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.

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A318587 Heinz numbers of integer partitions whose sum of reciprocals squared is 1.

Original entry on oeis.org

2, 81, 8775, 64827, 950625, 1953125, 7022925, 9055935, 21781575, 36020025, 50124555, 51883209, 57909033, 102984375, 118978125, 760816875, 816747435, 981059625, 1206902781, 1265058675, 1387132263, 2359670625, 3902169375, 4868424351, 5222768733, 5430160125
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sequence of integer partitions with Heinz numbers in this sequence begins: (1), (2222), (633222), (4444222), (66333322).

Crossrefs

Cf. A001222, A051908, A056239, A112798, A289506, A289507, A316855, A316856, A316857.
Cf. A318584, A318585, A318586, A318588, A318589.

Programs

  • Mathematica
    Select[Range[100000],Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]==1&]

Extensions

a(6)-a(26) from Alois P. Heinz, Aug 30 2018

A318588 Heinz numbers of integer partitions whose sum of reciprocals squared is an integer.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 81, 128, 162, 256, 324, 512, 648, 1024, 1296, 2048, 2592, 4096, 5184, 6561, 8192, 8775, 10368, 13122, 16384, 17550, 20736, 26244, 32768, 35100, 41472, 52488, 64827, 65536, 70200, 82944, 104976, 129654, 131072, 140400, 165888, 209952
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Sequence of integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (11), (111), (1111), (11111), (111111), (2222), (1111111), (22221), (11111111), (222211), (111111111), (2222111).

Crossrefs

Cf. A001222, A056239, A058360, A112798, A289506, A289507, A316855, A316857.
Cf. A318584, A318585, A318586, A318587, A318589.

Programs

  • Mathematica
    Select[Range[10000],IntegerQ[Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

A318589 Heinz numbers of integer partitions whose sum of reciprocals squared is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Crossrefs

Cf. A001222, A051908, A056239, A058360, A112798, A289506, A289507, A316854, A316855.
Cf. A318584, A318585, A318586, A318587, A318588.

Programs

  • Mathematica
    Select[Range[2,1000],IntegerQ[1/Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

A325624 a(n) = prime(n)^(n!).

Original entry on oeis.org

2, 9, 15625, 191581231380566414401, 92709068817830061978520606494193845859707401497097037749844778027824097442147966967457359038488841338006006032592594389655201
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

A subsequence of A325619 (numbers whose prime indices have reciprocal factorial sum equal to 1).

Crossrefs

Cf. A056239, A112798.
Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325619.

Programs

  • Mathematica
    Table[Prime[n]^n!,{n,5}]

A318574 Denominator of the reciprocal sum of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 6, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 1, 12, 8, 3, 3, 13, 4, 14, 5, 3, 9, 15, 2, 2, 3, 14, 6, 16, 2, 15, 4, 8, 10, 17, 6, 18, 11, 4, 1, 2, 10, 19, 7, 18, 12, 20, 1, 21, 12, 6, 8, 20, 3, 22
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Crossrefs

Positions of 1's are A316856.
Cf. A051908, A056239, A058360, A112798, A289506, A289507, A296150, A316854, A316855, A316857, A318573.

Programs

  • Mathematica
    Table[Sum[pr[[2]]/PrimePi[pr[[1]]],{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]//Denominator

Formula

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) is the denominator of Sum y_i/x_i.

A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 7, 1, 5, 1, 25, 2, 4, 1, 2, 1, 13, 13, 121, 1, 7, 1, 721, 3, 49, 1, 5, 1, 5, 61, 5041, 5, 3, 1, 40321, 361, 19, 1, 37, 1, 241, 7, 362881, 1, 9, 1, 4, 2521, 1441, 1, 5, 7, 73, 20161, 3628801, 1, 8, 1, 39916801, 25, 6, 121, 181, 1
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the numerator of 1/i_1! + ... + 1/i_k!.

Links

Crossrefs

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.
Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325703.

Programs

  • Mathematica
    Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator
  • PARI
    A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

Formula

a(n) = A318573(A325709(n)).

A316890 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783
Offset: 1

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Includes 29888089, which is the first perfect power in the sequence and is absent from A316888.

Links

Crossrefs

Cf. A000837, A002966, A051908, A058360, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

  • Mathematica
    Select[Range[2,100000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.

Links

Crossrefs

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.
Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.

Programs

  • Maple
    f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024
  • Mathematica
    Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

Formula

a(n) = A318574(A325709(n)).

A316889 Heinz numbers of aperiodic integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 147, 195, 3185, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963, 414295, 442975, 444925, 455975
Offset: 1

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.

Examples

			Sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (4,4,2), (6,3,2), (6,4,4,3), (12,4,3,3), (6,6,6,2), (8,8,4,2), (12,6,4,2), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

Links

Crossrefs

Cf. A000837, A002966, A007916, A051908, A058360, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

  • Mathematica
    Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316900 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 36, 64, 72, 128, 144, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2592, 2646, 2916, 3120, 3185, 3510, 4000, 4096, 4500, 4608
Offset: 1

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (11), (111), (1111), (221), (11111), (2211), (111111), (22111), (1111111), (221111), (22221), (632), (3331), (11111111).

Links

Crossrefs

Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

  • Mathematica
    Select[Range[2,1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]
Previous Showing 11-20 of 23 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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