A000586 -id:A000586 - cp's OEIS frontend

A379304 Number of integer partitions of n with a unique prime part.

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379305 Number of strict integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379308 Number of integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867

Offset: 0

Gus Wiseman, Dec 26 2024

nonn

			The a(1) = 1 through a(11) = 5 partitions:
  (1)  (2)  (3)  .  (5)    (6)    (7)    .  (5,4)    (10)     (11)
                    (4,1)  (4,2)  (4,3)     (8,1)    (6,4)    (7,4)
                                            (4,4,1)  (8,2)    (8,3)
                                                     (9,1)    (9,2)
                                                     (4,4,2)  (4,4,3)

If all parts are squarefree we have A073576 (strict A087188), ranks A302478.
If no parts are squarefree we have A114374 (strict A256012), ranks A379307.
For composite instead of squarefree we have A379302 (strict A379303), ranks A379301.
For prime instead of squarefree we have A379304, (strict A379305), ranks A331915.
The strict case is A379309.
For old prime instead of squarefree we have A379314, (strict A379315), ranks A379312.
Ranked by A379316, positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A013928, A023895, A034891, A072284, A073247, A120327, A175804, A376657, A377430.

Table[Length[Select[IntegerPartitions[n],Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

A379309 Number of strict integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
The non-strict version is A379308, ranks A379316.
For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A046675 Expansion of Product_{i>0} (1-x^{p_i}), where p_i are the primes.

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, -1, 1, 1, 1, -1, 1, -1, -1, -1, 2, 0, 1, -1, 1, 0, 0, -3, 2, 1, 1, -2, 1, -2, 1, -2, 1, 0, 2, -3, 3, -1, 0, -2, 4, -1, 2, -4, 1, -1, 3, -5, 4, -1, 2, -3, 4, -4, 3, -5, 3, -1, 4, -8, 6, -1, 2, -7, 6, -4, 8, -6, 3

Offset: 0

N. J. A. Sloane

sign

The difference between the number of even partitions of n into distinct primes and the number of odd partitions of n into distinct primes. - T. D. Noe, Sep 08 2006

B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187 [broken link].

Cf. A000607, A000586.

CoefficientList[Series[Product[1 - x^Prime[i], {i, 1, 100}], {x, 0, 100}], x] (* Vaclav Kotesovec, Sep 13 2018 *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j]] -= poly[[j - p]], {j, nmax + 1, p + 1, -1}];, {k, 2, pmax}]; poly (* Vaclav Kotesovec, Sep 20 2018 *)

a(n) = A184171(n) - A184172(n). - R. J. Mathar, Jan 10 2011

Revised by N. J. A. Sloane, Jun 10 2012

A054845 Number of ways of representing n as the sum of one or more consecutive primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 2

Offset: 0

Jud McCranie, May 25 2000

Moser shows that the average order of a(n) is log 2, that is, Sum_{i=1..n} a(i) ~ n log 2. This shows that a(n) = 0 infinitely often (and with positive density); Moser asks if a(n) = 1 infinitely often, if a(n) = k is solvable for all k, whether these have positive density, and whether the sequence is bounded. - Charles R Greathouse IV, Mar 21 2011

			a(5)=2 because of 2+3 and 5. a(17)=2 because of 2+3+5+7 and 17.
41 = 41 = 11+13+17 = 2+3+5+7+11+13, so a(41)=3.

R. K. Guy, Unsolved Problems In Number Theory, C2.

T. D. Noe, Table of n, a(n) for n = 0..10000
Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161.
Carlos Rivera, Problem 9, The Prime Puzzles and Problems Connection.

Cf. A000586, A054859.

S:=[0]; for n in [1..104] do count:=0; for q in PrimesUpTo(n) do p:=q; s:=p; while s lt n do p:=NextPrime(p); s+:=p; end while; if s eq n then count+:=1; end if; end for; Append(~S, count); end for; S; // Klaus Brockhaus, Feb 17 2011

A054845 := proc(n)
    local a,mipri,npr,ps ;
    a := 0 ;
    for mipri from 1 do
        for npr from 1 do
            ps := add(ithprime(i),i=mipri..mipri+npr-1) ;
            if ps = n then
                a := a+1 ;
            elif ps >n then
                break;
            end if;
        end do:
        if ithprime(mipri) > n then
            break ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Nov 27 2018

f[n_] := Block[{p = Prime@ Range@ PrimePi@ n}, len = Length@ p; Count[(Flatten[#, 1] &)[Table[ p[[i ;; j]], {i, len}, {j, i, len}]], q_ /; Total@ q == n]]; f[0] = 0; Array[f, 102, 0](* Jean-François Alcover, Feb 16 2011 *) (* fixed by Robert G. Wilson v *)
nn=100; p=Prime[Range[PrimePi[nn]]]; t=Table[0,{nn}]; Do[s=0; j=i; While[s=s+p[[j]]; s<=nn,t[[s]]++; j++], {i,Length[p]}]; Join[{0},t]

