A240751 -id:A240751 - cp's OEIS frontend

A240755 Smallest prime which occurs in prime power factorization of A240751(n)! with exponent n.

2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 2, 5, 3, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 5, 2, 2, 3, 3, 7, 3, 2, 2, 5, 2, 2, 3, 5, 2, 2, 3, 2, 2, 5, 3, 3, 2, 2, 3, 2, 2, 5, 5, 2, 2, 3, 2, 2, 3, 3, 7, 3, 3, 2, 2, 5, 2, 2, 3, 5, 2, 2, 3, 2, 2, 3, 3, 5, 2, 2, 3, 2, 2, 5, 3, 2, 2, 5

Offset: 1

Vladimir Shevelev, Apr 12 2014

nonn

			A240751(4)=6, since 6!=2^4*3^2*5. Here, only the prime 2 is in power 4, thus A240755(4)=2.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751.

a(n)^n || A240751(n)!.

More terms from Peter J. C. Moses, Apr 12 2014

A240764 Least k such that prime power factorization of A240751(k)! contains p^k when the smallest such p equals prime(n), or a(n)=0 if there is no such k.

Original entry on oeis.org

1, 2, 12, 29, 186, 2865, 3265, 379852, 7172525

Offset: 1

Vladimir Shevelev, Apr 12 2014

The first position k in A240755 in which A240755(k) = prime(n), or a(n)=0 if prime(n) does not occur in A240755.

Conjecture: all a(n)>0.

			A240751(a(3))! = A240751(12)! = 50!. 50! is the least factorial having exponent 12 in its prime factorization. That exponent denotes the multiplicity of prime(3) = 5. - _David A. Corneth_, Mar 27 2017

David A. Corneth, A240751; a(n) is the smallest k such that in the prime power factorization of k! there exists at least one exponent n., Seqfan, (Mar 25 2017)

Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755.

a(5)-a(7) from Peter J. C. Moses, Apr 14 2014

a(8)-a(9) from David A. Corneth, Mar 27 2017

A286003 Numbers n such that 3^n is the highest power of 3 dividing A240751(n).

Original entry on oeis.org

2, 5, 6, 9, 13, 14, 17, 21, 27, 28, 30, 36, 40, 44, 45, 48, 55, 58, 59, 61, 62, 68, 72, 75, 76, 80, 84, 90, 93, 99, 103, 106, 107, 108, 111, 114, 115, 121, 122, 123, 125, 126, 129, 136, 139, 140, 144, 147, 148, 151, 155, 156, 157, 163, 167, 170, 171, 175, 178, 179

Offset: 1

David A. Corneth, Apr 30 2017

Is A005187 the same as values n such that 2^n||A240751(n)?

Cf. A005187, A240751.

A286010 Numbers n such that 23^n is the highest power of 23 dividing A240751(n).

Original entry on oeis.org

7172525, 15122988, 21210412, 48538612, 48752905, 51466002, 52658325, 55938150, 57611786, 73564837, 79352750, 83297618, 87015476, 96850742, 101001720, 112873406, 112907600, 122331745, 131633392, 132790999, 135376501, 141763112, 148650650, 156434774, 173413868, 177310781, 180684007, 190994768

Offset: 1

David A. Corneth, May 01 2017

nonn

No terms for "Numbers n such that 29^n is the highest power of 29 dividing A240751(n)." up to 10^8.

Cf. A005187, A240751, A286003-A286009.

A284050 a(n) = floor(A240751(n) / n), where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

Original entry on oeis.org

2, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 4, 1, 1, 2, 2, 6, 2, 1, 1, 4, 1, 1, 2, 4, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 2, 1, 1, 2, 2, 6, 2, 2, 1, 1, 4, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 2, 4, 1, 1, 2, 1, 1, 4, 2, 1, 1, 4

Offset: 1

David A. Corneth, Mar 19 2017

For n > 2, p = a(n) + 1 is the prime that has exponent n in A240751(n)! (see A240751 for an outline of a proof).

First occurrence of p-1: 1, 2, 12, 29, 186, 2865, 3265, 379852, 7172525, ..., (A240764). - Robert G. Wilson v, Apr 15 2017. Comment changed by David A. Corneth, Apr 15 2017

			For n = 5, p = a(n) + 1 = 3 is the prime such that A240751(5)! = 12! is the least factorial that has exponent 5.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A240751, A240755, A240764, A284051, A157928.

