A325755 -id:A325755 - cp's OEIS frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325755.

			The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

A325705 Number of integer partitions of n containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 4, 3, 7, 8, 16, 15, 24, 28, 39, 44, 68, 80, 98, 130, 167, 200, 259, 320, 396, 497, 601, 737, 910, 1107, 1335, 1631, 1983, 2372, 2887, 3439, 4166, 4949, 5940, 7043, 8450, 9980, 11884, 13984, 16679, 19493, 23162, 27050, 31937, 37334, 43926

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325706.

			The partition (4,2,1,1,1,1) has distinct multiplicities {1,4}, both of which belong to the partition, so it is counted under a(10).
The a(0) = 1 through a(10) = 16 partitions:
  ()  (1)  (21)  (22)   (41)   (51)    (61)    (71)     (81)     (91)
                 (31)   (221)  (321)   (421)   (431)    (333)    (541)
                 (211)         (2211)  (3211)  (521)    (531)    (631)
                               (3111)          (3221)   (621)    (721)
                                               (4211)   (3321)   (3322)
                                               (32111)  (4221)   (3331)
                                               (41111)  (5211)   (4321)
                                                        (32211)  (5221)
                                                                 (6211)
                                                                 (32221)
                                                                 (33211)
                                                                 (42211)
                                                                 (43111)
                                                                 (322111)
                                                                 (421111)
                                                                 (511111)

Cf. A109297, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325706, A325707, A325755.

Table[Length[Select[IntegerPartitions[n],SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   36: {1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

The first term that is not a prime power A000961 is 36.
The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by Robert Israel, Mar 10 2025
The non-recursive version is A325755, counted by A325702.
Removing all primes gives A353389.
These partitions are counted by A353426.
The version for compositions is A353431.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with all distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

pshadow:= proc(n) local F,i;
  F:= ifactors(n)[2];
  mul(ithprime(i),i=F[..,2])
end proc:
filter:= proc(n) local s;
  if isprime(n) then return true fi;
  s:= pshadow(n);
  n mod s = 0 and member(s,R)
end proc:
R:= {}:
for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,200],suQ[#]&]

Equals A353389 U A000040.

A353394 Product of prime shadows of prime indices of n (with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Gus Wiseman, May 17 2022

nonn

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.

			We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

Positions of first appearances are A353397.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Times@@red/@primeMS[n],{n,100}]

a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.

A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

Original entry on oeis.org

1, 2, 12, 20, 36, 44, 56, 68, 100, 124, 164, 184, 208, 236, 240, 268, 332, 436, 464, 484, 508, 528, 608, 628, 688, 716, 720, 752, 764, 776, 816, 844, 880, 964, 1108, 1132, 1156, 1168, 1200, 1264, 1296, 1324, 1344, 1360, 1412, 1468, 1488, 1584, 1604, 1616, 1724

Offset: 1

Gus Wiseman, May 17 2022

nonn

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.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    20: {1,1,3}
    36: {1,1,2,2}
    44: {1,1,5}
    56: {1,1,1,4}
    68: {1,1,7}
   100: {1,1,3,3}
   124: {1,1,11}
   164: {1,1,13}
   184: {1,1,1,9}
   208: {1,1,1,1,6}
   236: {1,1,17}
   240: {1,1,1,1,2,3}

Product of prime indices is A003963, counted by A339095.
The LHS (product of exponents) is A005361, counted by A266477.
The RHS (product of shadows) is A353394, first appearances A353397.
A related comparison is A353395, counted by A353396.
The partitions are counted by A353398.
Taking indices instead of exponents on the LHS gives A353503.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393
- recursive version counted by A353426
Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==Times@@Last/@FactorInteger[#]&]

A005361(a(n)) = A353394(a(n)).