{/* program gives values of a(n) for n=0..precprime(nn)-1 */
nn=2000;t=vector(nn+1);forprime(x=2,nn,s=x;
forprime(y=x+1,nn,if(s<=nn,t[s+1]++,break());s=s+y));t[1..precprime(nn)]} \\ Zak Seidov, Feb 17 2011, corrected by Michael S. Branicky, Feb 28 2022

use ntheory ":all"; my $n=10000; my @W=(0)x($n+1); forprimes { my $s=$; do { $W[$s]++; $s += ($=next_prime($)); } while $s <= $n; } $n; print "$ $W[$]\n" for 0..$#W;  # _Dana Jacobsen, Aug 22 2018

from sympy import primerange
def aupton(nn): # modification of PARI by Zak Seidov
    alst = [0 for n in range(nn+1)]
    for x in primerange(2, nn+1):
        s = x
        alst[s] += 1
        for y in primerange(x+1, nn+1):
            s += y
            if s <= nn:
                alst[s] += 1
            else:
                break
    return alst
print(aupton(101)) # Michael S. Branicky, Feb 17 2022

# alternate version for going to large n
import heapq
from sympy import prime
from itertools import islice
def agen(): # generator of terms
    p = v = prime(1); h = [(p, 1, 1)]; nextcount = 2; oldv = ways = 0
    while True:
        (v, s, l) = heapq.heappop(h)
        if v == oldv: ways += 1
        else:
            yield ways
            for n in range(oldv+1, v): yield 0
            ways = 1
        if v >= p:
            p += prime(nextcount)
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        oldv = v
        v -= prime(s); s += 1; l += 1; v += prime(l)
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 102))) # Michael S. Branicky, Feb 17 2022

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^prime(k). - Ilya Gutkovskiy, Apr 18 2019

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of Jake M. Foster

A070215 Number of ways to write the n-th prime as a sum of distinct primes.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 3, 5, 7, 9, 11, 14, 15, 19, 26, 35, 39, 50, 61, 67, 87, 102, 130, 178, 204, 224, 257, 278, 320, 522, 595, 724, 776, 1064, 1136, 1364, 1634, 1836, 2192, 2601, 2761, 3645, 3863, 4294, 4549, 6262, 8558, 9453, 9964, 11001, 12774, 13438

Offset: 1

Lekraj Beedassy, May 07 2002

nonn

			With the 10th prime 29, for instance, we have a(10)=7 distinct-prime partitions, viz. 29 = 2 + 3 + 7 + 17 = 2 + 3 + 5 + 19 = 2 + 3 + 11 + 13 = 3 + 7 + 19 = 5 + 7 + 17 = 5 + 11 + 13.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1201 terms from Seth Troisi)

Cf. A000586, A056768 (parts may repeat).

a070215 = a000586 . a000040  -- Reinhard Zumkeller, Aug 05 2012

nn = PrimePi[300]; t = CoefficientList[Series[Product[(1 + x^Prime[k]), {k, nn}], {x, 0, Prime[nn]}], x]; t[[1 + Prime[Range[nn]]]] (* T. D. Noe, Nov 13 2013 *)

a(n) = A000586(prime(n)). - R. J. Mathar, Apr 30 2007

More terms from Naohiro Nomoto and Don Reble, May 11 2002

Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009

A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

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:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, 0, -1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -3, 3, -3, 4, -4, 5, -6, 6, -6, 8, -9, 9, -11, 12, -13, 14, -16, 19, -19, 21, -25, 26, -28, 32, -36, 38, -41, 46, -50, 55, -60, 65, -70, 77, -85, 91, -99, 108, -116, 126, -138, 149, -160, 174, -188, 202, -219, 237, -255, 274, -296

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000040, A000586, A000607, A001221, A008683, A083399, A088705, A280954, A305614.

nn=20;
ser=Product[1/(1+x^p),{p,Select[Range[nn],PrimeQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}] (* Gus Wiseman, Jun 06 2018 *)

a(n) = A184198(n) - A184199(n). - Vaclav Kotesovec, Jan 11 2021

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset: 1

Gus Wiseman, Dec 29 2024

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:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

cp's OEIS Frontend

A379304 Number of integer partitions of n with a unique prime part.

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A379305 Number of strict integer partitions of n with a unique prime part.

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A379308 Number of integer partitions of n with a unique squarefree part.

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A379309 Number of strict integer partitions of n with a unique squarefree part.

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A046675 Expansion of Product_{i>0} (1-x^{p_i}), where p_i are the primes.

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A054845 Number of ways of representing n as the sum of one or more consecutive primes.

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A070215 Number of ways to write the n-th prime as a sum of distinct primes.

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A379312 Positive integers whose prime indices include a unique 1 or prime number.

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