Table[k = 2; While[! MemberQ[FactorInteger[k!][[All, -1]], n], k++]; Floor[k/n], {n, 87}] (* Michael De Vlieger, Mar 24 2017 *)

a(n) = A240751(n)\n \\ (for computation of A240751(n), see A240751)

A240751(n) = n*a(n) + A284051(n). - Antti Karttunen, Mar 22 2017

a(n) = A240755(n) - 1 for n > 2 and a(n) = A240755(n) for n < 3. I.e., A240755(n) - A157928(n+1). - David A. Corneth, Mar 27 2017

A284051 a(n) = A240751(n) mod n, where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, Mar 19 2017

			A240751(5) = 12 so a(5) = 12 mod 5 == 2.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A240751, A284050.

Table[k = 2; While[! MemberQ[FactorInteger[k!][[All, -1]], n], k++]; Mod[k, n], {n, 87}] (* Michael De Vlieger, Mar 24 2017 *)

a(n) = A240751(n)%n \\ (For computation of A240751(n), see A240751)

A240751(n) = n*A284050(n) + a(n). - Antti Karttunen, Mar 22 2017

A286005 Numbers n such that 7^n is the highest power of 7 dividing A240751(n).

Original entry on oeis.org

29, 60, 91, 92, 141, 154, 185, 204, 217, 241, 254, 279, 285, 342, 403, 441, 473, 497, 510, 528, 541, 572, 603, 622, 666, 697, 715, 753, 771, 779, 780, 811, 841, 873, 922, 936, 954, 973, 1022, 1047, 1053, 1066, 1091, 1122, 1234, 1291, 1347, 1404, 1422, 1453, 1496, 1561

Offset: 1

David A. Corneth, May 01 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005187, A240751, A286003, A286004, A286006-A286010.

A286006 Numbers n such that 11^n is the highest power of 11 dividing A240751(n).

Original entry on oeis.org

186, 447, 635, 765, 1035, 1092, 1378, 1435, 1540, 2015, 2553, 2740, 2808, 3027, 4154, 4465, 4497, 4603, 4766, 4816, 4897, 5084, 5166, 5265, 5403, 5590, 5666, 5747, 5828, 6245, 6515, 6572, 6759, 6809, 7029, 7559, 7690, 7991, 8459, 8810, 8859, 9202, 9234, 9340, 9821

Offset: 1

David A. Corneth, May 01 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005187, A240751, A286003-A286005, A286007-A286010.

A286004 Numbers n such that 5^n is the highest power of 5 dividing A240751(n).

Original entry on oeis.org

12, 20, 24, 33, 37, 43, 51, 52, 65, 69, 77, 83, 87, 96, 100, 118, 124, 132, 133, 160, 164, 172, 199, 226, 230, 234, 238, 245, 249, 253, 267, 275, 298, 306, 307, 315, 320, 338, 346, 347, 351, 355, 361, 362, 363, 376, 380, 384, 388, 402, 411, 415, 433, 437, 442, 443, 451

Offset: 1

David A. Corneth, May 01 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005187, A240751, A286003, A286005-A286010.

A286007 Numbers n such that 13^n is the highest power of 13 dividing A240751(n).

Original entry on oeis.org

2865, 3640, 3942, 4922, 6677, 7959, 10972, 11577, 11928, 12859, 18233, 19213, 22153, 30295, 30646, 31977, 34664, 35620, 36527, 44440, 49764, 51520, 54182, 55439, 55839, 56045, 56951, 60340, 60920, 63909, 64308, 64890, 66995, 68278, 70540, 72246, 76795, 77195, 77595

Offset: 1

David A. Corneth, May 01 2017

nonn

Cf. A005187, A240751, A286003-A286006, A286008-A286010.

cp's OEIS Frontend

A240755 Smallest prime which occurs in prime power factorization of A240751(n)! with exponent n.

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A240764 Least k such that prime power factorization of A240751(k)! contains p^k when the smallest such p equals prime(n), or a(n)=0 if there is no such k.

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A286003 Numbers n such that 3^n is the highest power of 3 dividing A240751(n).

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A286010 Numbers n such that 23^n is the highest power of 23 dividing A240751(n).

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A284050 a(n) = floor(A240751(n) / n), where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

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A284051 a(n) = A240751(n) mod n, where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

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A286005 Numbers n such that 7^n is the highest power of 7 dividing A240751(n).

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A286006 Numbers n such that 11^n is the highest power of 11 dividing A240751(n).

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A286004 Numbers n such that 5^n is the highest power of 5 dividing A240751(n).

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A286007 Numbers n such that 13^n is the highest power of 13 dividing A240751(n).

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