cosh(x) + 7*(sin(x) + cos(x) + (3*x - 4)*sinh(x)))/4. - Ilya Gutkovskiy, Jun 07 2016

A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 5, 6, 1, 1, 4, 9, 1, 1, 7, 8, 1, 6, 10, 1, 1, 1, 9, 10, 1, 1, 5, 16, 1, 7, 15, 1, 11, 12, 1, 1, 9, 16, 1, 1, 13, 14, 1, 1, 8, 21, 1, 1, 6, 10, 15, 16, 21, 25, 1, 1, 1, 12, 22, 1, 17, 18, 1, 15, 21, 1, 9, 28, 1, 1, 19, 20, 1, 13

Offset: 2

Eric M. Schmidt, Aug 05 2012

The n-th row has length A034444(n) - 1.
If m appears in row n, then gcd(n,m) appears in the n-th row of A077610. Moreover, if m', distinct from m, also appears in row n, then gcd(n, m) does not equal gcd(n, m').
For odd n and any integer m, m^2 == m (mod n) iff m^2 == m (mod 2n).
Let P(1)={1} and for integers x > 1, let P(x) be the set of distinct prime divisors of x. We can define an equivalence relation ~ on the set of elements in the ring (Z_n, +mod n,*mod n): for all a,b in Z_n (where a,b are the least nonnegative residues modulo n) a ~ b iff P(gcd(a,n)) intersect P(n) is equal to P(gcd(b,n)) intersect P(n). If we include 0 in each row then these elements can represent the equivalence classes. They form a commutative monoid. - Geoffrey Critzer, Feb 13 2016

			Triangle begins:
1;
1;
1;
1;
1, 3, 4;
1;
1;
1;
1, 5, 6;
1;
1, 4, 9;
1;
1, 7, 8;
1, 6, 10;
1;
1;
1, 9, 10; etc.  - _Bruno Berselli_, Aug 06 2012

Eric M. Schmidt, Rows 2..2000, flattened

For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13: A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984; n=100: A008852; n=1000: A008853.

[m: m in [1..n-1], n in [2..40] | m^2 mod n eq m]; // Bruno Berselli, Aug 06 2012

Table[Select[Range[n], Mod[#^2, n] == # &], {n, 2, 30}] // Grid (* Geoffrey Critzer, May 26 2015 *)

def A215202(n) : return [m for m in range(1, n) if m^2 % n == m];

A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 50, 60, 76, 100, 105, 110, 500, 501, 505, 506, 600, 605, 756, 760, 826, 1000, 1001, 1050, 1100, 5000, 5010, 5050, 5060, 5941, 6000, 6050, 7560, 7600, 8260, 10000, 10005, 10010, 10500, 10505, 11000, 12731

Offset: 1

Riccardo Pietro Giovambattista Mazzei and Matteo Albanese, Sep 28 2019

If k is in the sequence then so is 10*k. - David A. Corneth, Sep 29 2019

No term starts with the digit 2. - Chai Wah Wu, Apr 04 2023

			5^2 = 25, whose first digit is 5, hence 5 is a term of the sequence.
11^2 = 121, whose first and third digit are (1, 1), hence 11 is a term of the sequence.
756^2 = 571536, whose digits in odd positions - starting from the least significant one - are (7, 5, 6), hence 756 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..2361 (terms < 10^15; first 485 terms from Charles R Greathouse IV)

Subsequence of A008851, A029772, A046829, A064827, A052212, A189056.

Select[Range[0, 13000], Reverse@ #[[-Range[1, Length@ #, 2]]] &@ IntegerDigits[#^2] === IntegerDigits[#] &] (* Michael De Vlieger, Oct 06 2019 *)

isok(n) = my(d=Vecrev(digits(n^2))); fromdigits(Vecrev(vector((#d+1)\2, k, d[2*k-1]))) == n; \\ Michel Marcus, Oct 01 2019

def ok(n): s = str(n*n); return n == int("".join(s[1-len(s)%2::2]))
print(list(filter(ok, range(13000)))) # Michael S. Branicky, Sep 10 2021

A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

Original entry on oeis.org

0, 1, 6, 10, 15, 45, 55, 66, 91, 120, 136, 231, 276, 300, 406, 435, 496, 561, 595, 630, 741, 780, 820, 861, 1081, 1225, 1326, 1431, 1830, 2016, 2080, 2145, 2211, 2415, 2485, 2701, 2850, 3240, 3321, 3486, 3655, 3916, 4005, 4465, 4560, 4950, 5050, 5356, 5460, 5565

Offset: 1

Stefano Spezia, Oct 02 2022

Subset of even terms of A357505, divided by 2. - Michel Marcus, Nov 05 2022

Cf. A000217 (supersequence), A002378.
Half of the complement of A357504 in A020756.
Half of the complement of A020757 in A357505.
Subsequence of A008851.

TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=0, k<=105, k++, ok=1; For[h=0, h<2k, h++, If[TriangularQ[2*A000217[k] - A000217[h]] && k!=h, ok=0]]; If[ok==1, AppendTo[a,k(k+1)/2]]]; a (* Stefano Spezia, Nov 05 2022 *)

A370753 Antidiagonal products of A319840.

