A023893 -id:A023893 - cp's OEIS frontend

A377055 Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.

0, 0, 1, 1, 4, 48, 61, 83, 29, 57, 290, 121, 7115, 14207, 68320, 14652, 149979, 122704, 481540, 980376, 632441, 29973, 25343678, 50577935, 7512418, 210836403, 67253056, 224083553, 910629561, 931524323, 452509699, 2880227533, 396690327, 57954538325, 77572935454, 35395016473

Offset: 0

Gus Wiseman, Oct 22 2024

nonn

			The fourth differences of A246655 begin: 1, -3, 3, 0, -2, 2, ... so a(4) = 4.

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree numbers we have A377042, nonsquarefree A377050.
These are the positions of first zeros in each row of A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For leaders we have A377054, for primes A007442 or A030016.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A246655 lists the prime-powers, differences A057820 (except first term).
Cf. A025475, A053707, A093555, A174965, A175804, A361102, A376340, A376596, A377056.

nn=10000;
u=Table[Differences[Select[Range[nn],PrimePowerQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]], {k,mnrm[Union[First/@Position[u,0]]]}]

a(12)-a(27) from Pontus von Brömssen, Oct 22 2024
a(28)-a(30) from Chai Wah Wu, Oct 23 2024
a(31)-a(35) from Lucas A. Brown, Nov 03 2024

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 34, 35, 43, 47, 57, 58, 73, 74, 91, 96, 112, 113, 139, 141, 163, 168, 197, 198, 235, 236, 272, 279, 317, 321, 378, 379, 427, 436, 501, 502, 575, 576, 653, 666, 742, 743, 851, 853, 952, 963, 1080, 1081, 1211, 1216, 1361

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322912 at a(12) = 34, A322912(12) = 33.

			a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(1) = 1 through a(8) = 15 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (71)
                    (211)   (311)    (51)      (421)      (422)
                    (1111)  (2111)   (222)     (511)      (611)
                            (11111)  (411)     (2221)     (2222)
                                     (2211)    (4111)     (3311)
                                     (3111)    (22111)    (4211)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Cf. A000123, A005704, A005705, A005706, A072721.

Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@DeleteCases[#,1]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n) = a(n-1) + A072721(n). a(p) = a(p-1)+1 for p prime.

A280543 Expansion of 1/(1 - x - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 62, 123, 244, 483, 958, 1899, 3765, 7463, 14794, 29329, 58141, 115258, 228486, 452949, 897922, 1780031, 3528716, 6995293, 13867402, 27490602, 54497104, 108034531, 214166610, 424561814, 841647229, 1668473323, 3307565365, 6556885566, 12998306479, 25767716954, 51081672682

Offset: 0

Ilya Gutkovskiy, Jan 05 2017

nonn

Number of compositions (ordered partitions) of n into prime powers (1 included).

			a(3) = 4 because we have [3], [2, 1], [1, 2] and [1, 1, 1].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A000961, A001221, A023893, A280195.

nmax = 37; CoefficientList[Series[1/(1 - x - Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x - Sum_{k>=2} floor(1/omega(k))*x^k).

A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 7, 2, 11, 2, 9, 5, 11, 2, 16, 2, 18, 6, 17, 2, 27, 3, 23, 6, 30, 2, 38, 2, 37, 8, 39, 5, 58, 2, 49, 10, 66, 2, 74, 2, 78, 14, 77, 2, 109, 3, 100, 12, 118, 2, 131, 6, 146, 15, 143, 2, 190, 2, 169, 20, 203, 6, 224, 2, 242, 18, 248

Offset: 0

Gus Wiseman, Dec 30 2018

nonn

Such a partition contains either no 1's or only 1's.
A proper power of n is a number n^k for some positive integer k.
Also integer partitions whose parts all have the same radical base (A052410).

			The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
      (11) (111) (22)   (11111) (33)     (1111111) (44)       (333)
                 (1111)         (42)               (422)      (111111111)
                                (222)              (2222)
                                (111111)           (11111111)
.
  (A)          (B)           (C)            (D)             (E)
  (55)         (11111111111) (66)           (1111111111111) (77)
  (82)                       (84)                           (842)
  (442)                      (93)                           (4442)
  (4222)                     (444)                          (8222)
  (22222)                    (822)                          (44222)
  (1111111111)               (3333)                         (422222)
                             (4422)                         (2222222)
                             (42222)                        (11111111111111)
                             (222222)
                             (111111111111)

a(n) = A072721(n) + 1.

Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]

A105420 Number of partitions of n into 3-smooth parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 23, 31, 38, 53, 63, 82, 100, 128, 152, 194, 228, 284, 336, 410, 478, 586, 678, 814, 947, 1127, 1296, 1539, 1761, 2070, 2372, 2764, 3146, 3667, 4153, 4796, 5437, 6249, 7044, 8080, 9080, 10358, 11636, 13208, 14778, 16762, 18698

Offset: 0

Reinhard Zumkeller, Apr 07 2005

nonn

See A062051 for partitions into distinct 3-smooth numbers.

			n=10: there are 11 partitions of 10 with at least one part not of the form 2^i*3^j: 10, 7+3, 7+2+1, 7+1+1+1, 5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 5+2+1+1+1 and 5+1+1+1+1+1, therefore a(10) = A000041(10) - 11 = 42 - 11 = 31.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 121 terms from Jean-François Alcover)

Cf. A000041, A003586.
Cf. A117220, A117221.
Cf. A062051, A023893, A131995.

nmax = 120;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
P[n_] := IntegerPartitions[n, All, TakeWhile[S, # <= n &] ];
a[n_] := a[n] = P[n] // Length;
Table[Print[n, " ", a[n]]; a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 13 2021 *)

A117222(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006

A321378 Number of integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 3, 2, 3, 0, 6, 1, 5, 3, 6, 1, 11, 2, 9, 6, 12, 5, 19, 4, 17, 11, 23, 9, 32, 10, 31, 22, 39, 17, 55, 21, 57, 37, 67, 33, 92, 44, 97, 65, 114, 63, 154, 78, 162, 113, 191, 117, 250, 138, 269, 194, 320

Offset: 0

Gus Wiseman, Dec 11 2018

nonn

			The a(30) = 11 integer partitions:
  (30)
  (24,6)
  (15,15)
  (18,12)
  (20,10)
  (18,6,6)
  (12,12,6)
  (14,10,6)
  (10,10,10)
  (12,6,6,6)
  (6,6,6,6,6)

Cf. A000607, A000688, A000961, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321347, A321665, A322452, A322454.

nn=100;
ser=Product[If[PrimePowerQ[n],1,1/(1-x^n)],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A131995 Number of partitions of n into powers of 2 or of 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 26, 32, 42, 50, 62, 74, 92, 108, 131, 153, 184, 213, 251, 288, 339, 387, 448, 511, 589, 666, 761, 857, 976, 1095, 1237, 1384, 1561, 1737, 1946, 2161, 2415, 2672, 2971, 3281, 3640, 4007, 4425, 4860, 5359, 5869, 6446, 7049, 7729, 8428

Offset: 0

Reinhard Zumkeller, Aug 06 2007

			a(10) = #{9+1, 8+2, 8+1+1, 4+4+2, 4+4+1+1, 4+3+3, 4+3+2+1,
4+3+1+1+1, 4+2+2+2, 4+2+2+1+1, 4+2+1+1+1+1, 4+1+1+1+1+1+1, 3+3+3+1,
3+3+2+2, 3+3+2+1+1, 3+3+1+1+1+1, 3+2+2+2+1, 3+2+2+1+1+1,
3+2+1+1+1+1+1, 3+1+1+1+1+1+1+1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1,
2+2+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 26.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A018819, A062051, A023893, A000041, A131996.

g:=(1-x)/(product((1-x^(2^k))*(1-x^(3^k)),k=0..10)): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..53); # Emeric Deutsch, Aug 26 2007

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

Prepended a(0) = 1, Joerg Arndt and David A. Corneth, Sep 06 2020

A322902 Numbers whose prime indices are all proper powers of the same number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
A proper power of n is a number n^k for some positive integer k.
Also the union of A322903 and A000079.

			The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (6), (1111), (7), (8), (42), (9), (33), (222).

Cf. A001597, A018819, A023893, A023894, A052410, A056239, A072720, A072721, A302242, A302593, A322900, A322901, A322903.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Select[Range[100],SameQ@@radbase/@primeMS[#]&]

cp's OEIS Frontend

A377055 Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.

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A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

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A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

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A280543 Expansion of 1/(1 - x - Sum_{k>=2} floor(1/omega(k))*x^k), where omega(k) is the number of distinct prime factors (A001221).

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A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

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A105420 Number of partitions of n into 3-smooth parts.

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A321378 Number of integer partitions of n containing no 1's or prime powers.

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A354911 Number of factorizations of n into relatively prime prime-powers.

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A131995 Number of partitions of n into powers of 2 or of 3.

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