proposed

approved

editing

proposed

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

allocated for Gus WisemanOdd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601

1,2

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 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

The sequence of multiset partitions whose MM-numbers belong to the sequence begins:

1: {}

7: {{1,1}}

9: {{1},{1}}

23: {{2,2}}

25: {{2},{2}}

97: {{3,3}}

121: {{3},{3}}

151: {{1,1,2,2}}

161: {{1,1},{2,2}}

169: {{1,2},{1,2}}

175: {{2},{2},{1,1}}

183: {{1},{1,2,2}}

185: {{2},{1,1,2}}

195: {{1},{2},{1,2}}

207: {{1},{1},{2,2}}

225: {{1},{1},{2},{2}}

227: {{4,4}}

289: {{4},{4}}

541: {{1,1,3,3}}

661: {{5,5}}

679: {{1,1},{3,3}}

687: {{1},{1,3,3}}

781: {{3},{1,1,3}}

841: {{1,3},{1,3}}

847: {{1,1},{3},{3}}

873: {{1},{1},{3,3}}

957: {{1},{3},{1,3}}

961: {{5},{5}}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[1, 100, 2], Or[#==1, SameQ[##, 2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A320325, A322526, A322527, A322530.

allocated

nonn

Gus Wiseman, Dec 17 2018

approved

editing

allocated for Gus Wiseman

recycled

allocated

reviewed

approved

proposed

reviewed

editing

proposed

a(n) = primepi(A059785(n + 1)) - primepi(A059785(n)).

1, 2, 11, 314, 339730, 797177366934

1,2

a(n) = primepi(A059785(n)) - Sum_{k=1..n-1} a(i). - David A. Corneth, Oct 01 2018

Differences@ PrimePi@ NestList[NextPrime[#^2, -1] &, 2, 5] (* Michael De Vlieger, Nov 07 2018 *)

(PARI) first(n) = my(res = vector(n)); res[1] = 1; t = 1; p = 2; for(i = 2, n, p = precprime(p^2); c = primepi(p) - t; res[i] = c; t+=c); res \\ David A. Corneth, Oct 01 2018

Cf. A059785.

nonn,more,changed

recycled

Lear Young, Sep 30 2018

a(6) from Amiram Eldar, Nov 08 2018

proposed

editing