A239885 -id:A239885 - cp's OEIS frontend

A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

3, 10, 28, 66, 70, 88, 208, 228, 306, 340, 364, 490, 495, 525, 544, 550, 675, 744, 870, 966, 1160, 1216, 1242, 1254, 1288, 1326, 1330, 1332, 1672, 1768, 1785, 1870, 2002, 2064, 2145, 2295, 2457, 2900, 2944, 3250, 3280, 3430, 3468, 3540, 3724, 4125, 4144, 4248

Offset: 1

Gus Wiseman, May 23 2019

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, with sum A056239(n).

			The term 70 is in the sequence because it has 8 divisors {1, 2, 5, 7, 10, 14, 35, 70} and its sum of prime indices is also 1 + 3 + 4 = 8.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    70: {1,3,4}
    88: {1,1,1,5}
   208: {1,1,1,1,6}
   228: {1,1,2,8}
   306: {1,2,2,7}
   340: {1,1,3,7}
   364: {1,1,4,6}
   490: {1,3,4,4}
   495: {2,2,3,5}
   525: {2,3,3,4}
   544: {1,1,1,1,1,7}
   550: {1,3,3,5}
   675: {2,2,2,3,3}
   744: {1,1,1,2,11}
   870: {1,2,3,10}
   966: {1,2,4,9}

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

Positions of 0's in A325794.
Contains A239885 except for 1.
Cf. A000005, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325792, A325795, A325796, A325797, A325798.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(numtheory:-pi(t[1])*t[2],t=F) = mul(t[2]+1,t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 16 2023

Select[Range[100],DivisorSigma[0,#]==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A110295 a(n) = 2^(n-1) * prime(n).

Original entry on oeis.org

2, 6, 20, 56, 176, 416, 1088, 2432, 5888, 14848, 31744, 75776, 167936, 352256, 770048, 1736704, 3866624, 7995392, 17563648, 37224448, 76546048, 165675008, 348127232, 746586112, 1627389952, 3388997632, 6912212992, 14361296896

Offset: 1

Ryan Propper, Sep 07 2005

nonn

Eric Angelini, "Array with primes." Pers. comm. on the SeqFan mailing list, Sep. 7 2005.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000079, A239885.

[2^(n-1)*NthPrime(n): n in [1..50]]; // G. C. Greubel, Jan 04 2023

Mathematica
```
Table[2^(n-1)*Prime[n], {n, 30}]
```

[2^(n-1)*nth_prime(n) for n in range(1,50)] # G. C. Greubel, Jan 04 2023

a(n) = 2*A239885(n). - G. C. Greubel, Jan 04 2023

A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

Original entry on oeis.org

3, 10, 28, 66, 88, 156, 208, 306, 340, 408, 544, 570, 684, 760, 912, 966, 1216, 1242, 1288, 1380, 1656, 1840, 2208, 2436, 2610, 2900, 2944, 3132, 3248, 3480, 3906, 4092, 4176, 4340, 4640, 4650, 5022, 5208, 5456, 5568, 5580, 6200, 6696, 6944, 7326, 7424, 7440

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325793 in lacking 70.
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, with sum A056239(n). A subset-sum of an integer partition is any sum of a submultiset of it.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.

			340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    88: {1,1,1,5}
   156: {1,1,2,6}
   208: {1,1,1,1,6}
   306: {1,2,2,7}
   340: {1,1,3,7}
   408: {1,1,1,2,7}
   544: {1,1,1,1,1,7}
   570: {1,2,3,8}
   684: {1,1,2,2,8}
   760: {1,1,1,3,8}
   912: {1,1,1,1,2,8}
   966: {1,2,4,9}
  1216: {1,1,1,1,1,1,8}
  1242: {1,2,2,2,9}
  1288: {1,1,1,4,9}
  1380: {1,1,2,3,9}

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

Positions of 1's in A325799.
Includes A239885 except for 1.
Cf. A002033, A056239, A108917, A112798, A126796, A299701, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325801, A325802.

filter:= proc(n) local F,t,S,i,r;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = add(t[1]*t[2],t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 30 2024

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],hwt[#]==Length[Union[hwt/@Divisors[#]]]&]

A056239(a(n)) = A299701(a(n)) = A304793(a(n)) + 1.

A135483 a(n) = Sum_{j=1..n} prime(j)*2^(j-2).

Original entry on oeis.org

1, 4, 14, 42, 130, 338, 882, 2098, 5042, 12466, 28338, 66226, 150194, 326322, 711346, 1579698, 3513010, 7510706, 16292530, 34904754, 73177778, 156015282, 330078898, 703371954, 1517066930, 3211565746, 6667672242, 13848320690, 28478053042, 58811259570

Offset: 1

Ctibor O. Zizka, Feb 07 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000079, A092844, A110299.

Partial sums of A239885.

Table[Sum[2^(i-2) * Prime[i], {i, 1, n}], {n, 1, 10}] (* G. C. Greubel, Oct 15 2016 *)
Accumulate[Table[Prime[i]*2^(i-2),{i,30}]] (* Harvey P. Dale, Aug 14 2019 *)

a(n) = sum(k=1, n, prime(k)*2^(k-2)); \\ Michel Marcus, Oct 15 2016

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A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

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A110295 a(n) = 2^(n-1) * prime(n).

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A325800 Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.

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Formula

A135483 a(n) = Sum_{j=1..n} prime(j)*2^(j-2).

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