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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A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

Crossrefs

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, 0, -1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -3, 3, -3, 4, -4, 5, -6, 6, -6, 8, -9, 9, -11, 12, -13, 14, -16, 19, -19, 21, -25, 26, -28, 32, -36, 38, -41, 46, -50, 55, -60, 65, -70, 77, -85, 91, -99, 108, -116, 126, -138, 149, -160, 174, -188, 202, -219, 237, -255, 274, -296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000005, A000040, A000586, A000607, A001221, A008683, A083399, A088705, A280954, A305614.

Programs

  • Mathematica
    nn=20;
ser=Product[1/(1+x^p),{p,Select[Range[nn],PrimeQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}] (* Gus Wiseman, Jun 06 2018 *)

Formula

a(n) = A184198(n) - A184199(n). - Vaclav Kotesovec, Jan 11 2021

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211
Offset: 1

Views

Author

Gus Wiseman, Dec 29 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

Crossrefs

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

A046343 Sum of the prime factors of the composite numbers (counted with multiplicity).

Original entry on oeis.org

4, 5, 6, 6, 7, 7, 9, 8, 8, 8, 9, 10, 13, 9, 10, 15, 9, 11, 10, 10, 14, 19, 12, 10, 21, 16, 11, 12, 15, 11, 25, 11, 14, 12, 20, 17, 11, 16, 13, 22, 31, 12, 33, 13, 12, 18, 16, 21, 26, 14, 12, 39, 13, 23, 18, 18, 13, 12, 43, 14, 22, 45, 32, 17, 13, 20, 27, 34, 49, 24, 13, 16, 17
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

The number of partitions of k into prime parts smaller than itself gives the number of times that a(n) = k. - Gionata Neri, Jun 11 2015
That number of partitions is A000607(k) if k is not prime, and A000607(k) - 1 if k is prime. - Robert Israel, Jun 11 2015

Examples

			a(31)=25 because 46 = 2 * 23 and 25 = 2 + 23.

Links

Crossrefs

Cf. A002808, A001414.
Cf. A000607, A046344, A046345.

Programs

  • Maple
    count:= 0:
for n from 2 while count < 200 do
  if not isprime(n) then
    count:= count+1;
    a[count]:= add(t[1]*t[2],t=ifactors(n)[2])
  fi
od:
seq(a[i],i=1..count); # Robert Israel, Jun 11 2015
  • Mathematica
    Total@ Flatten[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Select[Range@ 120, CompositeQ] (* Michael De Vlieger, Jun 11 2015 *)
t = {}; Do[If[! PrimeQ[n], AppendTo[t, Apply[Dot, Transpose[FactorInteger[n]]]]], {n, 4, 245}]; t (* Zak Seidov, Jul 03 2015 *)

Formula

a(n) = A001414(A002808(n)). - Michel Marcus, Jun 11 2015

A084993 Total number of parts in all partitions of n into prime parts.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 9, 12, 16, 20, 27, 33, 42, 53, 64, 80, 96, 117, 141, 169, 201, 239, 282, 333, 390, 456, 532, 617, 715, 826, 951, 1094, 1253, 1435, 1636, 1864, 2119, 2404, 2723, 3078, 3473, 3915, 4403, 4947, 5549, 6215, 6952, 7767, 8665, 9656, 10748
Offset: 1

Views

Author

Vladeta Jovovic, Jul 17 2003

Keywords

Examples

			Partitions of 9 into primes are 2+2+2+3=3+3+3=2+2+5=2+7; thus a(9)=4+3+3+2=12.

Links

Crossrefs

Cf. A000607, A024938.

Programs

  • Maple
    g:=sum(x^ithprime(j)/(1-x^ithprime(j)),j=1..20)/product(1-x^ithprime(j),j=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..57); # Emeric Deutsch, Mar 07 2006
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; local g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i=1 then `if`(irem(n, 2)=0, [1, n/2], [0, 0])
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i));
         b(n, i-1) +g +[0, g[1]]
      fi
    end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 30 2012
  • Mathematica
    nn=40;a=Product[1/(1-y x^i),{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[Series[D[a,y]/.y->1,{x,0,nn}],x],1]  (* Geoffrey Critzer, Oct 30 2012 *)
b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i == 1, If[EvenQ[n], {1, n/2}, {0, 0}], True, g = If[Prime[i] > n, {0, 0}, b[n - Prime[i], i]]; b[n, i - 1] + g + {0, g[[1]]}]];
a[n_] := b[n, PrimePi[n]][[2]];
Array[a, 52] (* Jean-François Alcover, Dec 30 2017, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n],AllTrue[#,PrimeQ]&]]],{n,60}] (* Harvey P. Dale, Jul 11 2023 *)
  • PARI
    sumparts(n, pred)={sum(k=1, n, 1/(1-pred(k)*x^k) - 1 + O(x*x^n))/prod(k=1, n, 1-pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, isprime), -n)} \\ Andrew Howroyd, Dec 28 2017

Formula

G.f.: sum(x^p(j)/(1-x^p(j)),j=1..infinity)/product(1-x^p(j), j=1..infinity), where p(j) is the j-th prime. - Emeric Deutsch, Mar 07 2006

A085755 Number of partitions of n into a prime number of prime parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 8, 9, 9, 12, 12, 16, 16, 19, 19, 26, 24, 31, 29, 39, 35, 50, 44, 61, 55, 74, 67, 93, 80, 111, 99, 136, 119, 166, 145, 197, 179, 239, 213, 292, 255, 342, 310, 409, 365, 492, 436, 577, 524, 682, 614, 814, 724, 947, 865, 1113, 1007, 1314
Offset: 4

Views

Author

Vladeta Jovovic, Jul 21 2003

Keywords

Examples

			a(20) = 12 because there are 12 partitions of 20 into a prime number of prime parts: 2+3+3+3+3+3+3 = 2+2+2+3+3+3+5 = 2+2+2+2+2+5+5 = 2+2+2+2+2+3+7 = 2+3+5+5+5 = 2+3+3+5+7 = 2+2+2+7+7 = 2+2+2+3+11 = 2+7+11 = 2+5+13 = 7+13 = 3+17.

