A152622
Tetrahedral numbers n*(n+1)*(n+2)/6 with n, n+1 and n+2 nonprime.
Original entry on oeis.org
120, 560, 1540, 2600, 2925, 3276, 5984, 6545, 7140, 9880, 15180, 19600, 20825, 22100, 27720, 29260, 30856, 41664, 43680, 45760, 54740, 70300, 73150, 76076, 88560, 102340, 105995, 109736, 125580, 129766, 134044, 138415, 142880, 161700
Offset: 1
-
tet := proc(n) n*(n+1)*(n+2)/6 ; end: for n from 1 to 300 do if not isprime(n) and not isprime(n+1) and not isprime(n+2) then printf("%d,",tet(n)) ; fi; od: # R. J. Mathar, Dec 10 2008
-
p=7;forprime(q=11,1e2,for(n=p+1,q-3,print1(binomial(n+2,3)", "));p=q) \\ Charles R Greathouse IV, Dec 21 2011
20400 replaced by 19600, 20625 replaced by 20825,
R. J. Mathar, Dec 10 2008
A144486
Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n = number of prime factors in n+1. (Prime factors are counted with multiplicity.)
Original entry on oeis.org
45, 105, 231, 325, 378, 561, 595, 741, 990, 1653, 2850, 3655, 3741, 4371, 4465, 4851, 6786, 7021, 7381, 7503, 7750, 8911, 9180, 10011, 10153, 10585, 10878, 11781, 12561, 13530, 14535, 14706, 15225, 15753, 20301, 20503, 21115, 22791, 23005, 23653
Offset: 1
The first 11 values of n that satisfy the definition are 9, 14, 21, 25, 27, 33, 34, 38, 44, 57 and 75, so
a(1) = 9*10/2 = 45; 9 = 3*3, 10 = 2*5.
a(2) = 14*15/2 = 105; 14 = 2*7, 14 = 3*5.
a(3) = 21*22/2 = 231; 21 = 3*7, 22 = 2*11.
a(4) = 25*26/2 = 325; 25 = 5*5, 26 = 2*13.
a(5) = 27*28/2 = 378; 27 = 3*3*3, 28 = 2*2*7.
a(6) = 33*34/2 = 561; 33 = 3*11, 34 = 2*17.
a(7) = 34*35/2 = 595; 34 = 2*17, 35 = 5*7.
a(8) = 38*39/2 = 741; 38=2*19, 39=3*13.
a(9) = 44*45/2 = 990; 44=2*2*11, 45=3*3*5.
a(10) = 57*58/2 = 1653; 57=3*19, 58=2*29.
a(11) = 75*76/2 = 2850; 75=3*5*5, 76=2*2*19.
-
isA045920 := proc(n) if numtheory[bigomega](n) = numtheory[bigomega](n+1) then true; else false; fi; end: A045920 := proc(n) option remember ; local a; if n =1 then 2; else for a from procname(n-1)+1 do if isA045920(a) then RETURN(a) ; fi; od: fi; end: A000217 := proc(n) n*(n+1)/2 ; end: A144486 := proc(n) A000217(A045920(n+1)) ; end: for n from 1 to 100 do printf("%d,",A144486(n)) ; od: # R. J. Mathar, Dec 10 2008
-
Times@@#/2&/@Select[Partition[Range[500],2,1],!PrimeQ[#[[1]]] && !PrimeQ[#[[2]]] && PrimeOmega[#[[1]]]==PrimeOmega[#[[2]]]&] (* Harvey P. Dale, May 23 2013 *)
-
for(n=2, 1e3, if(bigomega(n+1) == bigomega(n+2) && k = (n+1)*(n+2)/2, print1(k", "))) \\ Altug Alkan, Oct 18 2015
A144519
Triangular numbers n*(n+1)/2 with n prime and n+1 nonprime.
Original entry on oeis.org
6, 15, 28, 66, 91, 153, 190, 276, 435, 496, 703, 861, 946, 1128, 1431, 1770, 1891, 2278, 2556, 2701, 3160, 3486, 4005, 4753, 5151, 5356, 5778, 5995, 6441, 8128, 8646, 9453, 9730, 11175, 11476, 12403, 13366, 14028, 15051, 16110, 16471, 18336, 18721, 19503, 19900, 22366, 24976, 25878
Offset: 1
If n=3(prime) and n=4(nonprime), then 3*4/2=6=a(1). If n=5(prime) and n=6(nonprime), then 5*6/2=15=a(2). If n=7(prime) and n=8(nonprime), then 7*8/2=28=a(3). If n=11(prime) and n=12(nonprime), then 11*12/2=66=a(4). If n=13(prime) and n=14(nonprime), then 13+14/2=91=a(5), etc.
-
Table[(p(p+1))/2,{p,Prime[Range[2,50]]}] (* Harvey P. Dale, Dec 28 2023 *)
Corrected definition. Inserted 2701, extended beyond 11175. -
R. J. Mathar, Dec 19 2008
A144523
Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n > number of prime factors in n+1.
Original entry on oeis.org
36, 210, 300, 528, 1035, 1176, 1275, 1485, 1596, 2080, 2346, 2926, 3240, 3321, 3570, 4095, 4278, 5460, 5565, 6105, 6555, 6903, 7260, 8256, 8778, 9870, 10440, 11628, 11935, 12880, 13695, 14196, 15576, 16653, 17020, 17391, 17955, 20100, 20910, 21736, 22578, 23436, 24310, 25200, 25425
Offset: 1
If n=8=2*2*2(number of prime factors = 3) and n+1=9=3*3(number of prime factors = 2), then 8*9/2=36=a(1). If n=20=2*2*5(number of prime factors = 3) and n+1=21=3*7(number of prime factors = 2), then 20*21/2=210=a(2). If n=24=2*2*2*3(number of prime factors = 4) and n+1=25=5*5(number of prime factors = 2), then 24*25/2=300=a(3), etc.
-
(Times@@#)/2&/@Select[Partition[Range[250],2,1],AllTrue[ #,CompositeQ] && PrimeOmega[#[[1]]]>PrimeOmega[#[[2]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2020 *)
Corrected definition. 2926 inserted and extended. -
R. J. Mathar, Jan 17 2009
Showing 1-4 of 4 results.
Comments