A298156 Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1.

35, 42, 50, 60, 64, 72, 76, 81, 86, 93, 136, 145, 153, 159, 164, 174, 253, 273, 289, 325, 365, 385, 390, 416, 438, 462, 468, 488, 494, 497, 549, 550, 555, 559, 592, 644, 658, 660, 664, 666, 686, 703, 704, 710, 737, 747, 792, 836, 852, 884, 885, 891, 920, 940, 944, 946, 980

Offset: 1

Author

Daniel Blaine McBride, Jan 13 2018

Keywords

• base
• nonn

Comments

Composite numbers n, of which A001414(n) (sum of prime divisors of n with repetition, sopfr(n)) is in sequence A001704 (numbers m which are the concatenation of k and k+1).

Examples

			35 = 5*7, sopfr(35) = 5+7 = 12, 12 =k||k+1 when k = 1.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
• Eric Weisstein's World of Mathematics, Sum of Prime Factors

Crossrefs

Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).

Cf. A001704 (numbers which are the concatenation of k and k+1).

Programs

Mathematica

Select[Range[10^3], And[CompositeQ@ #, Subtract @@ Map[FromDigits, TakeDrop[#, Floor[Length[#]/2]]] == -1 &@ IntegerDigits@ Total[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]]]] &] (* Michael De Vlieger, Jan 14 2018 *)

PARI

is(n)=my(f = factor(n), sopfr = sum(i = 1, #f~, f[i, 1] * f[i, 2]); d = digits(sopfr), v); if((#d) % 2 == 0, v = vector(#d / 2); v[#v] = -1; return(vector(#d / 2, j, d[j]) - vector(#d / 2, #d / 2 + j, d[j]) == v), return(d == concat(digits(10^(#d \ 2) - 1), digits(10^(#d \ 2))))) \\ David A. Corneth, Jan 13 2018

PARI

sopfr(n,f=factor(n))=sum(i=1,#f~, f[i,1]*f[i,2])
has(n)=my(d=digits(n),k=#d); digits(fromdigits(d[1..k\2])+1) == d[k\2+1..k]
list(lim)=my(v=List()); forfactored(n=35,lim\1, if(n[2][,2]!=[1]~ && has(sopfr(0,n[2])), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 15 2018

Formula

sopfr(n) = k||k+1 when n is not prime and k is a positive integer.