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.

Previous Showing 11-20 of 24 results. Next

A085756 Number of partitions into a prime number of distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 47, 53, 62, 71, 82, 93, 108, 123, 141, 161, 184, 209, 239, 271, 309, 350, 397, 449, 509, 575, 649, 732, 825, 928, 1044, 1172, 1315, 1474, 1650, 1845, 2061, 2300, 2563, 2854, 3174, 3526, 3912, 4337
Offset: 3

Views

Author

Vladeta Jovovic, Jul 21 2003

Keywords

Examples

			a(15)=20 because there are 20 partitions of 15 into a prime number of distinct parts: 1+2+3+4+5=4+5+6=3+5+7=2+6+7=3+4+8=2+5+8=1+6+8=7+8=2+4+9=1+5+9=6+9=2+3+10=
1+4+10=5+10=1+3+11=4+11=1+2+12=3+12=2+13=1+14.

Crossrefs

Cf. A038499.

Formula

a(n) = A004526(n-1) +A001399(n-6) +A001401(n-15) +A008636(n-28) + .... - R. J. Mathar, Feb 13 2019

A372887 Number of integer partitions of n whose distinct parts are the binary indices of some prime number.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 8, 12, 14, 21, 29, 36, 48, 56, 74, 94, 123, 144, 195, 235, 301, 356, 456, 538, 679, 803, 997, 1189, 1467, 1716, 2103, 2488, 2968, 3517, 4185, 4907, 5834, 6850, 8032, 9459, 11073, 12933, 15130, 17652, 20480, 24011, 27851, 32344, 37520
Offset: 0

Views

Author

Gus Wiseman, May 19 2024

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

Examples

			The partition y = (4,3,1,1) has distinct parts {1,3,4}, which are the binary indices of 13, which is prime, so y is counted under a(9).
The a(2) = 1 through a(9) = 14 partitions:
  (2)  (21)  (22)   (221)   (51)     (331)     (431)      (3321)
             (31)   (311)   (222)    (421)     (521)      (4221)
             (211)  (2111)  (321)    (511)     (2222)     (4311)
                            (2211)   (2221)    (3221)     (5211)
                            (3111)   (3211)    (3311)     (22221)
                            (21111)  (22111)   (4211)     (32211)
                                     (31111)   (5111)     (33111)
                                     (211111)  (22211)    (42111)
                                               (32111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (321111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)

Crossrefs

For odd instead of prime we have A000041, even A002865.
The strict case is A372687, ranks A372851.
Counting not just distinct parts gives A372688, ranks A277319.
These partitions have Heinz numbers A372850.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000040, A000041, A029837, A035100, A038499, A372429, A372471.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^(Union[#]-1)]]&]],{n,0,30}]

A316154 Number of integer partitions of prime(n) into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 12, 19, 39, 50, 93, 136, 166, 239, 409, 682, 814, 1314, 1774, 2081, 3231, 4272, 6475, 11077, 14270, 16265, 20810, 23621, 30031, 68251, 85326, 118917, 132815, 226097, 251301, 342448, 463940, 565844, 759873, 1015302, 1117708, 1787452, 1961624
Offset: 1

Views

Author

Gus Wiseman, Jun 25 2018

Keywords

Examples

			The a(7) = 9 partitions of 17 into a prime number of prime parts: (13,2,2), (11,3,3), (7,7,3), (7,5,5), (7,3,3,2,2), (5,5,3,2,2), (5,3,3,3,3), (5,2,2,2,2,2,2), (3,3,3,2,2,2,2).

Links

Crossrefs

Cf. A000040, A000586, A000607, A038499, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316185, A344782.

Programs

  • Maple
    b:= proc(n, p, c) option remember; `if`(n=0 or p=2,
      `if`(n::even and isprime(c+n/2), 1, 0),
      `if`(p>n, 0, b(n-p, p, c+1))+b(n, prevprime(p), c))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 26 2018
  • Mathematica
    Table[Length[Select[IntegerPartitions[Prime[n]],And[PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,20}]
(* Second program: *)
b[n_, p_, c_] := b[n, p, c] = If[n == 0 || p == 2, If[EvenQ[n] && PrimeQ[c + n/2], 1, 0], If[p>n, 0, b[n - p, p, c + 1]] + b[n, NextPrime[p, -1], c]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 50] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
  • PARI
    seq(n)={my(p=vector(n,k,prime(k))); my(v=Vec(1/prod(k=1, n, 1 - x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

