cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A128673 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 3.

Original entry on oeis.org

94556602, 141834903, 189113204, 283669806, 450820422
Offset: 1

Views

Author

Alexander Adamchuk, Apr 18 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by Max Alekseyev, ((16843-1)/2)*16843^m found by Max Alekseyev, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by Max Alekseyev, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.

Links

Crossrefs

Cf. A001008, A002805, A058313, A058312.
Cf. A007406, A007407, A119682, A007410, A120296, A099828.
Cf. A125581, A126196, A126197, A128672, A128674, A128675, A128676, A128670, A128671.
Cf. A088164 (Wolstenholme primes).

Programs

  • Mathematica
    k=3; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n, 1, 450820422} ]

A128676 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 6.

Original entry on oeis.org

20, 100, 110, 156, 161, 272, 342, 345, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2162, 2500, 2756, 3051, 3422, 3660, 3703, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 7935, 9312, 9605, 10100, 10506, 11342, 11638, 11772, 12500, 12656, 13310
Offset: 1

Views

Author

Alexander Adamchuk, Mar 20 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Sequence contains all terms of geometric progressions of the form (p-1)*p^k, k > 0, for some primes p >= 5, such as 4*5^k, 7*23^k, 15*23^k, 27*113^k, etc. Note the factorization of initial terms of {a(n)} = {4*5, 4*5^2, 10*11, 12*13, 7*23, 16*17, 18*19, 15*23, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 46*47, 4*5^4, 52*53, 27*113, 58*59, 60*61, 7*23^2, ...}. See more details in Comments at A128672 and A125581.

Links

Crossrefs

Cf. A001008, A002805, A058313, A058312.
Cf. A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

Programs

  • Mathematica
    k=6; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,3703} ]

Extensions

Edited and extended by Max Alekseyev, May 08 2010

A128675 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the n-th alternating generalized harmonic number H'(m,k), for k = 5.

Original entry on oeis.org

444, 666, 888, 1332, 16428, 24642, 32856, 49284, 607836, 911754, 1215672, 1823508
Offset: 1

Views

Author

Alexander Adamchuk, Mar 20 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Sequence contains all terms of geometric progressions 37^k*(37-1)/3, 37^k*(37-1)/2, 37^k*(37-1)*2/3, 37^k*(37-1) for k > 0. Note the factorization of initial terms of {a(n)} = {37*12, 37*18, 37*24, 37*36, ...}. See more details in Comments at A128672 and A125581.

Links

Crossrefs

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

Programs

  • Mathematica
    k=5; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,2000} ]

Extensions

Eight more terms from Max Alekseyev, May 08 2010

A128670 Least number k > 0 such that k^n does not divide the denominator of generalized harmonic number H(k,n) nor the denominator of alternating generalized harmonic number H'(k,n).

Original entry on oeis.org

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, 77, 42, 12246, 20, 104, 42
Offset: 1

Views

Author

Alexander Adamchuk, Mar 24 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m}1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Some apparent periodicity in {a(n)} (not without exceptions): a(n) = 20 for n = 2 + 4m, a(n) = 42 for n = 4 + 12m and 8 + 12m, a(n) = 76 for n = 9 + 18m, a(n) = 77 for n = 1 + 10m, a(n) = 104 for n = 7 + 12m, a(n) = 110 for n = 12m, a(n) = 136 for n = 25 + 32m, etc.
See more details in Comments at A128672 and A125581.

Links

Crossrefs

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676, A128671, A128670.

Extensions

More terms and b-file from Max Alekseyev, May 07 2010

A128674 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 4.

Original entry on oeis.org

42, 110, 156, 272, 294, 342, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11342, 11638, 11772, 12656, 13310, 14406, 16002, 17030, 18632, 19182, 22052, 22650, 23548, 24492, 26364
Offset: 1

Views

Author

Alexander Adamchuk, Mar 20 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Sequence contains geometric progressions of the form (p-1)*p^k for k > 0 and some prime p > 5. Note the factorization of initial terms of {a(n)} = {6*7, 10*11, 12*13, 16*17, 6*7^2, 18*19, 22*23, 28*29, 30*31, 10*11*2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, ...}. See more details in Comments at A128672 and A125581.

Links

Crossrefs

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128675, A128676.

Programs

  • Mathematica
    k=4; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,2000} ]

Extensions

Edited and extended by Max Alekseyev, May 09 2010

A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

Original entry on oeis.org

20, 94556602, 444, 104, 77, 3504, 1107, 104, 2948, 903, 77, 1752, 77, 104, 370
Offset: 1

Views

Author

Alexander Adamchuk, Mar 24 2007, Mar 26 2007

Keywords

Comments

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.
a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.
a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.

Examples

			a(2) = A128673(1) = 94556602.

Links

Crossrefs

Cf. A128670, A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676.

Formula

a(n) = A128670(prime(n)).

Extensions

a(9) = 2948 and a(12) = 1752 from Max Alekseyev
Edited by Max Alekseyev, Feb 20 2019

A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

Original entry on oeis.org

6, 20, 110, 272, 506, 812, 2162, 2756, 3422, 4970, 6806, 7832, 11342, 12656, 17030, 18632, 22052, 27722, 29756, 31862, 36290, 38612, 51302, 54056, 56882, 62750, 65792, 68906, 72092, 85556, 96410, 100172, 120062, 124256, 128522
Offset: 1

Views

Author

Charles R Greathouse IV, Dec 28 2016

Keywords

Links

Crossrefs

Primitive elements of A228870.
Subsequence of A002943. Also a subsequence of A028689, A036689, A053198, A068377, A079143, A128672, A220211 and other sequences ...- Paolo P. Lava, Jan 10 2017

Programs

  • PARI
    has(n)=my(f=factor(n)[,1]); for(i=1,#f, if(n%(f[i]-1)==0 && f[i]>2, return(1))); 0
is(n)=if(n%2, return(0)); if(n%3==0, return(n==6)); if(n%20==0, return(n==20)); if(!has(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if(has(n/f[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 28 2016
Showing 1-7 of 7 results.

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