A302491 -id:A302491 - cp's OEIS frontend

A072774 Powers of squarefree numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
   (1, 1, 1) : f (tail a005117_list) empty where
   f vs'@(v:vs) m
    | Map.null m || xx > v = (v, v, 1) :
                             f vs (insert (v^2) (v, 2) m)
    | otherwise = (xx, bx, ex) :
                  f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

A302478 Products of prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 66, 67, 68, 72, 73, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 90, 93, 94

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01:  {}
02:  {{}}
03:  {{1}}
04:  {{},{}}
05:  {{2}}
06:  {{},{1}}
08:  {{},{},{}}
09:  {{1},{1}}
10:  {{},{2}}
11:  {{3}}
12:  {{},{},{1}}
13:  {{1,2}}
15:  {{1},{2}}
16:  {{},{},{},{}}
17:  {{4}}
18:  {{},{1},{1}}
20:  {{},{},{2}}
22:  {{},{3}}
24:  {{},{},{},{1}}
25:  {{2},{2}}
26:  {{},{1,2}}
27:  {{1},{1},{1}}
29:  {{1,3}}
30:  {{},{1},{2}}
31:  {{5}}
32:  {{},{},{},{},{}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A007716, A050320, A056239, A063834, A076610, A270995, A275024, A281113, A296119, A301753, A302242, A302243, A302491.

Select[Range[100],Or[#===1,And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

ok(n)={!#select(p->!issquarefree(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A277098 Finitary primes. Primes of finitary index.

Original entry on oeis.org

2, 3, 5, 11, 13, 29, 31, 41, 47, 79, 101, 109, 113, 127, 137, 167, 179, 211, 257, 271, 293, 313, 317, 397, 401, 421, 449, 487, 491, 547, 599, 601, 617, 677, 709, 733, 773, 811, 823, 829, 907, 929, 977, 991, 1033, 1063, 1109, 1187, 1231, 1259, 1297, 1361, 1429, 1483, 1489, 1559, 1609, 1621, 1741, 1759, 1831, 1871, 1889

Offset: 1

Gus Wiseman, Sep 29 2016

nonn

Definition: prime(n) is a finitary prime iff n is a product of distinct finitary primes, where prime = A000040. This sequence could be described as a "multiplicative Aronson transform" of A005117. (Aronson transforms such as A079000 satisfy "n is in a if and only if a(n) is in b".

The composite bijection (finitary primes -> finitary numbers -> finite sets of finitary primes) can be used to construct a natural linear extension SET : N -> F where F is the partially ordered inverse limit of all finite Boolean algebras of finite sets of positive integers. Then a(n) = prime(Prod_p a(p)) where the product is over SET(n).

			The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:
0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,
g,bd,be,h,i,cd,af,ag,ce,abd,abe,
j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,
o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,
bce,ak,ch,r,al,am,ci,bj,abh,an,s,
t,ao,abi,ade,acf,u,bk,acg,v,w,df,
bl,abcd,ap,bm,dg,aq,ef,bn,abce,
cj,x,y,eg,ach,bo,z,ar,bde,bcf,*

Gus Wiseman, Table of n, a(n) for n = 1..10000

Subsequence of A302491.

Cf. A000040, A005117, A079000, A079254, A276625.

has(p)=if(p<7, 1, my(f=factor(primepi(p))); if(vecmax(f[,2])>1, return(0)); for(i=1,#f~, if(!has(f[i,1]), return(0))); 1)
is(n)=isprime(n) && has(n) \\ Charles R Greathouse IV, Aug 03 2023

a(n) = A000040(A276625(n)).

A318400 Numbers whose prime indices are all powers of 2 (including 1).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 19, 21, 24, 27, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 128, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 256, 262

Offset: 1

Gus Wiseman, Dec 16 2018

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 sequence of all integer partitions whose parts are all powers of 2 (including 1) begins: (), (1), (2), (11), (21), (4), (111), (22), (211), (41), (1111), (221), (8), (42), (2111), (222), (411), (11111), (2211), (81), (421), (21111), (44).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003963, A018819, A033844, A056239, A106349, A112798, A302242, A302491, A322551.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pow2Q[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,_}}]];
Select[Range[100],And@@pow2Q/@primeMS[#]&]

Sum_{n>=1} 1/a(n) = 1/Product_{k>=0} (1 - 1/prime(2^k)) = 3.81625872357742992578... . - Amiram Eldar, Dec 03 2022

A322551 Primes indexed by squarefree semiprimes.

Original entry on oeis.org

13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 983, 1051, 1061, 1093

Offset: 1

Gus Wiseman, Dec 15 2018

nonn

A squarefree semiprime is a product of two distinct prime numbers.

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of non-loop edges.

			The sequence of edges whose MM-numbers belong to the sequence begins: {{1,2}}, {{1,3}}, {{1,4}}, {{2,3}}, {{2,4}}, {{1,5}}, {{1,6}}, {{2,5}}, {{1,7}}, {{3,4}}, {{1,8}}, {{2,6}}, {{1,9}}, {{2,7}}, {{3,5}}, {{2,8}}.

Cf. A001358, A003963, A006881, A056239, A085156, A106349, A112798, A302242, A302491, A320458, A320459, A320461.

