A060462 -id:A060462 - cp's OEIS frontend

A090542 Incorrect version of A060462.

1, 3, 5, 7, 8, 11, 13, 14, 15

Offset: 1

dead

A072668 Numbers one less than composite numbers.

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Henry Bottomley, Apr 11 2001

Complement of A006093 (primes minus 1).
Numbers which can be written as i*j+i+j, 0A072670(a(n))>0 for n>1.
a(n)! is divisible by a(n)*(a(n)+1)/2, see A060462.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A060462, A066938, A072672.

Cf. A002808, A006093, A079696.

[n-1: n in [2..120] | not IsPrime(n)]; // Vincenzo Librandi, Jun 09 2015

Select[Range[4, 96], CompositeQ] - 1 (* Michael De Vlieger, Dec 10 2020 *)

for(n=2,100,if(!isprime(n),print1(n-1,", "))) \\ Derek Orr, Jun 08 2015

from sympy import composite
def A072668(n): return composite(n)-1 # Chai Wah Wu, Aug 02 2024

a(n) = A002808(n) - 1.
a(n) = 2*A002808(n) - A079696(n). - Juri-Stepan Gerasimov, Oct 22 2009
a(n) = A060462(n).

A061743 Numbers k such that k! is divisible by (k+1)^2.

Original entry on oeis.org

11, 14, 15, 17, 19, 20, 23, 24, 26, 27, 29, 31, 32, 34, 35, 38, 39, 41, 43, 44, 47, 48, 49, 50, 51, 53, 54, 55, 56, 59, 62, 63, 64, 65, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 107, 109, 110, 111, 113, 114

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A001751, A060462, A178156, A264848.

Select[Range[120], IntegerQ[ #!/(# + 1)^2] &]

{ n=0; f=1; for (a=1, 2588, f*=a; if (f%(a + 1)^2 == 0, write("b061743.txt", n++, " ", a)) ) } \\ Harry J. Smith, Jul 27 2009

isok(k) = !(k! % (k+1)^2); \\ Michel Marcus, Jul 01 2018

from sympy import primepi
def A061743(n):
    def f(x): return int(n+2+primepi(x+1)+primepi(x+1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 17 2024

a(n) = A264828(n+3)-1. Complement of {A178156} - 1. - Chai Wah Wu, Oct 17 2024

A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

Original entry on oeis.org

1, 7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, 55, 57, 59, 61, 62, 64, 67, 71, 73, 76, 77, 79, 80, 84, 85, 87, 91, 92, 93, 94, 97, 101, 103, 104, 107, 109, 110, 115, 117, 118, 121, 122, 123, 124, 127, 129, 132, 133, 137, 139, 142, 143, 144, 145, 147

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

Product_{i=1..k} i^2 = (k!)^2 and Sum_{i=1..k} i^2 = k*(k+1)*(2*k+1)/6. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

			a(2) = A125314(2) = 7.

J. Mulder, Table of n, a(n) for a(n) below 20000

Cf. A000330, A001044, A125294, A125314, A060462, A166604, A166605, A166606, A166607, A166608, A166609, A166610, A334735.

Cf. A067656. - R. J. Mathar, Oct 23 2009

q:= k-> is(irem(k!^2, k*(k+1)*(2*k+1)/6)=0):
select(q, [$1..200])[];  # Alois P. Heinz, May 09 2020

Cases[Range[2, 5000], k_ /; Divisible[Factorial[k - 1]^2, 1/6 (-1 + k) k (-1 + 2 k)]] - 1 (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

isok(k) = ((k!)^2 % (k*(k+1)*(2*k+1)/6)) == 0; \\ Michel Marcus, May 09 2020

Terms below 5000 by J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms copied from the b-file by R. J. Mathar, Feb 14 2010

A166604 Numbers k such that Sum_{i=1..k} i^4 divides Product_{i=1..k} i^4.

Original entry on oeis.org

1, 31, 59, 94, 104, 122, 133, 181, 206, 223, 244, 248, 283, 298, 318, 342, 356, 401, 406, 421, 422, 439, 444, 449, 451, 469, 479, 493, 496, 507, 528, 532, 536, 541, 555, 597, 631, 637, 643, 668, 701, 706, 712, 717, 721, 722, 754, 762, 795, 797, 801, 815, 842

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(4) = 31.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A125314, A166602, A060462, A166605, A166606, A166607, A166608, A166609, A166610, A181426, A334734.

k = s = 1; p = 1; lst = {}; While[k < 1000, If[ Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^4; p = p*k^4]; lst (* Robert G. Wilson v, Nov 02 2009 *)
Module[{nn=1000,c},c=Range[nn]^4;Select[Range[nn],Divisible[Times@@ Take[ c,#], Total[Take[c,#]]]&]] (* Harvey P. Dale, Dec 18 2013 *)

a(15)-a(53) from Robert G. Wilson v, Nov 02 2009

A166605 Numbers k such that Sum_{i=1..k} i^5 divides Product_{i=1..k} i^5.