Links

Crossrefs

Cf. A000607, A038499.

Programs

  • Maple
    b:= proc(n, i, t) if n<0 then 0 elif n=0 then `if`(isprime(t), 1, 0) elif i=1 then `if`(irem(n,2)=0 and isprime(t +n/2), 1, 0) else b(n,i,t):= b(n -ithprime(i), i, t+1) +b(n, i-1, t) fi end: a:= proc(n) local i; for i while ithprime(i)Alois P. Heinz, Apr 30 2009
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],PrimeQ[Length[#]]&&AllTrue[ #, PrimeQ]&]],{n,4,70}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 18 2016 *)

A099773 Number of partitions of n into odd prime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 9, 10, 11, 12, 14, 15, 17, 20, 21, 24, 26, 29, 33, 35, 40, 44, 47, 53, 58, 64, 70, 77, 84, 91, 101, 110, 120, 130, 142, 155, 168, 184, 199, 215, 234, 254, 275, 298, 323, 348, 376, 407, 439, 474, 511, 551, 592
Offset: 0

Views

Author

Vladeta Jovovic, Nov 11 2004

Keywords

Links

Crossrefs

Cf. A000607, A024939.

Programs

  • Haskell
    a099773 = p a065091_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012
  • Mathematica
    CoefficientList[ Series[ Product[1/(1 - x^Prime[i]), {i, 2, 25}], {x, 0, 70}], x] (* Robert G. Wilson v, Jun 14 2006 *)

Formula

G.f.: 1/Product_{k>1} (1-x^prime(k)).

A184198 Number of partitions of n into an even number of primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 5, 8, 7, 11, 10, 15, 13, 20, 17, 26, 23, 34, 29, 43, 38, 55, 49, 69, 62, 88, 78, 109, 97, 135, 122, 167, 150, 205, 186, 251, 227, 306, 277, 371, 337, 448, 407, 539, 492, 647, 591, 773, 707, 922, 845, 1096, 1005, 1298, 1193, 1535, 1412, 1809, 1667, 2127
Offset: 0

Views

Author

R. J. Mathar, Jan 10 2011

Keywords

Examples

			n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=11 are even, namely 11+7, 13+5, 5+5+5+3, 7+5+3+3, 3+3+3+3+3+3, 7+7+2+2, 11+3+2+2, 5+3+3+3+2+2, 5+5+2+2+2+2, 7+3+2+2+2+2, 3+3+2+2+2+2+2+2.
The remaining A184199(18)=8 are odd.

Links

Crossrefs

Cf. A000607, A048165, A184199, A184171.

Programs

Formula

a(n) = (A000607(n)+A048165(n))/2.

Extensions

a(31)-a(69) corrected by Andrew Howroyd, Dec 28 2017

A184199 Number of partitions of n into an odd number of primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 4, 3, 5, 4, 7, 6, 10, 8, 13, 11, 17, 15, 23, 20, 29, 26, 38, 34, 49, 43, 62, 55, 78, 69, 97, 88, 122, 109, 150, 135, 186, 167, 227, 205, 277, 251, 337, 306, 407, 371, 492, 448, 591, 539, 707, 647, 845, 773, 1005, 922, 1193, 1096, 1412, 1298, 1667, 1535, 1963, 1809, 2305, 2127
Offset: 0

Views

Author

R. J. Mathar, Jan 10 2011

Keywords

Examples

			n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=8 are odd, namely  11+5+2, 13+3+2, 5+5+3+3+2, 7+3+3+3+2, 7+5+2+2+2, 3+3+3+3+2+2+2, 5+3+2+2+2+2+2, 2+2+2+2+2+2+2+2+2.
The remaining A184198(18)=11 are even.

Links

Crossrefs

Cf. A000607, A048165, A184198, A184172.

Programs

  • Mathematica
    a[n_] := a[n] = Count[IntegerPartitions[n, All, Prime[Range[PrimePi[n]]]], p_ /; OddQ[Length[p]]];
Reap[Do[Print[n, " ", a[n]]; Sow[a[n]], {n, 0, 200}]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)
  • PARI
    parts(n, pred, y)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=80); (Vec(parts(n, isprime, 1)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017

Formula

a(n) = (A000607(n)-A048165(n))/2.

Extensions

a(31)-a(70) corrected by Andrew Howroyd, Dec 28 2017

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759
Offset: 0

Views

Author

Gus Wiseman, Jun 07 2018

Keywords

Comments

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

Examples

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

Links

Crossrefs

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

Programs

  • Maple
    q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018
  • Mathematica
    nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]
Previous Showing 31-40 of 198 results. Next

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