Formula

a(n) = A085755(A000040(n)). - Alois P. Heinz, Jun 26 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jun 26 2018

A316185 Number of strict integer partitions of the n-th prime into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 2, 3, 5, 5, 6, 8, 10, 13, 18, 20, 26, 32, 34, 45, 54, 66, 90, 106, 117, 135, 142, 165, 269, 311, 375, 398, 546, 579, 689, 823, 938, 1107, 1301, 1352, 1790, 1850, 2078, 2153, 2878, 3811, 4241, 4338, 4828, 5495, 5637, 7076, 8000, 9032
Offset: 1

Views

Author

Gus Wiseman, Jun 25 2018

Keywords

Examples

			The a(14) = 8 partitions of 43 into a prime number of distinct prime parts: (41,2), (31,7,5), (29,11,3), (23,17,3), (23,13,7), (19,17,7), (19,13,11), (17,11,7,5,3).

Links

Crossrefs

Cf. A000586, A000607, A038499, A045450, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316154.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n=0, 0,
     `if`(isprime(n), n, h(n-1)))
    end:
b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(isprime(c), 1, 0), `if`(i<2, 0, b(n, h(i-1), c)+
      `if`(i>n, 0, b(n-i, h(min(n-i, i-1)), c+1))))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..56);  # Alois P. Heinz, May 26 2021
  • Mathematica
    Table[Length[Select[IntegerPartitions[Prime[n]],And[UnsameQ@@#,PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,10}]
(* Second program: *)
h[n_] := h[n] = If[n == 0, 0, If[PrimeQ[n], n, h[n - 1]]];
b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[PrimeQ[c], 1, 0], If[i < 2, 0, b[n, h[i - 1], c] +
     If[i > n, 0, b[n - i, h[Min[n - i, i - 1]], c + 1]]]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 56] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)
  • PARI
    seq(n)={my(p=vector(n, k, prime(k))); my(v=Vec(prod(k=1, n, 1 + x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

Formula

a(n) = A045450(A000040(n)).

Extensions

More terms from Alois P. Heinz, Jun 26 2018

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

Original entry on oeis.org

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987
Offset: 1

Views

Author

Gus Wiseman, Mar 24 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 (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.

Links

Crossrefs

These partitions are counted by A352493.
This is the restriction of A346068 to numbers that are not a prime power.
The prime-power version is A352519, counted by A230595.
A000040 lists the primes.
A000961 lists prime powers.
A001694 lists powerful numbers, counted by A007690.
A038499 counts partitions of prime length.
A053810 lists all numbers p^q for p and q prime, counted by A001221.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Programs

  • Mathematica
    Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Formula

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024

A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

Original entry on oeis.org

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2022

Keywords

Comments

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

Examples

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}

Links

Crossrefs

Numbers of the form p^q for p and q prime are A053810, counted by A001221.
These partitions are counted by A230595.
This is the prime power case of A346068.
For numbers that are not a prime power we have A352518, counted by A352493.
A000040 lists the primes.
A000961 lists prime powers.
A001597 lists perfect powers.
A001694 lists powerful numbers, counted by A007690.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A164336 lists all possible power-towers of prime numbers.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

Programs

  • Maple
    N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022
  • Mathematica
    Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

A352493 Number of non-constant integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 4, 5, 3, 1, 3, 5, 7, 3, 5, 6, 8, 8, 11, 7, 6, 8, 15, 14, 14, 10, 15, 17, 21, 18, 23, 20, 28, 25, 31, 27, 35, 32, 33, 37, 46, 41, 50, 45, 58, 56, 63, 59, 78, 69, 76, 81, 85, 80, 103, 107, 111, 114, 127
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2022

Keywords

Examples

			The a(n) partitions for selected n (B = 11):
n = 10    16       19        20         25          28
   ---------------------------------------------------------------
    3322  5533     55333     7733       77722       BB33
          55222    55522     77222      5533333     BB222
          3322222  3333322   553322     5553322     775522
                   33322222  5522222    55333222    55533322
                             332222222  55522222    772222222
                                        333333322   3322222222222
                                        3333322222