Original entry on oeis.org

1, 13, 64, 95, 111, 118, 123, 133, 134, 140, 151, 177, 199, 217, 229, 242, 255, 264, 274, 281, 302, 305, 325, 333, 338, 354, 359, 365, 376, 394, 411, 414, 431, 433, 440, 472, 475, 477, 489, 514, 525, 528, 529, 537, 547, 569, 574, 583, 584, 585, 589, 594, 615

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(5) = 13.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A125314, A166602, A060462, A166604, A166606, A166607, A166608, A166609, A166610.

k = s = 1; p = 1; lst = {}; While[k < 100000, If[Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^5; p = p*k^5]; lst  (* G. C. Greubel, May 18 2016 *)
Module[{nn=1000,i5},i5=Range[nn]^5;Position[Table[Times@@Take[i5,n]/Total[Take[i5,n]],{n,nn}],?IntegerQ]]//Flatten (* _Harvey P. Dale, Jan 18 2025 *)

More terms from Max Alekseyev, Sep 29 2010

A166609 Numbers k such that Sum_{i=1..k} i^9 divides Product_{i=1..k} i^9.

Original entry on oeis.org

1, 1441, 1715, 11706, 16741, 18435, 23793, 29927, 32071, 33932, 45768, 45831, 47103, 47215, 48257, 55743, 56007, 61976, 62773, 64841, 68561, 70853, 70880, 81624, 83526, 86243, 87529, 88162, 91054, 91395, 92288, 92933, 94211, 98982

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(9) = 1441.

David A. Corneth, Table of n, a(n) for n = 1..3003

Cf. A125314, A166602, A060462, A166604, A166605, A166606, A166607, A166608, A166610.

More terms from Max Alekseyev, Sep 30 2010

A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

Original entry on oeis.org

1, 1, 8, 180, 1120, 8064, 604800, 68428800, 830269440, 10897286400, 2324754432000, 640237370572800, 11585247657984000, 221172909834240000, 93666727314800640000, 2068161339110798131200, 47726800133326110720000, 1148978521728221184000000, 28806532937614688256000000

Offset: 1

Rick L. Shepherd, Jun 09 2005

nonn

A000142(n)/A000217(n) = n!/(n*(n+1)/2) = 2*(n-1)!/(n+1) is an integer iff n = 1 or n + 1 is composite; i.e., iff n is a term of A060462.

Cf. A060462 (corresponding k), A000142 (factorials), A000217 (triangular numbers).

select(x-> denom(x)=1, [k!/(k*(k+1)/2)$k=1..30])[];  # Alois P. Heinz, Dec 11 2020

Select[Table[(n - 1)!/((n (n - 1))/2), {n, 2, 50}], IntegerQ[#] &] (* Geoffrey Critzer, May 02 2015 *)

for(n=1,50, r=2*(n-1)!/(n+1); if(denominator(r)==1, print1(r,",")))

a(m) = 2*(A060462(m)-1)!/(A060462(m)+1) = A000142(A060462(m))/A000217(A060462(m)).

Offset corrected by Alois P. Heinz, Dec 11 2020

A166606 Numbers k such that Sum_{i=1..k} i^6 divides Product_{i=1..k} i^6.

Original entry on oeis.org

1, 1556, 1640, 3907, 5642, 6205, 7238, 8311, 10350, 11551, 12499, 13371, 13812, 17524, 17589, 18162, 18790, 21569, 21573, 22381, 22544, 23809, 24312, 24416, 24598, 24629, 25247, 25463, 26093, 26583, 26829, 27091, 27098, 28646, 28804

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(6) = 1556.

Cf. A125314, A166602, A060462, A166604, A166605, A166607, A166608, A166609, A166610.

k = s = 1; p = 1; lst = {}; While[k < 5000, If[Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^6; p = p*k^6]; lst (* G. C. Greubel, May 18 2016 *)

More terms from Max Alekseyev, Sep 30 2010

A166607 Numbers k such that Sum_{i=1..k} i^7 divides Product_{i=1..k} i^7.

Original entry on oeis.org

1, 733, 1637, 2096, 2367, 4231, 5674, 5839, 7585, 8344, 13719, 13753, 14983, 15151, 15197, 15257, 15757, 16595, 17305, 18791, 20701, 21442, 23652, 23738, 24519, 24789, 25474, 25916, 25933, 27474, 27487, 29185, 31455, 32846, 32950, 33421

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(7) = 733.

Cf. A123314, A166602, A060462, A166604, A166605, A166606, A166608, A166609, A166610.

With[{c=Range[40000]^7},Flatten[Position[#[[1]]/#[[2]]&/@Thread[ {Rest[ FoldList[ Times,1,c]],Accumulate[c]}],?IntegerQ]]] (* _Harvey P. Dale, Nov 16 2014 *)

More terms from Max Alekseyev, Sep 30 2010

cp's OEIS Frontend

A090542 Incorrect version of A060462.

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A072668 Numbers one less than composite numbers.

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A061743 Numbers k such that k! is divisible by (k+1)^2.

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A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

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A166604 Numbers k such that Sum_{i=1..k} i^4 divides Product_{i=1..k} i^4.

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A166605 Numbers k such that Sum_{i=1..k} i^5 divides Product_{i=1..k} i^5.

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A166609 Numbers k such that Sum_{i=1..k} i^9 divides Product_{i=1..k} i^9.

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A108552 Integer values of (1*2*...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

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A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.