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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A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

Original entry on oeis.org

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193
Offset: 1

Views

Author

Gus Wiseman, Jul 25 2022

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.
These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

Examples

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}

Crossrefs

Heinz numbers of the partitions counted by A023893.
Allowing prime index 1 gives A356066.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A023894 counts partitions into prime-powers, strict A054685.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@PrimePowerQ/@DeleteCases[primeMS[#],1]&]

A137793 Number of partitions of n into distinct parts with no prime gaps in their factorization.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 31, 38, 43, 50, 58, 66, 76, 88, 100, 113, 130, 146, 165, 188, 211, 237, 267, 298, 334, 375, 417, 464, 517, 573, 636, 706, 781, 862, 954, 1050, 1157, 1276, 1401, 1539, 1689, 1851, 2027, 2222, 2427
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 11 2008

Keywords

Examples

			a(16)=A000009(16)-#{14+2,10+6,10+5+1,10+4+2,10+3+2+1}=32-5=27.

Crossrefs

Cf. A073491, A000041, A000586, A106244, A137792, A137791.

A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101
Offset: 1

Views

Author

Gus Wiseman, Jul 31 2022

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}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

Crossrefs

The complement is A355743, counted by A023894.
The squarefree complement is A356065, counted by A054685.
Allowing prime index 1 gives A356064, complement A302492.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@PrimePowerQ/@primeMS[#]&]

Formula

Union of A299174 and A356064.

A131996 Number of partitions of n into distinct powers of 2 or of 3.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 06 2007

Keywords

Comments

a(A081601(n+1)) = n+1 and a(m) < n+1 for m < A081601(n+1).

Examples

			a(10) = #{9+1,8+2,4+3+2+1}=3;
a(20) = #{16+4,16+3+1,9+8+3,9+8+2+1}=4;
a(30) = #{27+3,27+2+1,16+9+4+1,16+9+3+2,16+8+4+2,16+8+3+2+1}=6.

Links

Crossrefs

Cf. A000009, A062051, A106244, A131995.

Programs

  • Maple
    g:=(product((1+x^(2^k))*(1+x^(3^k)),k=0..10))/(1+x): gser:=series(g,x=0,111): seq(coeff(gser,x,n),n=1..108); # Emeric Deutsch, Aug 26 2007
  • Mathematica
    max = 100; Product[((1 + x^(2^k)) (1 + x^(3^k))), {k, 0, Log[2, max] // Ceiling}]/(1 + x) + O[x]^max // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 30 2016 *)

Formula

G.f.: Product_{k>=0} ((1+x^(2^k))(1+x^(3^k)))/(1+x) (offset 0). - Emeric Deutsch, Aug 26 2007

A307825 Number of partitions of n into 3 distinct prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 4, 4, 5, 4, 6, 5, 7, 6, 8, 8, 10, 8, 10, 9, 12, 11, 12, 11, 15, 12, 15, 14, 17, 17, 20, 18, 19, 19, 19, 22, 23, 20, 21, 24, 23, 24, 24, 24, 27, 28, 24, 27, 28, 28, 28, 33, 27, 33, 29, 31, 30, 35, 27, 35, 33, 34, 31, 40, 32, 42, 35, 39, 35, 47, 32
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

Examples

			a(15) = 4 because we have [9, 4, 2], [8, 5, 2], [8, 4, 3] and [7, 5, 3].

Links

Crossrefs

Cf. A000961, A106244, A125688, A246655, A306433, A307727.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[n, {3}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 78}]

Formula

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^A246655(k)).

A111902 Number of partitions of n into distinct parts that are primes or squares of primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 4, 4, 4, 6, 4, 8, 5, 9, 7, 10, 9, 11, 12, 12, 15, 14, 17, 17, 20, 20, 23, 24, 26, 28, 30, 32, 35, 36, 40, 41, 46, 47, 52, 54, 58, 62, 65, 71, 73, 80, 82, 90, 93, 101, 104, 113, 117, 125, 132, 139, 148, 154, 165, 171, 183, 191
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 20 2005

Keywords

Examples

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 3*x^7 + x^8 + 4*x^9 + 2*x^10 + ...
a(12) = #{3^2+3, 7+5, 7+3+2, 5+2^2+3} = 4.

Crossrefs

Cf. A000586, A111900, A106244, A111901.

Programs

  • PARI
    {a(n) = if(n < 0, 0, polcoeff( prod(k=1, primepi(n), (1 + x^prime(k)^2 + x*O(x^n)) * (1 + x^prime(k))), n))}; /* Michael Somos, Dec 26 2016 */

Formula

G.f.: Product_{k>=1} (1 + x^prime(k))*(1 + x^(prime(k)^2)). - Ilya Gutkovskiy, Dec 26 2016

A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 4, 4, 3, 2, 4, 3, 3, 4, 4, 3, 5, 3, 5, 4, 6, 4, 7, 2, 4, 4, 6, 3, 5, 3, 5, 5, 5, 2, 7, 3, 6, 4, 6, 2, 7, 3, 7, 4, 5, 2, 7, 3, 5, 4, 6, 2, 9, 2, 7, 5, 7, 2, 9, 3, 6, 6, 7, 3, 9, 2, 8, 4, 5, 4, 10, 3, 8, 4, 7, 3, 11, 4, 8, 3, 6, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

Examples

			a(12) = 3 because we have [9, 3], [8, 4] and [7, 5].

Links

Crossrefs

Cf. A000961, A071330, A106244, A117929, A246655, A307726, A307825.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[n, {2}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 95}]

Formula

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^A246655(k)).

A358635 Number of partitions of n into at most 2 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 3, 3, 5, 4, 4, 5, 5, 4, 6, 4, 7, 5, 6, 4, 7, 3, 5, 4, 6, 4, 6, 4, 6, 5, 5, 3, 8, 4, 7, 4, 6, 3, 8, 3, 7, 4, 5, 3, 8, 4, 6, 4, 7, 3, 9, 3, 8, 5, 7, 3, 10, 4, 7, 6, 7, 3, 9, 3, 9, 5, 6, 5, 11, 3, 8, 4, 7, 4, 12, 4
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2022

Keywords

Crossrefs

Cf. A000961, A106244, A341132, A347643, A347762, A358636, A358637.

A358636 Number of partitions of n into at most 3 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 9, 9, 10, 9, 12, 11, 12, 12, 15, 14, 15, 14, 17, 16, 17, 17, 21, 19, 21, 20, 25, 22, 25, 24, 28, 27, 27, 26, 29, 29, 28, 31, 32, 30, 31, 32, 33, 35, 34, 34, 37, 37, 34, 37, 38, 39, 37, 41, 37, 44, 38, 40, 41, 44, 38, 47, 43, 46, 43, 50
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2022

Keywords

Crossrefs

Cf. A000961, A106244, A341140, A347644, A347763, A358635, A358637.

A358637 Number of partitions of n into at most 4 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 13, 17, 18, 19, 21, 24, 25, 27, 29, 32, 35, 35, 38, 42, 45, 46, 50, 54, 57, 57, 63, 65, 72, 70, 78, 79, 87, 82, 93, 93, 101, 97, 107, 107, 116, 112, 123, 122, 133, 127, 139, 137, 149, 140, 156, 154, 166, 158, 171, 168, 180, 174, 186
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2022

Keywords

Crossrefs

Cf. A000961, A106244, A341141, A347586, A347645, A347764, A358635, A358636.
Previous Showing 11-20 of 26 results. Next

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

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