Select[Range[100],PrimeOmega[#]==1&&PrimeOmega[PrimePi[#]]==2&&SquareFreeQ[PrimePi[#]]&]

isok(p) = isprime(p) && (ip=primepi(p)) && (omega(ip)==2) && (bigomega(ip) == 2); \\ Michel Marcus, Dec 16 2018

A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 377, 611, 1363, 1937, 2021, 2117, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 143663, 146653, 147533, 153023, 159659, 167243, 170839, 203087, 237679, 243893, 265369, 271049, 276877, 290029, 301129, 315433, 467711

Offset: 1

Gus Wiseman, Oct 13 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1363: {{1,3},{2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
  16211: {{1,2},{1,3},{1,4}}
  17719: {{1,2},{1,3},{2,3}}
  26273: {{1,2},{1,4},{2,3}}
  27521: {{1,2},{1,3},{2,4}}
  44603: {{1,2},{2,3},{2,4}}
  56173: {{1,2},{1,3},{3,4}}
  58609: {{1,3},{1,4},{2,3}}
  83291: {{1,2},{1,4},{3,4}}
  91031: {{1,3},{1,4},{2,4}}
  91039: {{1,2},{2,3},{3,4}}
  99499: {{1,3},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A001222, A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A302491, A305052.

Cf. A320456, A320459, A320461, A320462, A320463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A302493 Prime numbers of prime-power index.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 31, 41, 53, 59, 67, 83, 97, 103, 109, 127, 131, 157, 179, 191, 211, 227, 241, 277, 283, 311, 331, 353, 367, 401, 419, 431, 461, 509, 547, 563, 587, 599, 617, 661, 691, 709, 719, 739, 773, 797, 859, 877, 919, 967, 991, 1009, 1031

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

Cf. A000040, A000961, A001222, A005117, A007097, A056239, A063834, A275024, A281113, A302242, A302491, A302492.

Prime/@Select[Range[100],Or[#===1,PrimePowerQ[#]]&]

forprime(p=1, 500, if(p==2 || isprimepower(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Apr 10 2018

a(n) = A000040(A000961(n)).

A322554 Numbers whose product of prime indices is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80

Offset: 1

Gus Wiseman, Dec 15 2018

nonn

The complement is {35, 37, 39, 45, 61, 65, ...}.

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.

			Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MM-numbers do not belong to this sequence begins:
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  45: {{1},{1},{2}}
  61: {{1,2,2}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  70: {{},{2},{1,1}}
  71: {{1,1,3}}
  74: {{},{1,1,2}}
  75: {{1},{2},{2}}
  77: {{1,1},{3}}
  78: {{},{1},{1,2}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  90: {{},{1},{1},{2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}
  99: {{1},{1},{3}}

Cf. A003963, A056239, A062503, A072774, A112798, A302242, A302491, A320325, A320698, A322553.

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

A303365 Number of integer partitions of the n-th squarefree number using squarefree numbers.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 28, 36, 60, 76, 96, 150, 228, 342, 416, 504, 877, 1484, 1759, 2079, 2885, 3387, 3968, 5413, 6304, 7328, 9852, 11395, 13159, 20082, 23056, 39532, 51385, 66488, 85660, 97078, 109907, 140465, 158573, 226918, 255268, 286920, 361606, 405470

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(5) = 9 partitions are (6), (51), (33), (321), (3111), (222), (2211), (21111), (111111).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000051, A005117, A072774, A073576, A078374, A302491, A302796, A303283, A303364.

nn=80;
sqf=Select[Range[nn],SquareFreeQ];
ser=Product[1/(1-x^sqf[[n]]),{n,Length[sqf]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,sqf}]

a(n) = A073576(A005117(n)).

A320635 MM-numbers of simple labeled connected graphs spanning an initial interval of positive integers.

Original entry on oeis.org

13, 377, 611, 1363, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 147533, 203087, 301129, 315433, 467711, 761917, 1183403, 1280669, 1293487, 1917929, 2075567, 2174159, 2220907, 2415439, 2640131

Offset: 1

Gus Wiseman, Oct 18 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
       13: {{1,2}}
      377: {{1,2},{1,3}}
      611: {{1,2},{2,3}}
     1363: {{1,3},{2,3}}
    16211: {{1,2},{1,3},{1,4}}
    17719: {{1,2},{1,3},{2,3}}
    26273: {{1,2},{1,4},{2,3}}
    27521: {{1,2},{1,3},{2,4}}
    44603: {{1,2},{2,3},{2,4}}
    56173: {{1,2},{1,3},{3,4}}
    58609: {{1,3},{1,4},{2,3}}
    83291: {{1,2},{1,4},{3,4}}
    91031: {{1,3},{1,4},{2,4}}
    91039: {{1,2},{2,3},{3,4}}
    99499: {{1,3},{2,3},{2,4}}
   141401: {{1,2},{2,4},{3,4}}
   147533: {{1,4},{2,3},{2,4}}
   203087: {{1,3},{2,3},{3,4}}
   301129: {{1,4},{2,3},{3,4}}
   315433: {{1,3},{2,4},{3,4}}
   467711: {{1,4},{2,4},{3,4}}
   761917: {{1,2},{1,3},{1,4},{2,3}}
  1183403: {{1,2},{1,3},{1,4},{2,4}}
  1280669: {{1,2},{1,3},{1,4},{1,5}}

Cf. A001222, A007717, A055932, A056239, A112798, A255906, A290103, A302242, A302491, A305078, A320456, A320458.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#]),Length[zsm[primeMS[#]]]==1]&]

cp's OEIS Frontend

A072774 Powers of squarefree numbers.

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A302478 Products of prime numbers of squarefree index.

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A277098 Finitary primes. Primes of finitary index.

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A318400 Numbers whose prime indices are all powers of 2 (including 1).

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A322551 Primes indexed by squarefree semiprimes.

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A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

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Mathematica

A302493 Prime numbers of prime-power index.

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A322554 Numbers whose product of prime indices is a power of a squarefree number (A072774).

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