A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 5, 4, 6, 5, 6, 6, 7, 8, 10, 12, 12, 14, 13, 13, 18, 15, 16, 19, 20, 20, 32, 37, 53, 74, 105

Offset: 0

Gus Wiseman, May 16 2022

a(n) is number of integer partitions of n whose Heinz number belongs to A353393, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(n) partitions for selected n (A..M = 10..22):
  n=1: n=4:  n=14:     n=16:     n=17:     n=18:        n=22:
------------------------------------------------------------------
  (1)  (4)   (E)       (G)       (H)       (I)          (M)
       (22)  (5522)    (4444)    (652211)  (7722)       (9922)
             (532211)  (6622)    (742211)  (752211)     (972211)
                       (642211)  (832211)  (842211)     (A62211)
                       (732211)            (932211)     (B52211)
                                           (333222111)  (C42211)
                                                        (D32211)

The non-recursive version is A325702, ranked by A325755.
The version for compositions is A353391, non-recursive A353390.
These partitions are ranked by A353393, nonprime A353389.
A047966 counts uniform partitions, compositions A329738.
A239455 counts Look-and-Say partitions, ranked by A351294.
Cf. A000041, A002033, A074761, A181819, A181821, A323014, A325534, A333755.

oosQ[y_]:=Length[y]<=1||MemberQ[Subsets[Sort[y],{Length[Union[y]]}],Sort[Length/@Split[y]]]&&oosQ[Sort[Length/@Split[y]]];
Table[Length[Select[IntegerPartitions[n],oosQ]],{n,0,30}]

A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

Original entry on oeis.org

1, 2, 12, 36, 40, 112, 352, 832, 960, 1296, 2176, 2880, 4864, 5376, 11776, 12544, 16128, 29696, 33792, 34560, 38400, 63488, 64000, 101376, 115200, 143360, 151552, 159744, 335872, 479232, 704512, 835584, 1540096, 1658880, 1802240

Offset: 1

Gus Wiseman, May 17 2022

nonn

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. A number's prime signature (row n A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   352: {1,1,1,1,1,5}
   832: {1,1,1,1,1,1,6}
   960: {1,1,1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  2176: {1,1,1,1,1,1,1,7}
  2880: {1,1,1,1,1,1,2,2,3}
  4864: {1,1,1,1,1,1,1,1,8}
  5376: {1,1,1,1,1,1,1,1,2,4}

For shadows instead of exponents we get A003586, counted by A008619.
The LHS (product of prime indices) is A003963, counted by A339095.
The RHS (product of prime exponents) is A005361, counted by A266477.
The version for shadows instead of indices is A353399, counted by A353398.
These partitions are counted by A353506.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

from itertools import count, islice
from math import prod
from sympy import primepi, factorint
def A353503_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p,e in f.items()), count(max(startvalue,1)))
A353503_list = list(islice(A353503_gen(),20)) # Chai Wah Wu, May 20 2022

A003963(a(n)) = A005361(a(n)).

A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 53, 27, 49, 57, 131, 63, 311, 159, 133, 81, 719, 147, 1619, 171, 371, 393, 3671, 189, 361, 933, 343, 477, 8161, 399, 17863, 243, 917, 2157, 1007, 441, 38873, 4857, 2177, 513, 84017, 1113, 180503, 1179, 931, 11013, 386093, 567, 2809, 1083

Offset: 1

Gus Wiseman, May 17 2022

			The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     19: {8}
     21: {2,4}
     53: {16}
     27: {2,2,2}
     49: {4,4}
     57: {2,8}
    131: {32}
     63: {2,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..397 (calculated using the b-file at A033844)

These are the positions of first appearances in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A033844 lists primes indexed by powers of 2.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, firsts A181821, relatively prime A325131.
Equivalent sequence with prime(2*k) instead of prime(2^k): A297002.
Cf. A003586, A005117, A130091, A182850, A289509, A324850, A325755, A353393, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@(2^primeMS[n]),{n,100}]

a(n) = my(f=factor(n)); for(k=1, #f~, f[k,1] = prime(2^primepi(f[k,1]))); factorback(f); \\ Michel Marcus, May 20 2022

from math import prod
from sympy import prime, primepi, factorint
def A353397(n): return prod(prime(2**primepi(p))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 20 2022

If n = prime(e_1)...prime(e_k), then a(n) = prime(2^(e_1))...prime(2^(e_k)).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2^k)) = 1.90812936178871496289... . - Amiram Eldar, Dec 09 2022

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 53.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

cp's OEIS Frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

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A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

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A325705 Number of integer partitions of n containing all of their distinct multiplicities.

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A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

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A353394 Product of prime shadows of prime indices of n (with multiplicity).

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A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

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A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

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A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

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