Crossrefs

Constant partitions are counted by A001221, ranked by A000961.
Non-constant partitions are counted by A144300, ranked A024619.
The constant version is A230595, ranked by A352519.
This is the non-constant case of A351982, ranked by A346068.
These partitions are ranked by A352518.
A000040 lists the primes.
A000607 counts partitions into primes, ranked by A076610.
A001597 lists perfect powers, complement A007916.
A038499 counts partitions of prime length.
A053810 lists primes to primes.
A055923 counts partitions with prime multiplicities, ranked by A056166.
A257994 counts prime indices that are themselves prime.
A339218 counts powerful partitions into prime parts, ranked by A352492.
Cf. A000005, A007690, A031368, A035444, A052485, A056239, A066208, A089723, A114639, A320628, A330945.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&And@@PrimeQ/@#&& And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A102623 Number of compositions into a prime number of distinct parts.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 12, 18, 26, 32, 40, 52, 60, 72, 206, 218, 352, 490, 744, 1002, 1382, 1760, 2380, 3004, 3864, 4728, 5954, 12218, 13804, 20554, 27660, 39930, 52682, 75632, 99184, 132940, 172332, 227088, 287606, 373562, 465280, 587602, 725880, 899802, 1094846
Offset: 1

Views

Author

Vladeta Jovovic, Jan 31 2005

Keywords

Links

Crossrefs

Cf. A085756, A052467, A038499.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(n>i*(i+1)/2, [], zip((x, y)->x+y, b(n, i-1),
      `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= proc(n) local l; l:= b(n$2);
      add(`if`(isprime(i), l[i+1]*i!, 0), i=2..nops(l)-1)
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2012
  • Mathematica
    CoefficientList[ Series[ Sum[ Prime[k]!* x^(Prime[k]^2/2 + Prime[k]/2)/Product[1 - x^j, {j, Prime[k]}], {k, 44}], {x, 0, 44}], x] (* Robert G. Wilson v, Feb 04 2005 *)

Formula

G.f.: Sum(prime(k)!*x^(1/2*prime(k)^2+1/2*prime(k))/Product(1-x^j, j = 1 .. prime(k)), k = 1 .. infinity).

Extensions

More terms from Robert G. Wilson v, Feb 04 2005

A316153 Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.

Original entry on oeis.org

15, 33, 45, 93, 153, 177, 275, 327, 369, 405, 425, 537, 603, 605, 775, 831, 891, 1025, 1059, 1125, 1413, 1445, 1475, 1641, 1705, 1719, 1761, 2057, 2075, 2319, 2511, 2577, 2979, 3175, 3179, 3189, 3459, 3485, 3603, 3609, 3825, 3925, 4299, 4475, 4497, 4565, 4581
Offset: 1

Views

Author

Gus Wiseman, Jun 25 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 of prime numbers into a prime number of prime parts, preceded by their Heinz numbers, begins:
   15: (3,2)
   33: (5,2)
   45: (3,2,2)
   93: (11,2)
  153: (7,2,2)
  177: (17,2)
  275: (5,3,3)
  327: (29,2)
  369: (13,2,2)
  405: (3,2,2,2,2)
  425: (7,3,3)
  537: (41,2)
  603: (19,2,2)
  605: (5,5,3)
  775: (11,3,3)
  831: (59,2)
  891: (5,2,2,2,2)

Crossrefs

Cf. A000586, A000607, A038499, A056239, A056768, A064688, A070215, A085755, A302590, A316092, A316151.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[PrimeQ[PrimeOmega[#]],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A286141 Number of partitions of n into a squarefree number of parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 30, 40, 53, 70, 92, 120, 154, 199, 254, 324, 409, 517, 648, 811, 1008, 1253, 1549, 1911, 2347, 2880, 3519, 4294, 5219, 6338, 7671, 9273, 11173, 13451, 16147, 19359, 23151, 27656, 32958, 39231, 46594, 55276, 65444, 77391, 91341, 107689, 126734
Offset: 0

Views

Author

Ilya Gutkovskiy, May 07 2017

Keywords

Comments

Also number of partitions of n such that the largest part is a squarefree (A005117).

Examples

			a(6) = 9 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [2, 2, 2], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions into a squarefree number of parts).
Also a(6) = 9 because we have [6], [5, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions such that the largest part is a squarefree).

Links

Crossrefs

Cf. A005117, A038499, A073576, A089333, A102848, A178927.

Programs

  • Mathematica
    Join[{1}, Table[Length@Select[IntegerPartitions@n, SquareFreeQ@Length@# &], {n, 50}]]
nmax = 50; CoefficientList[Series[1 + Sum[MoebiusMu[i]^2 x^i/Product[1 - x^j, {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 + Sum_{i>=1} x^A005117(i) / Product_{j=1..A005117(i)} (1 - x^j).
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