Listed below are the powerful numbers (A001694) between k^2 and (k+1)^2 for each term of A119242 up to a(12).

Donovan Johnson, Sep 01 2013


a(0) = 1
No powerful numbers between 1^2 and (1+1)^2

a(1) = 2
Exactly 1 powerful number between 2^2 and (2+1)^2:
8 = 2^3

a(2) = 5
Exactly 2 powerful numbers between 5^2 and (5+1)^2:
27 = 3^3
32 = 2^5

a(3) = 31
Exactly 3 powerful numbers between 31^2 and (31+1)^2:
968 = 2^3*11^2
972 = 2^2*3^5
1000 = 2^3*5^3

a(4) = 234
Exactly 4 powerful numbers between 234^2 and (234+1)^2:
54872 = 2^3*19^3
54925 = 5^2*13^3
55112 = 2^3*83^2
55125 = 3^2*5^3*7^2

a(5) = 1822
Exactly 5 powerful numbers between 1822^2 and (1822+1)^2:
3319756 = 2^2*11^2*19^3
3321125 = 5^3*163^2
3321188 = 2^2*13^2*17^3
3321216 = 2^7*3^3*31^2
3322336 = 2^5*47^3

a(6) = 3611
Exactly 6 powerful numbers between 3611^2 and (3611+1)^2:
13041125 = 5^3*17^2*19^2
13041675 = 3^3*5^2*139^2
13042575 = 3^2*5^2*7^3*13^2
13043800 = 2^3*5^2*7^2*11^3
13045131 = 3^4*11^5
13045832 = 2^3*1277^2

a(7) = 17329
Exactly 7 powerful numbers between 17329^2 and (17329+1)^2:
300295863 = 3^3*7^2*61^3
300300075 = 3^3*5^2*23^2*29^2
300304000 = 2^7*5^3*137^2
300306875 = 5^4*11^3*19^2
300312500 = 2^2*5^7*31^2
300321032 = 2^3*11^2*557^2
300321637 = 7^2*11^2*37^3

a(8) = 1511067
Exactly 8 powerful numbers between 1511067^2 and (1511067+1)^2:
2283323796675 = 3^5*5^2*19387^2
2283324664392 = 2^3*3^2*37^2*4813^2
2283325464500 = 2^2*5^3*67577^2
2283325517844 = 2^2*3^5*7^3*2617^2
2283325600788 = 2^2*3^4*37^3*373^2
2283326145792 = 2^8*3^2*997^3
2283326152312 = 2^3*19^2*47^2*71^3
2283326338300 = 2^2*5^2*7^3*41^2*199^2

a(9) = 524827
Exactly 9 powerful numbers between 524827^2 and (524827+1)^2:
275443408188 = 2^2*3^2*13^2*23^3*61^2
275443489692 = 2^2*3^2*7^3*4723^2
275443710584 = 2^3*7^3*43^2*233^2
275443920500 = 2^2*5^3*7^4*479^2
275444061373 = 13^3*11197^2
275444085600 = 2^5*3^3*5^2*3571^2
275444172000 = 2^5*3^3*5^3*1597^2
275444401500 = 2^2*3^3*5^3*4517^2
275444425625 = 5^4*761^3

a(10) = 180469424
Exactly 10 powerful numbers between 180469424^2 and (180469424+1)^2:
32569213011122000 = 2^4*5^3*23^2*175453^2
32569213036360275 = 3^3*5^2*11^2*97^3*661^2
32569213069378629 = 3^3*7^3*1875317^2
32569213124550375 = 3^7*5^3*7^2*13^2*3793^2
32569213150077336 = 2^3*3^3*17^2*722317^2
32569213221959000 = 2^3*5^3*359^3*839^2
32569213250423432 = 2^3*11^2*73^2*181^2*439^2
32569213250580727 = 7^3*23^2*199^2*2129^2
32569213294659123 = 3^5*17^2*23^2*29^2*1021^2
32569213341055761 = 3^4*17^2*41^3*4493^2

a(11) = 472532614
Exactly 11 powerful numbers between 472532614^2 and (472532614+1)^2:
223287071342645376 = 2^7*3^3*11^3*53^2*4157^2
223287071467548672 = 2^12*3^3*1420921^2
223287071475926808 = 2^3*3^2*7^3*53^3*7793^2
223287071631836544 = 2^7*3^3*8037943^2
223287071706384512 = 2^7*37^2*1128821^2
223287071735772833 = 17^3*619^2*10891^2
223287071840761600 = 2^8*5^2*7^2*41^2*751^3
223287071887095003 = 3^3*41^3*346397^2
223287072008175000 = 2^3*3^2*5^5*7^3*19^4*149^2
223287072152787808 = 2^5*11^3*2289643^2
223287072165783500 = 2^2*5^3*7^3*53^2*21529^2

a(12) = 78102676912
Exactly 12 powerful numbers between 78102676912^2 and (78102676912+1)^2:
6100028140823118543328 = 2^5*7^3*17^3*61^2*3121^3
6100028140824773408000 = 2^8*5^3*11^2*41^2*971^2*997^2
6100028140829661293400 = 2^3*3^5*5^2*23^2*311^2*49529^2
6100028140844678043375 = 3^13*5^3*61^2*90697^2
6100028140877830109600 = 2^5*5^2*13^3*523^2*112643^2
6100028140892904714771 = 3^2*11^3*31^2*1907^2*12071^2
6100028140908690917348 = 2^2*7^4*17^3*211^2*53887^2
6100028140910838016000 = 2^11*5^3*1021^2*151189^2
6100028140925163836656 = 2^4*31^3*43^2*2630843^2
6100028140935631114800 = 2^4*3^5*5^2*7^2*35787779^2
6100028140963245260800 = 2^16*5^2*7^3*379^2*8693^2
6100028140975511521352 = 2^3*7^2*1291^2*3055601^2