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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A318979 Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 1, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 5, 0, 5, 1, 2, 1, 6, 0, 2, 0, 6, 0, 4, 0, 4, 2, 2, 0, 8, 0, 3, 1, 4, 0, 4, 1, 6, 0, 2, 0, 9, 0, 2, 0, 6, 0, 5, 0, 4, 1, 5, 0, 9, 0, 2, 2, 4, 1, 4, 0, 8, 0, 2, 0, 8, 1, 2, 0
Offset: 1

Views

Author

Gus Wiseman, Sep 06 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, corresponding to the prime index multisets (), (1), (2), (11), (12), (22), (112), (122), (1122) respectively. Of these, only (1), (11), (12), (112), (122), (1122) are relatively prime, corresponding to the divisors 2, 4, 6, 12, 18, 36, so a(36) = 6.

Links

Crossrefs

Cf. A000005, A000837, A001221, A018783, A056239, A289508, A289509, A296150, A298748, A318978, A327657.

Programs

  • Mathematica
    Table[Length[Select[Divisors[n],GCD@@PrimePi/@FactorInteger[#][[All,1]]==1&]],{n,100}]
  • PARI
    a(n) = sumdiv(n, d, gcd(apply(x->primepi(x), factor(d)[,1])) == 1); \\ Michel Marcus, Jan 09 2019

Formula

a(n) = A000005(n) - A327657(n). - Antti Karttunen, Dec 05 2021

A319152 Nonprime Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409, 10201
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A subsequence of A001597.
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence.
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 the sequence begins: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).

Crossrefs

Cf. A001597, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319151.

Programs

  • Mathematica
    supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[10000],And[!PrimeQ[#],supperQ[#]]&]

A319180 Perfect powers whose prime indices are relatively prime.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 100, 128, 144, 196, 216, 225, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1444, 1600, 1728, 1764, 1936, 2025, 2048, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3364, 3375, 3600, 3844, 4096
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (3311), (1111111), (221111), (4411), (222111), (3322), (11111111), (222211), (331111), (5511), (111111111), (22111111), (6611), (441111), (332211), (333111).

Crossrefs

Cf. A001597, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319181.

Programs

  • Mathematica
    Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,GCD@@FactorInteger[#][[All,2]]>1]&]

A319181 Numbers that are not perfect powers but whose prime indices have a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 171, 173, 179, 181, 183, 185, 189
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (6,2), (13), (14), (15), (16), (8,2), (17), (18), (4,2,2), (6,3).

Crossrefs

Cf. A007916, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319180.

Programs

  • Mathematica
    Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]>1,GCD@@FactorInteger[#][[All,2]]==1]&]

A319270 Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.

Original entry on oeis.org

1, 2, 6, 12, 18, 24, 26, 48, 52, 54, 72, 74, 78, 96, 104, 108, 122, 148, 156, 162, 178, 192, 202, 208, 222, 234, 244, 288, 296, 312, 338, 356, 366, 384, 404, 416, 432, 444, 446, 468, 478, 486, 488, 502, 534, 592, 606, 624, 648, 666, 702, 712, 718, 732, 746
Offset: 1

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Comments

Also Matula-Goebel numbers of series-reduced locally non-intersecting aperiodic rooted trees.

Examples

			The sequence of Matula-Goebel trees of elements of this sequence begins:
   1: o
   2: (o)
   6: (o(o))
  12: (oo(o))
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  48: (oooo(o))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  72: (ooo(o)(o))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  96: (ooooo(o))

Crossrefs

Cf. A000081, A007097, A061775, A276625, A298748, A301700, A303431, A316470, A319271.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ain[n_]:=Or[n==1,And[GCD@@primeMS[n]==1,GCD@@Length/@Split[primeMS[n]]==1,And@@ain/@primeMS[n]]];
Select[Range[100],ain]
Previous Showing 21-25 of 25 results.

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