Original entry on oeis.org

1, 1, 4, 36, 576, 12800, 360000, 12192768, 481890304, 21743271936, 1101996057600, 61952000000000, 3824628881965056, 257164113195565056, 18704075505689706496, 1462975070062038220800, 122444006400000000000000, 10918111308394619734065152, 1033255398127440061257744384

Offset: 0

Stefano Spezia, Jun 22 2024

nonn

a(n) has trailing zeros iff n is congruent to 0 or 1 mod 5. Cf. A008851.
a(n) is a square iff n = 1 or congruent to {1, 3, 4} mod 5. Cf. A047206.
It appears that: (Start)
a(n) is a cube iff n = 0, 1, or is of the form (3*m - 4)^3 with m > 1 (A016791);
the only fourth powers in the sequence are 1 and a(9) = 21743271936 = 384^4;
the only fifth powers in the sequence are 1 and a(32) = 227200942336^5;
a(n) is a sixth power iff n = 0, 1, or is of the form (6*m - 10)^3 with m > 1;
the only seventh powers in the sequence are 1 and a(128) = 77458109039896212820250015287665035595218944^7. (End)

Cf. A000079, A000169, A000290, A008851, A016791, A047206, A319840.

a[0]=a[1]=1; a[n_]:=n^2*2^(n-2)*(n-1)^(n-2); Array[a,19,0]

a(0) = a(1) = 1, and a(n) = n^2*2^(n-2)*(n - 1)^(n-2) for n > 1.

A222170 a(n) = n^2 + 2*floor(n^2/3).

Original entry on oeis.org

0, 1, 6, 15, 26, 41, 60, 81, 106, 135, 166, 201, 240, 281, 326, 375, 426, 481, 540, 601, 666, 735, 806, 881, 960, 1041, 1126, 1215, 1306, 1401, 1500, 1601, 1706, 1815, 1926, 2041, 2160, 2281, 2406, 2535, 2666, 2801, 2940, 3081, 3226, 3375, 3526, 3681, 3840

Offset: 0

Bruno Berselli, Aug 08 2013

Also, a(n) = n^2 + floor(2*n^2/3), since 2*floor(n^2/3) = floor(2*n^2/3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tadeusz Dorozinskis, Pentagonpolyhedra Doro
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Subsequence of A008851.

Cf. A004773 (numbers of the type n+floor(n/3)), A008810 (numbers of the type n^2-2*floor(n^2/3)), A047220 (numbers of the type n+floor(2*n/3)), A184637 (numbers of the type n^2+floor(n^2/3), except the first two).

Magma
```
[n^2+2*Floor(n^2/3): n in [0..50]];
```

I:=[0,1,6,15,26]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[n^2 + 2 Floor[n^2/3], {n, 0, 50}]
CoefficientList[Series[x (1 + x) (1 + 3 x + x^2) / ((1 + x + x^2) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 15, 26}, 50] (* Hugo Pfoertner, Jan 17 2023 *)

G.f.: x*(1+x)*(1 + 3*x + x^2)/((1 + x + x^2)*(1-x)^3).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = floor(5*n^2/3). - Wesley Ivan Hurt, Mar 16 2015
a(n) = a(n-3) + 5*(2n-3) [Tadeusz Dorozinski]. - Eduard Baumann, Jan 18 2023

A270343 Numbers k that end with ( sum of digits of k )^2.

Original entry on oeis.org

0, 1, 81, 3144, 3256, 6225, 6484, 6576, 7121, 7529, 7676, 9100, 9324, 9361, 9729, 9784, 12144, 12256, 15225, 15484, 15576, 16121, 16529, 16676, 18100, 18324, 18361, 18729, 18784, 21144, 21256, 24225, 24484, 24576, 25121

Offset: 1

Soumil Mandal, Mar 15 2016

All terms end with a digit from the set S = {0,1,4,5,6,9}.

The sum of the digits of the numbers repeat and also change with regular intervals. For example, the sum of the digits S1 = {12,16,15,22,24,11,23,26,10,18,19,27,28} which is followed by 3144 to 8784, 12144 to 18784, 21144 to 27784, 30144 to 36784. Again S2 = {21,25,15,22,24,11,23,26,10,18,19,27,28} is followed by 39441 to 45784, 48441 to 54784, 57441 to 67784, 66441 to 72784. It can be seen that a set containing 13 elements repeats itself for 4 consecutive ranges.

			For k=3256, sum of digits is 16 and 16^2 is 256.
For k=7121, sum of digits is 11 and 11^2 is 121.
For k=18784, sum of digits is 22 and 22^2 is 484.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
Soumil Mandal, Graph Of Sum Of Digits
Soumil Mandal, Graph Of Cumulative Sums

Cf. A003226, A008851, A018247, A118881.

Select[Range[0, 20000], Function[n, Function[k, If[n >= k, FromDigits@ Take[#, -IntegerLength@ k] == k, False]][Total[#]^2] &@ IntegerDigits@ n]] (* Michael De Vlieger, Mar 15 2016 *)
esdQ[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[ Total[ IntegerDigits[ n]]^2]},Take[ idn,-Length[idn2]]==idn2]; Select[ Range[ 0,26000],esdQ]//Quiet (* Harvey P. Dale, Jan 01 2022 *)

isok(n) = {sds = sumdigits(n)^2; nbs = #Str(sds); ((n - sds) % 10^nbs) == 0;} \\ Michel Marcus, Mar 16 2016

for i in range(0,200000):
    res = pow((sum(map(int,str(i)))),2)
    if(i%pow(10,len(str(res)))==res):print(i)
# Soumil Mandal, Mar 17 2016

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A151971 Numbers n such that n^2 - n is divisible by 21.

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A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).

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A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

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A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

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A370753 Antidiagonal products of A319840.

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A222170 a(n) = n^2 + 2*floor(n^2/3).

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A270343 Numbers k that end with ( sum of digits of k )^2.

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